Geometry of Quantum states: The adiabatic

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field is cyclic, due to the centripetal magnetic force. The motion in an ..... (1.2) are also translated by the same amount, being. 0. 2 y. AB k. X c. eB. BR π. Φ. = -. +. (1.3) .... inequivalent atoms, A and B, that belong to two different Bravais lattices. One can ... electron in one of the atoms of type A or B within the unit cell: 1. 1 . /2. †.
DEPARTMENT OF PHYSICS

TOPOLOGICAL AND CONVENTIONAL QUANTUM MATTER IN STRONG MAGNETIC AND ELECTRIC FIELDS: GROUND STATE THERMODYNAMIC AND TRANSPORT CONSIDERATIONS

KONSTANTINOU GEORGIOS

A Dissertation submitted to the University of Cyprus in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

May, 2016

Georgios Konstantinou, 2016

VALIDATION PAGE

Doctoral Candidate: Konstantinou Georgios

Doctoral Thesis Title: Topological and conventional quantum matter in strong magnetic and electric fields: Ground state thermodynamic and transport considerations

The present Doctoral Dissertation was submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy at the Department of Physics and was approved on the ……………….. by the members of the Examination Committee.

Examination Committee:

Research Supervisor: Konstantinos Moulopoulos (Department of Physics, University of Cyprus)

Committee Member: Grigorios Itskos (Department of Physics, University of Cyprus)

Committee Member: Nicolaos Toumbas (Department of Physics, University of Cyprus)

Committee Member: Ioannis Giapintzakis (Department of Mechanical and Manufacturing Engineering, University of Cyprus)

Committee Member: Thierry Champel (Universite Grenoble Alpes/CNRS, Laboratoire de Physique et Modelisation des Milieux Condenses)

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DECLARATION OF DOCTORAL CANDIDATE

The present Doctoral Dissertation was submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy of the University of Cyprus. It is a product of original work of my own, unless otherwise mentioned through references, notes, or any other statements.

Georgios Konstantinou (Γεώξγηνο Κσλζηαληίλνπ)

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ΠΕΡΙΛΗΨΗ Η παξνύζα δηαηξηβή επηθεληξώλεηαη ζηε κειέηε ηεο ζεξκνδπλακηθήο ειεθηξνληθώλ ζπζηεκάησλ (ηνπνινγηθώλ ή ζπκβαηηθώλ) ζηελ ζεκειηώδε ηνπο θαηάζηαζε, όηαλ κεξηθέο εμσηεξηθέο παξάκεηξνη κεηαβάιινληαη (π.ρ. ην πάρνο κηαο εηεξνδνκήο, ή εμσηεξηθά ειεθηξνκαγλεηηθά πεδία). Η ζπκπεξηθνξά απηή, ε νπνία ζηελ γεληθόηεηά ηεο αγλνεί ηηο αιιειεπηδξάζεηο, είλαη κηα ππόζρεζε όηη ε θπζηθή ελόο ειεθηξνλίνπ (αξθεηά επηηπρεκέλε ζε παιαηόηεξα πεηξάκαηα ζε δηάθνξεο θαη δηαθνξεηηθέο κεηαμύ ηνπο πεξηνρέο) είλαη αθόκα έλα ζεκαληηθό αληηθείκελν κειέηεο, ην νπνίν πνιιέο θνξέο νδεγεί ζηελ θαηαλόεζε ησλ ηνπνινγηθώλ ηδηνηήησλ νξηζκέλσλ πιηθώλ. Ξεθηλώ ηε δηαηξηβή κνπ κε ην πξώην Κεθάιαην λα ζπλνςίδεη νξηζκέλεο από ηηο ηδηόηεηεο ησλ ιεγόκελσλ ηνπνινγηθώλ κνλσηώλ, κηαο λέαο θαηεγνξίαο πιηθώλ ε νπνία ζηηο κέξεο καο βξίζθεηαη ζην πξνζθήλην ηεο ζεσξίαο θαη πεηξακάησλ ζηελ πεξηνρή ηεο Σπκππθλσκέλεο Ύιεο. Τα πιηθά απηά ραξαθηεξίδνληαη από κε-κεδεληθό αξηζκό Chern, ν νπνίνο είλαη έλαο ηνπνινγηθόο δείθηεο, όκνηνο κε ην ραξαθηεξηζηηθό Euler πνπ ζπλαληάκε ζηα Μαζεκαηηθά. Ο αξηζκόο Chern είλαη ππόινγνο γηα ηελ εκθάληζε επηθαλεηαθώλ ξεπκάησλ, αιιά θαη γηα ηελ επζηάζεηα θαη ζπλνρή ηνπο, θαζώο απηά ξένπλ ρσξίο απώιεηεο ελέξγεηαο. Σηα επόκελα θεθάιαηα παξνπζηάδεηαη ε δηδαθηνξηθή κνπ εξγαζία, κε κεγάιν κέξνο από ηα αλαγξαθόκελα απνηειέζκαηα λα νθείινληαη ζε δηθή κνπ πξσηνβνπιία θαη αλαιπηηθή εμαγσγή ηνπο, θαη κε έλα κέξνο ηνπο ήδε δεκνζηεπκέλν ζε επηζηεκνληθό άξζξν. Σην Κεθάιαην 2, ζπδεηείηαη δηεμνδηθά ε θπζηθή ηύπνπ Berry (ε νπνία νδεγεί ζηελ εύξεζε ηνπ αξηζκνύ Chern) κε άκεζε εθαξκνγή ηεο ζην ζεώξεκα HellmannFeynman, πνπ νδεγεί ζηελ θβάλησζε ηνπ καγλεηηθνύ θνξηίνπ ζηνλ 3D ρώξν κέζα από έλα εηθνληθό καγλεηηθό θνξηίν πνπ εκθαλίδεηαη ζηνλ ρώξν ησλ παξακέηξσλ. Δπηπξόζζεηα, ην θεθάιαην απηό αθνξά θαη ζε κε-εξκεηηαλέο επηξξνέο ζηα ζεσξήκαηα Hellmann-Feynman θαη Ehrenfest, πνπ εμαξηώληαη από ηηο εθάζηνηε ζπλνξηαθέο ζπλζήθεο ηνπ πξνβιήκαηνο θαη νδεγνύλ ζηε ζεξαπεία δηαθόξσλ παξαδόμσλ πνπ απνξξένπλ από ηα ζεσξήκαηα απηά. Τέινο, κέζα από ηε δνπιεηά απηή επαλεκθαλίδνληαη νξηζκέλνη αλαιινίσηνη ηειεζηέο, πνπ ππαθνύνπλ ζε κηα γεληθεπκέλε εμίζσζε ζπλέρεηαο. Η εμίζσζε απηή πεξηγξάθεη γεληθεπκέλα ξεύκαηα παξόκνηα κε ηα γλσζηά καο ξεύκαηα πηζαλόηεηαο κε ηνλ κε-νκνγελή ηεο όξν λα εκπεξηέρεη ηνλ αλαιινίσην ηειεζηή. Η καζεκαηηθή δνκή πνπ πξνθύπηεη έρεη ελδηαθέξνλ, θαη νη γεληθέο ηεο ζπλέπεηεο πξέπεη λα δηεξεπλεζνύλ ζε μερσξηζηή κειέηε ζην κέιινλ. Σην Κεθάιαην 3 παξνπζηάδεηαη κηα κειέηε πνπ αθνξά ειεθηξνληθά ζπζηήκαηα ηα νπνία δνπλ ζε εηεξνδνκέο (ζε νξζνγώληεο ή θπθιηθέο γεσκεηξίεο) θαη ππνβάιινληαη ζε ηζρπξά καγλεηηθά πεδία κε εύξεζε ησλ ζεξκνδπλακηθώλ ηνπο ηδηνηήησλ ζε ζπλάξηεζε κε κηα κεηαβαιιόκελε εμσηεξηθή παξάκεηξν, αιιά θαη κε πξνζδηνξηζκό ησλ ηδηνηήησλ κεηαθνξάο. Σπγθεθξηκέλα, ηα ειεθηξόληα βξίζθνληαη ζην εζσηεξηθό εηεξνδνκώλ πάρνπο d ζηελ ηξίηε δηάζηαζε (ζην ηέινο, ην όξην ηνπ άπεηξνπ πάρνπο αλαπαξάγεη ηα αληίζηνηρα απνηειέζκαηα ζηηο 3 δηαζηάζεηο). Μέζα από ηελ εύξεζε iii

ησλ ζεξκνδπλακηθώλ ηδηνηήησλ ζηε ζεκειηώδε θαηάζηαζε ππνινγίδνπκε νξηζκέλεο παξαβηάζεηο από ηηο πεξηνδηθόηεηεο de Haas – van Alphen θαη ηηο ηηκέο ηνπ καγλεηηθνύ πεδίνπ ή ηνπ πάρνπο όπνπ απηέο ζπκβαίλνπλ. Η κεζνδνινγία πνπ αθνινπζνύκε γηα ηελ εμαγσγή όισλ ησλ απνηειεζκάησλ απηνύ ηνπ Κεθαιαίνπ είλαη πξσηόηππε, πινύζηα ζε θπζηθό πεξηερόκελν θαη έρεη δεκνζηεπζεί ζε έλα κεγάιν ζε κέγεζνο άξζξν πνπ εθζέηεη θαη όιεο ηηο εθαξκνγέο ηεο κε ιεπηνκέξεηα. Σηνπο ππνινγηζκνύο πεξηιακβάλνληαη ηόζν ην ζπηλ ησλ ζσκαηηδίσλ, όζν θαη νη δηειεθηξνληαθέο αιιειεπηδξάζεηο, ηνπιάρηζηνλ ζε κηα εηθόλα κε αιιειεπηδξώλησλ Σύλζεησλ Φεξκηνλίσλ πνπ θηλνύληαη ζε δηεπηθάλεηα κε πάρνο. Παξνπζηάδεηαη επίζεο κηα πξώηε κειέηε ηνπνινγηθώλ κνλσηώλ ζε ζρέζε κε ην πάρνο ηνπο. Τέινο, ε κειέηε επεθηείλεηαη θαη ζε ζπζηήκαηα πνπ βξίζθνληαη εθηόο καγλεηηθνύ πεδίνπ, αιιά ζηελ παξνπζία δηαλπζκαηηθνύ δπλακηθνύ (δηαηάμεηο ηύπνπ Aharonov-Bohm). Σην Κεθάιαην 4 ε δνπιεηά κνπ επηθεληξώλεηαη ζε αιιαγέο πνπ πθίζηαηαη ε θάζε ηεο θπκαηνζπλάξηεζεο ειεθηξνλίνπ ην νπνίν θηλείηαη ζηελ παξνπζία καγλεηηθνύ πεδίνπ (νκνγελνύο αιιά θαη κε νκνγελνύο) ζε 2 δηαζηάζεηο (κε ηαπηόρξνλε γελίθεπζε θαη ζε 3 δηαζηάζεηο). Γύν πνζόηεηεο έξρνληαη ζηελ επηθάλεηα, κηα παιηά θαη „μεραζκέλε‟, ε ιεγόκελε ςεπδννξκή (ζε Καξηεζηαλέο ζπληεηαγκέλεο) θαη κηα ςεπδνζηξνθνξκή (ζε πνιηθέο ζπληεηαγκέλεο). Πξώηα ππνινγίδνπκε ηηο κεηαβνιέο ηεο θάζεο ηνπ ειεθηξνλίνπ όηαλ κεηαβάιιεηαη ε βαζκίδα δηαλπζκαηηθνύ δπλακηθνύ κέζσ ελόο ζπλήζνπο κεηαζρεκαηηζκνύ βαζκίδαο. Τόζν όκσο πξηλ ην κεηαζρεκαηηζκό όζν θαη κεηά, απαηηνύκε ε θπκαηνζπλάξηεζε λα εηλαη ηδηνζπλάξηεζε ηαπηόρξνλα ηεο Χακηιηνληαλήο θαη κηαο ζπγθεθξηκέλεο από ηηο ζπληζηώζεο ηεο ςεπδννξκήο (δηνξζώλνληαο έηζη παιαηόηεξε βηβιηνγξαθία όπνπ ν κεηαζρεκαηηζκόο βαζκίδαο κε ρξήζε ησλ ζπλεζηζκέλσλ θπκαηνζπλαξηήζεσλ Landau ήηαλ πνιύ πην πεξίπινθνο). Γεληθεύνληαο ηε κέζνδό καο, πξνρσξνύκε θαη ζε ρσξηθά κε-νκνγελή καγλεηηθά πεδία, όπνπ παξνπζηάδεηαη ε ρξεζηκόηεηα ησλ παξαπάλσ ςεπδo-πνζνηήησλ, κε ηε κέζνδν απηή λα νδεγεί θαη πάιη ζηελ θβάλησζή ηoπο, ππό νξηζκέλνπο πεξηνξηζκνύο πνπ αθνξνύλ ζηε κε νκνηνγέλεηα ηνπ καγλεηηθνύ πεδίνπ. Τέινο, ππνινγίδνληαη θαη ζπδεηώληαη θάπνηεο θάζεηο Berry πνπ πξνθύπηνπλ κε αδηαβαηηθή κεηαβνιή νξηζκέλσλ παξακέηξσλ. Σην ηειεπηαίν Κεθάιαην παξνπζηάδεηαη κηα επίζεο πξσηόηππε δνπιεηά πνπ αθνξά Κβαληηθά Σπζηήκαηα Hall, ζπκβαηηθά θαη κε (π.ρ. Γξαθίλε, κε ζρεηηθηζηηθό θάζκα Dirac), ζε ηζρπξό ειεθηξηθό πεδίν. Σπλέπεηα απηνύ ηνπ πεδίνπ είλαη ε αιιειεπηθάιπςε ησλ ζηαζκώλ Landau, ε νπνία βξίζθνπκε όηη κπνξεί λα νδεγήζεη ππό νξηζκέλεο πξνππνζέζεηο ζε θιαζκαηηθνύο παξάγνληεο θαηάιεςεο αθόκα θαη ρσξίο λα ιεθζνύλ ππόςηλ νη αιιειεπηδξάζεηο κεηαμύ ησλ ειεθηξνλίσλ (θάηη πνπ γίλεηαη ζην θιαζκαηηθό Κβαληηθό Φαηλόκελν Hall). Οη παξάγνληεο απηνί εκθαλίδνληαη σο ζπλέπεηα ηεο εμαθάληζεο ηνπ ελεξγεηαθνύ ράζκαηνο ιόγσ ηνπ ηζρπξνύ ειεθηξηθνύ πεδίνπ. Η αγσγηκόηεηα Hall όκσο κπνξεί λα εκθαλίζεη πιαηώ (πεξηνρέο ηνπ καγλεηηθνύ πεδίνπ όπνπ απηή παξακέλεη ακεηάβιεηε) ζηελ πεξίπησζε πνπ ιάβνπκε ππόςηλ θαη ην δπλακηθό πξνζκίμεσλ ηνπ πιηθνύ. Δπεηδή ζε θάζε ζηάζκε Landau ππάξρεη έλαο πεπεξαζκέλνο αξηζκόο εθηεηακέλσλ θαηαζηάζεσλ πνπ iv

βξίζθνληαη ζην κέζν ηεο θάζε ζηάζκεο Landau, αλεμάξηεηα από ην ειεθηξηθό πεδίν (αξθεί απηό λα κελ είλαη πάξα πνιύ ηζρπξό ώζηε λα έρνπκε ξήμε ζην θξπζηαιιηθό πιέγκα ή ζεξκηθά θαηλόκελα, ή θαη θαηάξξεπζε ηεο ίδηαο ηεο εηθόλαο ησλ ζηαζκώλ Landau!), απηέο νη θαηαζηάζεηο δύλαληαη λα εκθαλίζνπλ λέα πιαηώ ζηελ αγσγηκόηεηα Hall. Δπίζεο, νη ππνινγηκνί ζπκπεξηιακβάλνπλ θαη ππνινγηζκό ζεξκνδπλακηθώλ ηδηνηήησλ θαη λέα απνηειέζκαηα πνπ αθνξνύλ ζηηο πεξηνδηθόηεηεο de Haas – van Alphen, αλαδεηθλύνληαο ηνλ πινύην ηνπ πξνβιήκαηνο από άπνςε θπζηθήο - αθήλνληαο όκσο γηα ην κέιινλ ην εξώηεκα θαηά πόζνλ νη εμσηηθέο ηδηόηεηεο πνπ πξνθύπηνπλ κπνξνύλ λα εκθαληζζνύλ ζην εξγαζηήξην (όπνπ αλαπόθεπθηα έρνπκε θαη αηαμία ζην ζύζηεκά καο). Γηα απηό ην αλνηθηό εξώηεκα ρξεηάδεηαη εληειώο δηαθνξεηηθή αληηκεηώπηζε, ηζσο κε δηαθνξεηηθά όπια, θαη παξαπέκπεηαη ζην κέιινλ. Γίλεηαη θαη έλαο ζπλνιηθόο επίινγνο (Σπκπεξάζκαηα) ζην ηέινο ηεο Γηαηξηβήο πνπ ηε βάδεη ζε έλα γεληθόηεξν πιαίζην, θπξίσο από ηελ άπνςε ηνπ ηη πξέπεη λα γίλεη, σο δπλαηή βειηίσζε ηέηνησλ ππνινγηζκώλ ζε απηή ηε γλσζηηθή πεξηνρή.

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ABSTRACT This work focuses on the thermodynamic behavior of electronic systems (either topological or conventional) in their ground state, when some external parameters are varied, i.e. thickness of heterostructures or external electromagnetic fields. This behavior, which completely ignores interactions, is a promise that one-electron physics is still of great importance (quite successful in early experiments in various distinct areas) and quite often leads to topological properties. Starting my work, I review the field of topological insulators which is a milestone in physics today, and remind the reader that a topological insulator may be found from two distinct cases: the first one is that it is an emergent phenomenon resulting from conventional materials when placed in a magnetic field, or it is a pure phenomenon resulting from compounds with large spin-orbit coupling. These cases involve non-zero Chern number, which is the topological index analogous to well-known Euler characteristics in geometry. When a Chern number appears, a non-dissipative current flows in the edges (or surface, if 3D) of the material which has a topological origin. In the following Chapters my theoretical work is presented with part of my results already published in a long article in a physics journal. In Chapter 2, Berry physics (which directly leads to the Chern number) is thoroughly discussed, highlighting some new consequences on Hellmann – Feynman theorem, and the resulting quantization of real magnetic charge in 3-D space, through a fictitious magnetic charge that appears in the parameter space. In addition, we study some non-Hermitian influences on Hellmann – Feynman and Ehrenfest theorems that come out of boundary contributions, resulting in the resolution of some paradoxes associated with these important theorems of Quantum Mechanics. Invariant operators seem to appear in the sink term of a generalization of a continuity equation which contains a generalized current, a nontrivial extension of the quantum probability current, that cures all paradoxes. The mathematical structure that comes out is interesting, and its general consequences must be investigated in a self – contained study in the future. In Chapter 3 my theoretical work is presented, regarding electronic systems subjected to strong magnetic fields with thermodynamic and transport calculations with a simultaneous study of finite thickness effects on the energetics and on Hall properties. Specifically, I consider electronic systems that live in heterostructures of finite thickness d, whose ground state magnetic properties violate standard de-Haas van Alphen periodicities, with precise predictions on the critical magnetic field values where these violations occur. Also, both spin and interactions (through a Composite Fermions picture) are included in the calculations. A small part of Chapter 3 also examines properties and energy transitions that may occur in topological insulators by varying their thickness d. Finally, systems confined on cylinder surfaces and nanorings threaded by Aharonov – Bohm fluxes are thoroughly studied, with their analytical thermodynamic properties discussed in detail. vi

In Chapter 4, my work concentrates on quantum mechanical phase alterations in two dimensional systems that move under the influence of external magnetic fields, both homogenous and inhomogeneous. By using a forgotten conserved quantity, the so called pseudomomentum (introduced in Cartesian coordinates) and a pseudoangular momentum (in polar coordinates) we determine the nontrivial phase factors picked up by the wavefunction and we answer an old question in the literature regarding the correct gauge transformation in different gauge choices in the Landau Level problem. We also generalize the application of our method to inhomogeneous magnetic fields, where the usefulness of the conserved quantities is shown, leading to quantized pseudo-angular and pseudo-momentum under certain restrictions in the type of inhomogeneity of the magnetic field. Relevant Berry phases are also determined under adiabatic variation of certain parameters and discussed in connection to other problems. Last, but not least, in Chapter 5, a piece of also original work is presented regarding Quantum Hall systems (both conventional and unconventional e.g. Graphene with relativistic Dirac spectrum) in a very strong electric field with consequent interLandau Level overlaps, leading to emergent fractional filling factors without the introduction of interactions amongst electrons. The fractional filling factors appear as a consequence of energy gap elimination when the electric field becomes sufficiently strong. Plateaus formation, however, might be observable only upon inclusion of impurity potential which results in a further broadening of L.Ls. Also, thermodynamic properties are analytically determined, and deviations from de Haas-van Alphen periodicities are highlighted, revealing the richness of this problem – leaving, however, for the future the question whether the exotic properties that are revealed can really appear in the lab (where we must deal with the inevitable disorder). To this open question there is a whole different treatment that seems to be necessary, possibly with different tools, which we leave for the future. There is also a collective Epilogue (Thesis Conclusions) at the end of this Dissertation that places it in a broader perspective, mainly in regard to what lies ahead as a possible improvement of such calculations in this area.

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To the woman of my life, Dr. Panayiota Charalambous

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I would kindly like to thank my advisor prof. Kostas Moulopoulos for standing next to me all these years, for believing in me even after I stopped believing in myself, encouraged me and reminded me that life is never about giving up. Also, special thanks to the woman of my life, Dr. Panayiota Charalambous, whose love and endorsement strengthened my will and passion for science and lifted my major depression I fell in for several years, resulting in the completion of my research work as a Doctoral Candidate. I‟m also obliged to my beloved family, Dad, Mom and siblings, for their endless care, and to all my precious friends and relatives Christiana, for her kind support and encouragement every time I needed it, Marinos, Theodoros, Andreas, Aris, that stood next to me all these difficult years I„ve been through. Thank you all from the depths of my heart!

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Contents Introduction ............................................................................................................ 13

Chapter 1: Topological Insulators: An overview 1.1 Introduction .......................................................................................................... 18 1.2 The Quantum Hall Effect (QHE) .......................................................................... 18 1.3 The Quantum Spin Hall Effect ............................................................................. 23 1.4 Graphene ............................................................................................................... 23 1.5 Berry Physics: The beginning ............................................................................... 28 1.6 Time Reversal operator ......................................................................................... 31 1.7 Time reversal Bloch Hamiltonians ....................................................................... 32 1.8 The SSH chain model ........................................................................................... 33 Chapter References ................................................................................................... 37

Chapter 2: Geometry of quantum states: The adiabatic approximation 2.1 Berry‟s phase: a non-conventional method.......................................................... 39 2.2 Relation to topology.............................................................................................. 41 2.3 Gauss-Bonnet theorem .......................................................................................... 42 2.4 Generalization of Hellmann - Feynman theorem with Berry terms ..................... 43 2.5 The adiabatic limit ................................................................................................ 45 2.6 Applications .......................................................................................................... 47 2.7 Quantization of magnetic charge – Dirac monopole ............................................ 49 2.8 Non Hermitian corrections to Ehrenfest and Hellmann-Feynman theorems….....51 2.9 Examples ............................................................................................................... 55 2.10 Contributions of surface terms to Ehrenfest and Hellmann - Feynman theorems x

when the parameter has explicit time dependence .............................................. 59 2.11 Invariant Operators derived from Hellmann- Feynman theorem …..... ............. 61 Chapter References ................................................................................................... 62

Chapter 3: Electron systems in strong magnetic fields: Thermodynamic and hints of transport properties 3.1 Introduction........................................................................................................... 65 3.2 Nonrelativistic electron gas in 2D in a perpendicular magnetic field ................... 69 3.3 Finite-thickness interface (without magnetic field) .............................................. 73 3.4 Finite-thickness interface in a perpendicular magnetic field ................................ 82 3.5 Inclusion of Zeeman term ................................................................................... 103 3.6 Inclusion of electron-electron interactions: Composite Fermions ...................... 109 3.7 Electron gas inside a magnetic field in full 3D space ......................................... 112 3.8 Relevance and applicability to the dimensionality crossover in Topological Insulators …..... ......................................................................................................... 126 3.9 Aharonov – Bohm oscillations in nanorings and nanocylinders ........................ 130 3.10 Conclusions ....................................................................................................... 140 Chapter References ................................................................................................. 141

Chapter 4: Quantum phase alterations in response to proper changes of vector potentials 4.1 Introduction and Motivation ............................................................................... 144 4.2 The Landau - Level system ................................................................................. 145 4.3 Homogeneous magnetic field-Polar coordinates ................................................ 148 4.4 Inhomogeneous magnetic field ........................................................................... 150 4.5a Probability current ............................................................................................ 154 4.5b Berry‟s phase .................................................................................................... 155

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4.6 Inclusion of a homogeneous Electric field.......................................................... 156 4.7 Conclusions and discussion ................................................................................ 157 Chapter References ................................................................................................. 161

Chapter 5: High electric field influence on QHE systems 5.1 Conventional system - Low Electric field strength ............................................ 164 5.2 High Electric Field strength ................................................................................ 169 5.3 Generalizations to relativistic systems - Graphene ............................................. 181 5.4 Conclusions ......................................................................................................... 190 Chapter References ................................................................................................. 191

Thesis Conclusions ............................................................................................. 193

xii

Introduction

Introduction Since the history making discovery of1 the Quantum Hall Effect (QHE) by von Klitzing et. al, a new era emerges in the world of Condensed Matter Physics. It seems that beyond the physical properties of many Solid State systems, the Topology of Hilbert space plays a very important role, the exact and robust microscopic quantization of some macroscopic quantities as the Hall conductivity, charge transport, spin Hall conductivity and the number of conducting edge states in topological insulators, being indeed a reflection of intrinsic topological properties of the wavefunction. The Quantum Hall effect was a breakthrough discovery, because it helped physicists to increasingly realize with time the importance of the underlying topological nature of physical systems. Take as example the Hall conductivity: How is it possible that, in a certain range of values of the external magnetic field, it stays unchanged? Recall that it is the conductance (not conductivity) that is actually measured in experiments, in 3D samples with a small, finite thickness and usually in a „dirty‟ crystalline environment. In what exactly way a macroscopic quantity as the Hall conductance is quantized in universal values, when it is a unique (hence one would expect system-specific) property of electrons moving in the interior of the material? The answer of course is not simple, and it involves a quantum mechanical interpretation and associated sophisticated calculations but with the nontrivial topology of Hilbert space in which the wavefunctions live playing a major role. After the discovery of QHE, several theories have been proposed to explain the robustness of quantization. Some of them include edge states formation, while others involve charge transport by adiabatic pumping due to a variation of an external parameter, such as an Aharonov-Bohm flux2. Recent theories are much more precise: They connect the robust quantization with a topological index similar to the Euler characteristic of Mathematics, called the first Chern number, which is responsible for the insensitivity of the QHE under small adiabatic deformations, or under non-zero temperatures (as long as they are low enough). It has been recognized that this Chern number controls the topological properties of several physical systems (not only QHE), and it appears in modern topological insulating phases, that are a milestone in Solid State Physics. These phases are distinct from the known insulating phases in the sense that they allow the flow of electric current only along their surfaces, while they are insulators in the bulk. In addition, topological properties also depend on certain external parameters such as the thickness d of some heterostructures. By varying the thickness in a continuous 1

More precisely, this was the discovery of the Integral QHE (IQHE), although in this Thesis we mostly identify the two, as the interaction physics that is supposed to be responsible for the Fractional QHE is not really considered. 2 Q. Niu, D. J. Thouless, and Y.-S. Wu, Quantized Hall Conductance as a topological invariant, Phys. Rev. B 31, 6 (1985) 13

Introduction

manner (in the ideal case, without taking into account the crystalline environment) one may succeed in a change in the topological index describing the system. In this way, the thickness provides a useful method of altering the topological properties of some systems (as a control parameter). In Chapter 3, we present a method that takes advantage of the third direction, namely the thickness of a 3D strong topological insulator, to examine the passing from 2D (low thickness) to 3D (larger than a critical thickness value). Apart from topological variations, thickness considerations in some heterostructures often serve as a criterion to determine its magnetic properties. That is, by varying d the system may experience magnetic phase transitions, e.g. paramagnetism to diamagnetism and vice versa. Magnetic properties are usually controlled by magnetic susceptibility, which in turn depends on the heterostructure thickness. Similar results to this are also presented in Chapter 3, in which we are dealing with heterostructures when placed in a perpendicular homogeneous magnetic field. There are, however, cases where the external parameter may be the magnetic field (B) itself. Varying B will result in a change of a topological index (i.e. QHE). Using this information we examine the consequences of a homogeneous and an inhomogeneous magnetic field on the electron‟s wavefunction, by introducing two quantities that are constants of motion. The motion of a charged particle in a homogeneous magnetic field is cyclic, due to the centripetal magnetic force. The motion in an inhomogeneous magnetic field however is a more complicated task. The problem is solvable only in the case where the gradient of B is known, while in other cases there is no integrability. In the latter case, a partial solution involves a generalized constant of motion (even in the interior of an inhomogeneous magnetic field) that appears as a line integral of the inhomogeneous B. Therefore, by fixing the path of integration, one finds a partial solution (for the possible forms of the wavefunctions) under certain restrictive requirements (see Chapter 4). Apart from the real magnetic field, another fictitious quantity is important in modern Solid State Physics: the so called Berry curvature, that acts as a real magnetic field (but in crystal momentum or in other generalized parameter space), having all its properties except for one: its divergence is generally not zero. Naturally, Berry curvature‟s realization is done in a parameter space, in which a magnetic monopole density might be relevant (as a mathematical structure). Indeed, the existence of these monopoles is responsible for many fascinating physical phenomena, and it is „needed‟ for their interpretation. Berry‟s curvature also modifies the usual Hellmann-Feynman theorem, with effective force terms that are responsible for the transverse response of some systems to a longitudinal perturbation (such as a small electric field). Another important parameter that needs special attention appears in the case when the magnetic field is enclosed in some region of space and therefore cannot reach the particle (i.e. in a solenoid covered by a superconducting material). This is an Aharonov-Bohm arrangement, a term that refers to an influence on a charge particle‟s energy spectrum due to the presence of a non-zero vector potential, even though the 14

Introduction

magnetic field B is zero on the region where the particle moves. By varying the enclosed flux Φ in an adiabatic manner, we succeed in changing the system‟s underlying topology; i.e. in the Laughlin argument, the Aharonov-Bohm flux acts as a quantum pump, that when adiabatically varied by a flux quantum, it leads to a quantized charge transport along the cylindrical surface. This happens because by the end of the adiabatic process the multielectron system returns to a gauge equivalent system (although the Chern number may have changed). Localized states only change by a trivial phase factor, while extended states are allowed to move, in response to an electric field caused by induction. Thermodynamics is also a matter of great importance, with the measurable quantities (i.e. total energy, electric current) depending only on the Fractional Part of the flux, reflecting the previously mentioned periodicity with respect to a flux quantum. This is another matter under investigation in this Thesis (as it appears several times in intermediate steps of our calculations). In addition, when dealing with these systems, it is usual to exclude the effect of the electric field in the calculations, and treat it as a small perturbation in the Hamiltonian. In general, this is not safe, because an electric field might have measurable consequences on both transport and thermodynamics (and sometimes it directly leads to analytical, closed forms of solutions, without the need to be treated as a perturbation). For example, in the Quantum Hall case, where both (magnetic and electric) fields are present, the electric field determines the slope of Landau Levels, and when it is not arbitrarily low, it might yet cause an inter – Landau Level overlap, and therefore destruction of energy gap, in which case the topological stability is lost. Another important fact is that when dealing with the topological part of QHE, it is usual to forget about disorder. Disorder is the actual proven generator of the QHE, and has to be taken seriously in the calculations, as it is responsible for the broadening of the density of states, dividing the occupied states into two distinct categories: the localized and extended states, with the latter being responsible for the edge states formation (for more information see Chapter 1). This is more or less what this Thesis is dealing with. It is mostly about a thermodynamic calculation for a non-interacting electron system that moves in an exotic geometry in an electromagnetic field. This picture is justified by the historic fact that still, nowadays, one electron physics has much to tell about several phenomena, both topological and conventional. By completely ignoring the interactions, or the electron‟s spin, the theoretical results are quite successful in explaining the observations (and often they have predicted real experimental results). Even if we could exactly solve an arbitrary mixture of interacting particles in an electromagnetic field, we would still recover the same topological effects as we would have if we solved it ignoring the interactions, and other energy factors (although new, additional topological properties – now depending on interactions - might also show up)! For example, Hall conductivity has an error less than one part in a million, and can be derived using only one-electron Physics. The relevant pieces for this one-

15

Introduction

electron Physics will show up in detail in several places of this Dissertation (either in detailed parts of calculations or in focused discussions) which we now begin.

16

1. Topological Insulators: An overview

Chapter 1 Topological Insulators: An overview The first chapter reviews the main properties of some newly found materials, called topological insulators, which are considered a milestone in Solid State Physics. The main reason is that they are quantum coherent and robust, conserving their nondissipative currents even in conditions that tend to destroy quantum coherence, such as temperature, crystal imperfections, dirty materials etc. These currents are formed under certain topological conditions, such as conservation of time reversal symmetry and non-zero Chern number, which is the topological index for this kind of materials. Similar to Gaussian curvature of mathematics, which in return gives the Euler characteristics, in Physics a quantity called Berry curvature controls the topological properties of some electronic systems. It is now realized that the first topological insulator was the Quantum Hall Effect system, in which the amazing quantization of Hall conductivity stems from a Chern number and cannot be changed, as long as the temperature is low enough and impurity potential varies slowly within the sample. Also, the more recent Quantum Spin Hall Effect is a two dimensional topological insulator, with a non-dissipative spin current running only at its edges, and we also have 3D topological insulators where the electric currents run across their surfaces. To complete our overview, we also present some properties of Graphene, both transport and thermodynamic.

17

1. Topological Insulators: An overview

1.1 Introduction In Physics, many intriguing phenomena have been brought to light since the discovery of quantum mechanics. Some of them include the double slit experiment, which help in understanding the coexistence of wave and particle nature of quantum matter. Others involve the quantum Hall effect and the universal quantization of Hall conductivity  H in integer multiples of e 2 / h . The most fascinating fact about  H is its exact and robust quantization that does not weaken even in the presence of moderately dirty materials or not-so-low temperatures, factors that tend to destroy quantum coherence. The reason for this extraordinary robustness is topological, and lies in hidden properties of the wavefunctions in Hilbert space. These special wavefunctions are defined in a parameter space with non-trivial topology and have a non-zero Chern number, which as we„ll see below is a topological index. Whenever a Chern number appears, the material under consideration acquires exotic transport properties, i.e. it might be an insulator in its bulk, but has protected metallic states in its surface resulting in a so-called topological insulator. Or, it might be a semimetal in its bulk, with conducting channels in its surface, resulting in a topological semimetal. These materials follow a new classification that is characterized by a non-zero topological index, in contradistinction to conventional materials (i.e. semiconductors, metals, etc.).

1.2 The Quantum Hall Effect (Q.H.E) The “grandfather” of topological materials is the Quantum Hall Effect[1], discovered by K. von Klitzing et al. in 1980 in a high mobility two dimensional (2D) semiconductor under high magnetic fields. Temperature must be low enough to avoid thermal excitations and Landau quantization leads to vanishing longitudinal conductivity  xx (due to the lack of [3phase-space scattering) accompanied with a quantized Hall conductivity  H   e 2 / h (with π: integer) when the Fermi energy lies in the bulk energy gap (somewhere in the range between adjacent Landau Levels (L.Ls)). The quantization of  H is exact and it has a topological origin. Laughlin‟s argument (see below) interpret the integer π that appears in  H as the number of electrons transported from one edge to another, if the 2D rectangular sheet is folded to a cylindrical surface. TKNN[2] in 1982 showed that the crystal momentum space (the k-space) is mapped to a topologically non-trivial Hilbert space, whose topology can be specified by an integer, called the TKNN invariant or the first Chern number, or equivalently, a winding number (from the non-zero winding-number of the wavefunction‟s phase). Also, Halperin showed that the Q.H.E is always accompanied by conducting edge states that carry the electric current. For our generalization of the Hall conductivity, when the electric field exceeds a critical value, see Chapter 5 of this Thesis. To observe the Quantum Hall Effect, a planar metallic or semiconducting sample is placed in a strong magnetic field (B-field) perpendicular to its surface. The sample must have a very small thickness (order of nanometers) to prevent excitations of electrons in z-direction, and the motion is therefore restricted to two dimensions. Of course, this is not a strong criterion for the Q.H.E to occur. The case of a large 18

1. Topological Insulators: An overview

thickness in thoroughly studied in Chapter 3, where we propose the quantization of Hall conductivity in the large thickness limit as well. Next, with the help of two oppositely charged gates we create a small electric field (E-field) in the longitudinal direction, which in turn creates a small current passing through the sample (the oriented collective motion of electrons). Next, we connect via two other gates a voltmeter, counting the voltage drop in the direction across the E-field, while another voltmeter is connected in the opposite edges of the sample, counting the Hall voltage. This voltage is clearly a B-field property, as it emerges as a response to the transverse electromagnetic field, in contradistinction to longitudinal voltage which is a property only of the external E-field. These two voltages are of great importance in the Q.H.E regime. They both depend on the amount of current flowing through the sample (despite the fact that they are observed in transverse directions). To make them independent of current flow, we can divide each voltage by the current to define two resistances: the longitudinal one  L , which is the ratio of the longitudinal voltage drop to current, and the Hall resistance  H which is the ratio of Hall voltage to current flowing in the sample. In 1879, Edwin Hall found that the Hall resistance must increase linearly with B, which is the standard classical behavior. In 1980 Klaus von Klitzing discovered the integer Q.H.E., in which the Hall resistance is strongly quantized, and forms a series of plateaus (see Figure on the right). In these plateaus, varying B will not result in any variation of  H . Only at the end of plateaus (for special values of B)  H varies, just to move to another plateau where it stays constant again. On the other hand,  L is exactly zero (for the T=0 case, because in higher temperatures slightly differs from zero) when  H is on a plateau (no dissipations!), and is non-zero only between the B range of transition of  H to another plateau. The Hall resistance obeys a delicate and simple relation when located on a plateau: H1   H  ne2 / h where  H is the Hall conductance, n is an integer named filling factor, e is the electron‟s charge, and h is the Planck‟s constant. These values are independent of the sample‟s geometry, or other characteristics concerning the material composition (impurities, defects, etc.) and are extremely accurate with an error lower than 1 part in six million. Because of the integer values of π the effect is more precisely called Integral QHE (IQHE). In 1982, Tsui, Stormer and Gossard discovered the Fractional Quantum Hall Effect (FQHE), which occurs in cleaner samples and lower temperatures than its integral 19

1. Topological Insulators: An overview

form. In this regime, the filling factor can become fractional (e.g. 1/3, 2/3), and additional plateaus of  H appear. This new phenomenon rises when the interactions among the electrons are of significant importance. Indeed, because L.Ls are not fully occupied, interactions come into play and destroy the quantized transport. F.Q.H.E. can be explained using Composite Fermions picture[4], in which each electron is accompanied by an integer number of flux quanta, creating new quasiparticles that can be considered as non-interacting.

1.2.1 Role of impurities on the quantization of Hall conductance Impurities[5] and defects play an important role in solid state materials, because it is the main reason for the resistance of some metals and semiconductors. Electrons accelerating by a small external electric field will collide with defects dissipating energy or form a bound state with an impurity with an excess of positive charge, and therefore cannot carry an electric current anymore. In a „dirty‟ sample not all electrons are current carriers, because some of them will be localized in some region and therefore they are unable to contribute to the electric current. Others are described by an extended wavefunction, allowing electricity to flow. Paradoxically, in the Q.H.E regime this is not the case at all. It is the presence of impurities that vanish the longitudinal resistance! To understand this, we recall from elementary Quantum Mechanics the fact that each L.L. contains up to Φ/Φ0 spinless electrons, where Φ is the magnetic flux penetrating the system and Φ0 is the flux quantum (Φ0=hc/e). Some of them are completely localized in some region inside the sample, while others travel free from one end to the other. Because of the electric field, each L.L. becomes an energy band with Φ/Φ0 states, each one centered at a specific point. States that have low energy are located at the bottom of the band and they become localized, orbiting an impurity atom without escape. States with high energy, located at the top of a band are also localized, leaving only the middle states (the extended states) to contribute to the conduction. So, when the Fermi energy passes through a whole L.L, it also passes through extended and localized states. This means that the number of conducting states is varied, as the magnetic field changes. When the Fermi energy is located in the lower localized states, the number of conducting states does not change, and neither does Hall conductivity. Only when passing through the extend states region the number of conducting states varies, resulting in a corresponding variation of Hall conductivity (it moves from one plateau to its next adjacent one). This is the reason why in the Quantum Hall regime there is a large stability and exactness when measuring the Hall conductivity. For this reason, the Quantum Hall Effect has been proposed as a standard model for measuring other quantities with high precision, i.e. the fine structure constant, and the quantum of conductance that consists of electron‟s charge and Planck‟s constant.

20

1. Topological Insulators: An overview

1.2.2 Laughlin’s interpretation of QHE (universality of IQHE) Laughlin[6] considered the previous thin film to be folded into a cylinder; let‟s call the radius R and the height L. The previous transverse magnetic field is now also „folded‟ to preserve the perpendicularity on the cylinder‟s surface (although this is experimentally impossible: it violates .B  0 . For an interesting extension see[7]) We call the folded direction as y-direction, while the other, extended direction as x. In addition, Laughlin placed an inaccessible Aharonov-Bohm (AB) flux in the interior of the empty space producing a homogeneous (on the cylindrical surface) vector potential AAB   AB / 2 Ryˆ , where  AB is the AB flux. The total vector potential, given that the magnetic field is represented by the Landau gauge AB  Bxyˆ (along the folded direction) then reads: A   Bx   / 2 R  yˆ  B  x   / 2 BR  yˆ (1.1)

Laughlin‟s gendanken experiment: An AB flux is adiabatically inserted in the empty space of a cylindrical surface. When the system reaches equilibrium, the AB flux is varied by a flux quantum. The induced electric currents flow in the circular direction and in that time interval an integer number of electrons is adiabatically transported across the x-direction.

This relation (1.1) tells us that inserting the AB flux is equivalent to a shift on the positions of the electrons on the cylindrical surface[8]. This is a crucial point to the whole argument process: if all x are translated by this finite amount, then the position operators defined by

X 0  c

ky eB

(1.2)

are also translated by the same amount, being

X 0  c

ky eB



 AB (1.3) 2 BR

Hence, if we vary the AB flux  AB by an amount  AB in an adiabatic manner, such that the old adiabatic theorem is still valid and no energy excitations occur, all oscillation centers are also varied: X0  X0 

 AB (1.4) 2 BR

21

1. Topological Insulators: An overview

But what about if this variation equals exactly to one flux quantum  0  hc / e ? Then the system should return to its initial state (in the sense that the energetic occupancies do not change - see Chapter 3, Section 3.9 for the corresponding proof), but with all centers translated by a distance lB2 / R , where lB  hc / eB is the magnetic length. This distance covers a total area of 2 RlB2 / R  2 lB2 , which is exactly the area that corresponds to a single microstate of the system (Indeed, the flux passing through this area is B 2 lB2   0 ). In other words, by the time the flux variation procedure finishes, each electron has been transferred to its neighboring empty state. During this process, the external induced electric field results in the formation of a Hall electric field created in x-direction. This Hall field changes each electron‟s energy by an amount

  eEH lB2 / R , (1.5) So the change in the total energy of the system is just (for N electrons present on the surface)  E  NeEH lB2 / R  enA 2 LEH lB2 , where n A is the areal number density. Now, given that the Hall voltage is VH  LEH , and that when the Landau Levels are fully occupied (let v be the number of fully occupied L.Ls) the areal density can be written as nA  vB / 0 we have for the total energy change

 E  evVH (1.6) This relation means that the collective translation of all N oscillators is equivalent to a translation of only one state per Landau Level, which moves from one edge to the other and enters back at the other edge of the cylinder periodically. In other words, the AB flux acts as a charge pump. The induced electric current (it flows in the ydirection) is given by Faraday‟s law of induction: I  c

E (1.7)  AB

Substituting eq. (1.6) in (1.7) we find (with  AB   0 ):

e2vVH I  (1.8) h and the Hall resistance becomes simply    

h . (1.9) ve 2

In this manner, Laughlin proved the universality of the Hall resistance‟s character, and that it stems from gauge considerations. As a result, adding arbitrary impurity potential or a low temperature to the above argument, the result is not expected to change (much). This is a somewhat surprising result, because this exact quantization takes into account a microscopic quantum description but the Hall resistance is a quantity that is measured at a macroscopic level. A real life – material though is indeed dirty, filled with impurities and defects, and also includes interactions between electrons or electrons with phonons (due to the oscillatory nature of the atoms composing the material). We do know that electrons in 22

1. Topological Insulators: An overview

metals can be almost considered as free waves, propagating freely inside the material, and the electron-phonon interactions are weak. If we want to take into account the impurity potential, (as we did in the previous section) we must have in mind that the current is a special property of extended states, and that no matter how strong the impurity potential is, there will always be at least one extended state in the middle of a Landau Level, capable of contribute in the transport properties of the system. Reaching the end of this Section, we mention that the Laughlin argument can be extended in a way that the roles of Hall voltage and induced-electric current can also be interchanged. In a new consideration, the electric current flows in the x-direction (across the cylinder‟s height) and it is attributed to the electrons movement due to a change in AB flux, while the Hall voltage becomes now tangential (in the y-direction) and its created due to the rate of change of the magnetic flux. The Hall voltage becomes VH   AB / c t . The electric current is then I  ve /  t , where ve is the total transferred charge, and the Hall resistance becomes:

 

VH h   2 (1.10) I ve

We then conclude to the same result as before), but with a different point of view (and actually, in a much simpler way that Laughlin‟s). This reflects once again the universality of the quantization of Hall conductivity (and its invariance under the exchange of roles (of what a Hall voltage is and what a longitudinal current is).

1.3 The Quantum Spin Hall Effect (Q.S.H.E) Another recently discovered topologicallyoriginated and for applications phenomenon is the Quantum Spin Hall Effect[9], which can be useful in the generation and manipulation of spin currents can be applied in the area of spintronics. In this spin-version of Q.H.E, a longitudinal electric field will generate in the transverse direction a Hall spin current, curried by helical edge conducting states that flow without dissipation. Unlike Q.H.E, the formation of the edge states does not depend on any external magnetic field, but is a pure consequence of the non-trivial topological structure of the wavefunctions in the Hilbert space. Without magnetic field, or magnetic impurities, time reversal symmetry is preserved and is responsible for the underlying topological structure of the Hilbert space, which gives rise to robust and non-dissipative spin currents flowing in the edges of a material (without charge flow).

1.4 Graphene Graphene[10] is a purely two dimensional sheet of carbon atoms in a honeycomb lattice configuration. Many 2D graphene sheets are stack together to form graphite, the 3D analog of graphene. Until 2004, it was believed that one cannot isolate a single 23

1. Topological Insulators: An overview

sheet of carbon atoms out of graphite, because it would not be stable against external conditions such as temperature and that it would definitely break apart. In that year, Andre Geim, Novoselov and colleagues proved that Graphene could exist after isolating it using a scotch tape from a pencil on top of a SiO2 substrate. It has been long known (Wallace, The band theory of Graphite, 1947[11]) that graphene should exhibit unique electronic properties, both transport and thermodynamic, as its conducting electrons behave like relativistic massless particles moving with a constant Fermi velocity. We start our quick description with a single 2D sheet of Carbon atoms in a honeycomb lattice configuration:

Fig. 1.1: The honeycomb configuration of Carbon atoms in 2D Graphene. Lattice Bravais vectors and nearest-neighbor vectors are shown, and the corresponding 1st Brillouin zone (on the right) with its two special ponts K and K‟, called the Dirac points. For the honeycomb lattice, carbon atoms form three sp2 orbitals to match the 120 degrees spacing between adjacent ones, leaving one p orbital (perpendicular to the plane of the structure) to conduction. In the above graph, we can see that there are two inequivalent atoms, A and B, that belong to two different Bravais lattices. One can choose Bravais vectors a1 and a2 as a1 



a 3, 3 2



and a2 



a 3,  3 2



(1.11) where α is the nearest Carbon atoms

spacing  1.42  . Then, the reciprocal lattice vectors ai .b j  2 i , j are found to be: 2 2 1, 3 and b2  1,  3 . Using the reciprocal lattice vectors, one defines 3 3 the first Brillouin zone (bounded by the intersections of lines at the middle of each b1 









24

1. Topological Insulators: An overview

bond in the reciprocal space) as in Fig. 1.1, which resembles the direct honeycomb lattice but it is rotated by 90 degrees with respect to it. Because the reciprocal atoms are connected by reciprocal vectors forming two groups of 3 points, the six points at the Brillouin zone edges form two distinct categories: those named as K and those named as K‟. These points are responsible for the appearance of Dirac cones that we will see below, and the relativistic nature of conduction electrons, that we will later discuss. At this point one also defines the nearest – neighbor vectors, which are the distances between one type A atom and its three type B atoms as:

1 









a a 1, 3 ,  2  1,  3 and  3   a 1, 0  (1.12) 2 2

We are now ready to write down the tight binding Hamiltonian of our system. Considering that conduction electrons are in a π orbital, which is oriented normally to the plane of Carbon atoms, and denoting as (i,σ) the ith Carbon atom with a spin value ζ, the tight binding Hamiltonian in the nearest neighbor approximation is:

H  t

   b   HermitianConjugate  (1.13)

i , j  n.n  

† i,

j,

with t being the hopping matrix element, t  2.8eV ,  i†, creates an electron in the unit cell i, on the A atom, and b j , subtracts an electron from the atom B in the unit cell i. Now we are ready to find the eigenstates: as usual, they are written as Fourier transform that takes the variable i (that actually defines this site representation) to variable k , i.e. the crystal momentum space. Eigenstates must be expressed in the form of spinors, where each component corresponds to the amplitude to find the electron in one of the atoms of type A or B within the unit cell:  ai†e  ik .1 /2   ak  ik . Ri     e  †  ik . /2  (1.14) 1  bk  i  bi e 

In this manner, one moves from the site representation to momentum representation, which is uniquely connected with the reciprocal space resulting in the following Hamiltonian:  0 Hk   *  k

3 k     t eik . l , (1.15) with   k 0  l 1

and, upon diagonalisation of (1.15), one can get the eigenenergies (two distinct bands):

25

1. Topological Insulators: An overview

 k    k  t 1  4 cos

3k y a 3k x a 3 cos  4 cos 2 k y a (1.16) 2 2 2

The two bands meet each other in special points in the first Brillouin Zone, where the energies are equal to zero. This happens at:

3k y a 3k x a 1  2n , cos (1.17)  2 2 2 or

3k y a 3k x a 1   2n  1  , cos   (1.18) 2 2 2

For all integers n, the first choice takes us out of the range of the 1st B.Z. The second one is valid only for n=0, and takes us to the boundary of the 1st B.Z., at the points K and K ' discussed earlier. These special points are called the Dirac points, for a good reason that we shall see below. These points are:

K

2 3

2  1   1,  and K '  3 3 

1   1,   (1.19) 3 

1.4.1 Expansion around Dirac points We now discuss the form of energy and eigenvectors close to Dirac points K and K ' . We denote the difference k  K  q , and we expand eq. (1.15) for small values

of q to get: 3

 q  t  e





i q  K . l

l 1

3

3

l 1

l 1

 

 q. l  t  eiK .l  iq. l eiK .l   2 l 1  3

3

with

e

iK . l

e

i

2 3

1 e

i

2 3

  l eiK .  l

l 1

2

e

 2cos

l 1

3



 t  eiK .l eiq .l  t  eiK .l 1  iq. l iK . l

   

 (1.20)

2  1  0 and 3

i 2 i 2  a a 3a ˆ ˆ  1 3  1, 3 e 3  1,  3  a 1, 0  e 3  i  ij   i 2 2 2 2 2  















3a ˆ ˆ i 3 i  ij e 2





26

1. Topological Insulators: An overview  i 3a resulting in  q  it  qx  iq y  e 3  vF  qx  iq y  (1.21), where we have dropped 2 an unimportant phase factor, and (1.15) in this limit becomes

 0 H k  vF   qx  iq y

qx  iq y    vF  .q (1.22) 0 

i.e. it is linear to momentum q ! The eigenvalues of (1.22) are

 k   vF q

(1.23)

Note that this Hamiltonian describes an ultra-relativistic particle, or a relativistic massless particle, with the velocity of light now replaced by Fermi velocity v F . This is the reason that points K and K ' are called Dirac points, because for small crystal momentum values in their vicinity, the energy spectrum is linear with respect to k , and resembles Dirac equation. The corresponding eigenfunctions in the vicinity of K are  K  q  

q 1  ei /2  1 y (1.24)   i /2  with   tan qx 2  e 

Note that when the vector q moves in a circular path around a Dirac cone, it collects a total phase of π (the Berry‟s phase), as this is a usual fact regarding fermion systems. That is, when a small electric field is applied in Graphene, q gains time dependence, and moves around a Dirac cone collecting a trivial phase π. It is worth mentioning, however, that extra care must be given in the derivation of Berry‟s phase. As we will see in the next Chapter, which is devoted to geometric phase, one needs to deal with single-valued wavefunctions when trying to calculate Berry‟s phases. The above wavefunction (1.24) is not single-valued when the variable φ alters to φ+2π! With respect to the number of states per energy band (remember, we have two bands) we recall that we have N unit cells in our Graphene sheet, with two electrons per unit cell, therefore we have 2N electrons. The available number of states per energy band is exactly N, equal to the number of cells. Because in the absence of spin interaction terms (Zeeman, spin orbit coupling) two electrons can be placed in each state, we have the complete filling of the lower energy band. At zero temperature, Fermi energy is located at the two Dirac points (at zero energy). Additionally, because the density of states at Fermi energy is exactly zero (due to lack of available states in that region), Graphene can be considered as a perfect semimetal, or as a zero-gap semiconductor. Another thing we want to discuss here, that connects Graphene to the next Chapters, is the collective behavior of a Fermi gas that lives around a Dirac cone (see Chapter 5 for Graphene in an electromagnetic field). Things change when one applies a voltage 27

1. Topological Insulators: An overview

difference between graphene sheet and its substrate. Electron excitations can only take place in the region above the Dirac cones, filling those available states. What we have then is a free 2D electron gas! The positive energy band near the Dirac cone is

 k  vF q (1.25) We call N the number of electrons that occupy the Dirac cones. The 2D Fermi wavevector is

4S N 2

qF

 qdq  0

4S qF2  qF   nA (1.26) 2 2

with nA being the areal density and the factor 4 being due to valley and spin degeneracy (two and two accordingly). By „valley degeneracy‟ we mean the two Dirac cones (valleys), based on K and K‟, that both have degenerate energy values. The total energy of excited electrons is accordingly

Etot 

2S



qF

vF  q 2 dq  0

2 N  F (1.27) 3

in units of Fermi energy, in contradistinction to the conventional 2D energy which is N F / 2 .

1.5 Berry Physics: The beginning Consider a Hamiltonian that depends on a set of parameters described by the time







dependent vector R  t  , H R  t  . We make the assumption that, as long as R is relatively small, we can always write down the corresponding eigenvalue equation that will play a crucial role in the t-development of the system

   

H R n R

 

with n R

   

 n R n R

(1.28)

the instantaneous eigenvectors and  n the eigenvalues (all quantities are

R  t  - dependent). In fact, there is no assumption here; the old adiabatic theorem

guarantees that as long as the rate of change of the parameter vector is small, a quantum mechanical system that starts from an initial eigenstate will remain in that eigenstate forever (provided that there is no energy-degeneracy for every value of the parameter along the path). The time evolution of the eigenvectors is governed by time –dependent Schrödinger equation: H  i

d (1.29) dt

28

1. Topological Insulators: An overview

By then making the ansatz that the time-dependent wavefunction can be written as i

  ei nt e with

n

t

 dt ' n  Rt ' 0

n

(1.30)

an arbitrary phase factor that depends only on time, and by substituting eq.

(1.30) into the Schrödinger equation (1.29) one gets 



n i n n i n

where we have substituted

   n  i R . n  R n  R . AB (1.31) t

   R . R , because time dependence comes only through t

R  t  . AB is the Berry vector potential, or the Berry connection as it is called, and has

the meaning of a vector potential generated in the parameter space. The extra phase n

that in general had been overlooked in the past, then becomes t



 n   dt R . AB   dR.AB , (1.32) 0

where in the last line integral , explicit reference to t has been removed. Using Berry potential AB (see the next Chapter for our own Berry gauge potential) we may define the Berry magnetic field BB , or the Berry curvature generated in the parameter space by BB   R  AB

(1.33)

Physicists used to believe for several decades that this extra phase factor (eq. (1.32)) was un unimportant quantity with no physical meaning, as we always can multiply a wavefunction by an arbitrary phase without altering the underlying physics. This is not necessarily true, though, because if the whole time process followed a closed path, i.e. the vector R  t  returned to itself in the end of the process (an adiabatic cyclic process- the  n of (1.32) is called a Berry‟s phase) the Berry‟s phase cannot be gauged away. Hence, the Berry‟s phase is a real measurable quantity that have been overlooked for a long time! For more information on Berry‟s phase see the next Chapter. Indeed, in topological systems Berry curvature is a quantity of crucial importance, as it is directly related with the first Chern number, which is a topological index (i.e. it shows whether a system is topological or not).

29

1. Topological Insulators: An overview

1.5.1 Application of Berry curvature in solids One historic application of the above Berry-ology was in the Quantum Hall effect, resulting in the so called TKNN invariant[12]. There, by expanding the eigenvectors using time-dependent perturbation theory to take into account the response to the external in-plane electric field that is applied to a 2D crystal with a transverse magnetic field, namely n

E

 n

E 0



m n

m  eEy n

m

n m

(1.34)

they calculated the expectation value of the electric current in the Hall direction (the x direction) jx

E

  f  n  n n

E

evx n S

E



 n evx m m  eEy n n  eEy m m ev x n  1 jx E 0   f   n      S n n m n m mn  

(1.35)

where vx is the electron‟s velocity, f   n  is the Fermi-Dirac distribution, and S is the 2D crystal‟s area. The Hall conductivity can be written as:

 

jx

E

E



i e2 S

 n

 n vx m m v y n  n v y m m v x n  f  n    (1.36) 2    mn     n m 

Since from the Heisenberg picture we have m vy n  

i

 n   m 

m y n (1.37)

Now, we know that for the case of a crystal, the eigenvectors can be written in Bloch 1  u . Rewriting eq. form, um ,k  m and um,k ' v un ,k   n ,k   m,k ' um,k ' k n ,k





(1.36) one gets:

  

ie2 S

 

  f    k k

nm

nk



unk x

    unk  unk unk  (1.38) k y k y k x 

And by using the Berry connection for the Bloch states one finally obtains

 

e2

 

  BZ d 2kBB k 

ve2

(1.39)

n

30

1. Topological Insulators: An overview

This is because the closed integral of the Berry curvature (see next Chapter) in the 2D Brillouin zone must always be an integer-which is the Chern number- (and has the meaning of the number of magnetic monopoles inside the parameter space, for more information and other interpretations see next Chapter). This integer has a topological origin, as it can be written as an integral of a curvature (like the Gaussian curvature of Mathematics), and as a result the Berry curvature and cannot be changed under small perturbations/deformations. This is one of the reasons why the quantization of Hall conductivity is so stable and accurate.

1.6 Time Reversal Operator The next step of our quick overview of topological insulators is to introduce a certain kind of insulators that preserve the time reversal symmetry (TRS). The time reversal operator for electrons can be written as T  i y K , (1.40)

where  y is the spin operator and K is the complex conjugate operator and has the property: T 2  1 because T 2   i y K  i y K 

  i i y

* y

    . Note

also that it is an anti-unitary operator. As usual, the basis consists of eigenstates of the operator  z then, the following property holds:

  

 T     i y  *  

z,

  



z,



'z

  z*  z* i y  'z   '*z 

'z



 z , 'z

  z  z i y  'z  'z  *

   z,

 '

* z

 ' i  z  z  * z

† y

*

 z  *  '*z i y†  z   '*z (1.41)

'z

  i  † y

*

    i y K 

  T 

with

  

z

 z  1 (identity operator). In a similar way to this, one finds:

z

T T    T 2   

(1.42)

31

1. Topological Insulators: An overview

1.7 Time reversal Bloch Hamiltonians Let H be the Hamiltonian that governs a Bloch system. The eigenfunctions are  nk  r   eik .r unk  r  , (1.43)

with k the crystal momentum and unk  r  the periodic cell function. Using the above wavefunction we can write Bloch Hamiltonian as:

 

H k

p k  2m

2

 V  r  (1.44)

 

Now if this H preserves time reversal, then it must satisfy  H k , T  f  r   0 ,   meaning that

 

 

   

 

iH k  y f *  r   T  H k f  r   i y H * k f *  r   i y H k f *  r    (1.45)  H k  TH k T 1

 

When the above relation is valid, the eigenenergies always come in pairs (named Kramer‟s pairs), i.e. the wavevector k has the same energy with k .

Fig. 1.2: Kramers pairs of bands. Note that at the time reversal invariant

k

points that obey

k  k k  0,  ,   the bands touch, while in all other points in the interior of the B.Z. the bands are degenerate.

Perfect crystals without magnetic impurities and not placed in a magnetic field usually preserve TRS, resulting in the double degeneracies in the Brillouin Zone. Electrons occupying these bands move in opposite velocities at the edges of the topological material as we will see below. Because the edge motion is done in the absent of backscattering (i.e. no resistance) there are no energy-dissipations and the electric current channels are robust, and cannot be easily destroyed. The first Chern number, that involves Berry curvature is an integer that counts the number of available currentchannels on the surface (or on the edge) of the sample. It is important to notice here 32

1. Topological Insulators: An overview

that the non-dissipative current flows only on the surface; in the bulk there is no current and therefore the material behaves as an insulator in that region.

1.8 The Su-Schrieffer-Heeger (SSH) chain model A good example for the understanding of the underlying physics of topological insulators is the SSH model[12], which describes the energy bands in 1D polyacetylene chain. Consider a 1D chain, with 2N sites and with alternate amplitudes (staggered):

The chain is considered to be large, preserving its translational symmetry in the bulk, but not at the two edges, where the chain ends. Also, the two edges have disorder, resulting in non-fixed hopping amplitudes, while in the bulk the hopping amplitudes are considered fixed. Now, let‟s put our electrons on the chain; each site may accommodate one electron, such that the tight binding Hamiltonian is: M

H   tn cn†cn 1  h.c. (1.46) n 1

With M=2N, h.c. is the Hermitian conjugate, tn is the site – dependent hopping ablitude and cn† creates an electron at site n. Imposing periodic boundary conditions on our chain we have c2 N 1  c1 but if we want to study an open chain we may as well set t2 N  0 . Since our hoppings are alternate, we may use another notation: t2n  un , t2 n1  wn and the Hamiltonian becomes: N

H   un c2†n 1c2 n  wn c2†n c2 n 1  h.c (1.47) n 1

Now we make use of the fact that in each unit cell there are two types of atoms (sites); we call them 1 and 2. Because the number of sites is 2N, an even integer, we may introduce a cell index n=1,2…,N and a sublattice index j=1,2 (that is, in each cell there are two atoms that belong to two different sublattices). This means that the following relation holds:

33

1. Topological Insulators: An overview

cn , j  c2 n  j  2 (1.48)

In a more compact way, we group the operators in the unit cell as: cn†   cn†,1 , cn†,2  (1.49)

Rewriting the Hamiltonian in this new group-notation we have N

N

N

n 1

m 1

m 1

H   un cn†,1cn ,2  wn cn†,2 cn 1,1  h.c   cm† H m ,n cn  cm† H m,n cn (1.50)

with H m , n a 2  2 matrix. We also call U n  H n ,n the onsite energies, and Tn  H n ,n 1 the hopping matrices. All the other components vanish, because we suppose a short range hopping. The onsite and hopping matrices read 0 Un   *  un

un  0  and U n   0  wn

0  (1.51) 0

In the translation invariant bulk of the chain, we seek for Bloch wavefunctions obeying

 2n j 2  k   ei 2 kn / N  j  k  (1.52) These two-component Bloch vectors are eigenstates of the Hamiltonian written in k space

H  k   U  eikT †  eikT , (1.53) which, using (1.51) can be written in terms of Pauli spin matrices: H  k   h  k  . , (1.54)

and by using staggered hopping ablitudes, un  u , wn  w  w ei , to best describe the bulk, the vector h  k  can be written as:

hx  k   Re u  w cos  k    , hx  k    Im u  w sin  k    and hz  k   0 (1.55) such that the energy spectrum becomes:

E  k   u  e ik w* 

u  w  2 u w cos  k  arg u  arg w  (1.56) 2

2

The minima of the energy spectrum is   u  w , which means that the staggered hopping amplitudes open a finite gap. In this manner, we have two distinct energy

34

1. Topological Insulators: An overview

bands (due to two different types of atoms in the unit cell, just as we had in the case of Graphene). Now, a useful quantity that is directly connected with the chiral edge states is the winding number. Imagine that the above vector h  k  is forced to move in the k-space and return to its initial position. This can be achieved by varying k itself using an external small electric field. Because k runs over all possible values of k, the first and the last point being connected by a reciprocal lattice vector and therefore are equivalent, the whole procedure returns the system to itself. To a closed path on the k-plane that does not contain the origin (hence there is no gap there) we can associate a winding number v, which counts how many times the vector h  k  encircles the origin as the k is adiabatically varied from  to  as: 

1 d v dk log h  k  , (1.57)  2 i  dk

For the SSH model, the winding number is either +1 in the case w  v in the case of large intercell hopping, or v=0 in the case of small intercell hopping, w  v . If one wants to change the winding number (which is a topological invariant) then either he will pull the path through the origin in the h-space, or lift it out of the plane and put it back on the plane at a different position. The first method means that we must close the bulk gap; the second method means that we need to break the chiral symmetry of the edge states. Symmetries must be respected; if not, then the system loses its robust topological properties. Also, a finite bulk gap must exist in the energy spectrum, and it must remain through the whole process of an adiabatic deformation.

1.8.1 Edge states formation To study what happens with the edge states, we consider a finite open chain. This means that translational invariance is broken at the final cell, so we must set wN  0 there. Plotting the energy spectrum we can see that whenever the bulk winding number is nonzero, and the bulk gap is large enough, there exists a pair of low energy eigenstates located at the two edges of the chain. In 1D of course, these states are localized (0D), while if the chain was 2D, there would be an integer number of channels carrying the electric current around the 2D plane.

35

1. Topological Insulators: An overview

Figure 1.3: Energy spectrum of open SSH chain for the case of 40 sites (or 20 unit cells). When the bulk gap is open, a pair of midgap edge states that have winding number +1 appears, and they are localized at the edges of the material. In Fig b, intracell hopping u is gradually decreased to zero, leading once again to formation of a pair of edge states. In 2D time reversal invariant Topological Insulators (TRI), in which T 2  1 , one dimensional edge states appear at the edges of the material. In a similar way to the previous polyacetylene case, we separate the 2D sample into a translational invariant bulk with a finite energy gap that approaches zero, and the edges. Edge states are themselves eigenstates of the Hamiltonian that lie in the bulk gap and whose wavefunctions is mostly localized at the edges of our sample. However, there are also the so called Chern insulators, in which the proper topological invariant is the Chern number (the one we saw before) opposed to these Z2 Topological Insulators. Edge states are chiral, in the sense that equal number of them moves in opposite directions along the edges. If we now consider an adiabatic deformation, such as adding TRI impurity potential, or preserve the gap open through a control process, the total number of edge states cannot change. In the case of 3D Topological Insulators (see next Figure), edge states become surface states, propagating along the surfaces of the 3D material. These states once again lie in the bulk gap, wherever a bulk gap is open, and they have spin-momentum locking in the case of materials that have large spin-orbit coupling, e.g. the compounds Bi2Se3, Bi2Te3 etc.

36

1. Topological Insulators: An overview

Chapter References [1] K. von Klitzing, G.Dorda, M. Pepper, “New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance” Phys. Rev. Lett. 45, 494–497, (1980) [2] D. J. Thouless, Μ. Kohmoto, M.P Nightingale, M. den Nijs, Quantized Hall Conductance in a Two-Dimensional Periodic Potential, Phys. Rev. Lett. 49, 405 (1982) [3] M. Kohmoto, Topological Invariant and the Quantization of the Hall Conductance, Annals of Physics 160, 343-354 (1985) [4] J.K. Jain, Composite Fermions, Press.ISBN 978-0-521-86232-5 (2007)

New

York:

Cambridge

University

[5] Bertrand Halperin, The quantized Hall Effect, Scientific American, Volume 254, Issue 4 [6] R. B. Laughlin, Quantized Hall conductivity in two dimensions, Phys. Rev. B 23, 5632I (1980) [7] K. Moulopoulos, Topological Proximity Effect: A Gauge Influence from Distant Fields on Planar Quantum-Coherent Systems, International Journal of Theoretical Physics, Vol. 54: 1908-1925 (2015) [8] Benoit Ducot, Bertrand Duplantier, Vincent Pasquier, Vincent Rivasseaux, (Editors), The Quantum Hall Effect (Poincare Seminar 2004) [9] Markus Koenig, Hartmut Buhmann, Laurens W. Molenkamp, Taylor L. Hughes, Chao-Xing Liu, Xiao-Liang Qi, Shou-Cheng Zhang, The Quantum Spin Hall Effect: Theory and Experiment (2008) [10] A. J. Leggett, Lecture 5: Graphene: Electronic band structure and Dirac fermions [11] P. R. Wallace, The band theory of Graphite, Phys. Rev. 71, 9 (1947) [12] J. Asboth, L. Oroszlany, and A. Palyi, Topological insulators, Spring 2013 [13] Y. Ando,Topological Insulator Materials, Invited Review Papers, (2013)

37

2. Geometry of Quantum states: The adiabatic approximation

Chapter 2 Geometry of Quantum states: The adiabatic approximation The geometrical construction of quantum mechanical states allows us to measure and interpret useful quantities through adiabatic approximations. One such quantity is the Hall conductivity (that we overviewed in the previous Chapter), which is directly related with the first Chern number and therefore is established as a topological invariant. Not the geometry of the material, but a geometry of states that live in the projective Hilbert space is responsible for non-dissipative transport observed in recent topological systems. More than three decades ago Michael Berry discovered that when a system is driven out of equilibrium conditions by adiabatically varying external parameters, the wavefunction gains a geometrical phase factor (dependent only on the variation path) which cannot be eliminated by gauge transformations as long as the external parameter returns to itself by the end of the variation. This is Berry‟s geometric phase. After that, several works followed, extending Berry‟s work beyond adiabatic limit, or generalizing it to systems with multivalued wavefunctions. In this chapter we review Berry physics emphasizing its role on modern systems such as adiabatic transport and charge pumping. In addition, we present novel results regarding Hellmann- Feynman theorem which is enriched with Berry terms (a magnetic and an electric one) and certain non-Hermitian boundary terms. Berry curvature (an effective magnetic field generated in parameter space) is the physical relative to Gaussian curvature of Mathematics, and it leads to topologically non-trivial quantization (i.e. in charge transport, Quantum Hall Effect, Polarized Bloch crystals, etc.) justifying in a sense their universal and robust quantized character. We also show that parameter space has one to one mapping with real space, as long as we interpret an emergent fictitious particle of charge  that lives in the parameter space. The emergent magnetic and electric fields produced by external perturbations satisfy Maxwell equations in the presence of magnetic monopoles in parameter space.

38

2. Geometry of Quantum states: The adiabatic approximation

2.1 Berry’s phase: a non-conventional method



Consider a Hamiltonian H r , p, R



that depends on a set of time dependent

  parameters R t  , but not explicitly on time t. We fix R  t  to be much smaller than

typical frequency  E (energy-difference gap, in the moving spectra, hence no degeneracies allowed) so that the adiabatic conditions are valid [1]. Adiabatic theorem states that a system at an initial quantum state identical to a stationary state will remain in that state forever. First recalling the general form of Schrodinger equation:

 

H  R, t

i

 

d  R, t dt

, (2.1)

d     R . R being the material derivative which takes into account separate dt t  changes to variables R and t, we proceed to solution by substituting in (2.1) the following ansantz: with

i   i   dtE R t  n   R, t  e e n R , (2.2) 

 



with a geometric phase factor  n which depends only on time t, with the usual dynamic phase factor (the energy factor above) and with n being instantaneous  eigenkets (that depend explicitly on R ), and we assume n are single-valued under  close paths in R space (something forgotten to be state in Berry‟s original paper[1] – recall the related error in Graphene form previous Chapter). It turns out that, after some algebra, the geometric factor is: 



 n  i n n , (2.3) where the bold dot means total time derivative (covariant derivative). Because time dependence is only due to parameter R , we may substitute the total derivative with respect to t with   d  R . R (2.4) to find: dt 

n



         i n R . R n  i R . n  R n   n  i  dR. n  R n   dR. AB , (2.5)

  where AB  i n  R n (2.6) is Berry‟s vector potential (one can show here from   R n n  0 that n  R n is purely imaginary, resulting in a real vector potential 39

2. Geometry of Quantum states: The adiabatic approximation

and a real phase factor  n ). As already mentioned, for a long period of time scientists believed that this phase factor can be easily eliminated (hence it does not have any physical significance) just by adjusting the phase of the eigenkets n , which can be  chosen arbitrarily. Berry showed that this is not the case: if the path that R follows is   a closed path, meaning that Rt i   R t f , with ti and t f the initial and final time

 

respectively, Berry‟s phase factor becomes a gauge-invariant quantity, with a real impact on many physical systems, and cannot be gauged away. For example, redefining the eigenkets as 

n  e iX R  n the new Berry‟s vector potential is:





       AB  ie iX R  n  R e iX R  n  AB   R X R



And the new Berry‟s phase transforms to:  

 







   

 n   dR. AB   dR. AB   dR. R X   n  X Rt f   X Rti  (2.7).  For a single-valued gauge function X R (ordinary gauge transformation), given that



R  ti   R  t f  , Berry‟s phase is unchanged. Or, equivalently, for any multivalued  X R , whose final and initial value is an integer multiple of 2π, Berry‟s phase gains a



trivial additive phase of 2π, leaving it once again a gauge invariant quantity. Now, using Stokes theorem on eq. (2.5) we can rewrite Berry‟s phase as a surface integral of a Berry curvature:

 n   dS .BB , (2.8)  with B B being an effective magnetic field generated in parameter space (via some    “magnetic” sources, as shown below) defined as: B B   R  AB (2.9).

 dS .  A   Im  dS .  n   set of intermediate states  m m  1 , and noting that Rewriting (2.8) as:  n 

R

B

R

R

n , inserting a complete

m

m R n 

m R H n En  Em

,

for m  n , we can show that Berry curvature is (using (2.9)):  BB n    Im



m n

  n  R H m  m  R H n

Em  En 2

(2.10)

40

2. Geometry of Quantum states: The adiabatic approximation

First, we see that the Berry curvature is defined when there are no degeneracies (but in 1984 Wilczek and Zee generalized Berry‟s formula to include the case of degenerate Hamiltonians[3]). But one can now go further and note that the existence  of degeneracies acts as poles on function BB n  and therefore act as magnetic monopoles in parameters space. These are the sources of Berry‟s curvature. One other interesting and crucial fact regarding Berry curvature, is that its closed surface integral in parameter space is always quantized:

 dS .B

B

 2 n , (2.11)

with n=0,1,2… a positive integer that counts the number of magnetic monopoles enclosed by the surface. This is the first Chern number appearing in topological systems (and can be interpreted as the number of electric current channels that run along the boundaries of a topological insulator). Also, it has the meaning of the integer filling factor of fully occupied Landau Levels (see Chapters 3 and 5) in the Quantum Hall regime (if a “jellium” simple Landau Level picture is justified). An interesting point to stress at this point, is that eq. (2.11) should be exactly zero if one naively interpreted BB as a normal magnetic field, and was separating a closed surface (manifold) M in a sum of two open ends (M1+M2), namely:

 M1

and

M2

M

Bberry .dS  M1 Bberry .dS   M 2 Bberry .dS 



M1

A.dl  

M1

A.dl  0 ,

are the two component (open) surfaces of M . Both M 1 and M 2 share the

same boundary (path), and therefore, for any non-pathological analytic function A (the Berry vector potential) the result should indeed be zero. But this is not the case, since A is indeed pathological, because wavefunction is also pathological in parameter space, resulting in the quantization of the first Chern number (and the above closed integral is then non-zero-and equal to (2.11), with n being equal to this Chern number – note also that the above difference



M1

A.dl  

M1

A.dl being along

the same boundary (path) is the difference of two equal Berry‟s phases, hence this difference must indeed be equal to 2πn (since a phase is only defined modulo 2π)).

2.2 Relation to topology We will take here as main example the Bloch periodic crystal, whose parameter space is the crystal momentum space, (or the first Brillouin zone, B.Z.). When a small, external electric field is applied (or magnetic, or both) the crystal momentum gains time dependence according to the relation:  e k   E , (2.12)

41

2. Geometry of Quantum states: The adiabatic approximation

i.e. E forces k to move adiabatically (for small E field strength) across the boundary of the first Brillouin zone (from –π/α to π/α in 1D, for a direct lattice period α). Because this zone is a closed toroidal surface, due to periodicity of wavefunction with respect to k , 

n , k 

 

 x    n,k    x 

(2.13)



the two points k   / a and k   / a are actually the same point and therefore the first B.Z. has a circle (ring) topology:

On the other hand, in 2D, a rectangular B.Z has the same topology as a toroidal surface[4], and therefore initial and final crystal momentum points are the same, resulting in a closed surface in parameter space (a torus):

In crystal materials, Berry curvature is integrated along the boundary of the first B.Z., which is always a closed surface, resulting in a topological quantization of certain physical quantities (i.e. Hall conductivity), as is well known since the time of TKNN.

2.3 Gauss-Bonnet Theorem There is a delicate relation (or analogy) between the Gaussian curvature of the famous theorem Gauss-Bonnet of Mathematics[5] and the Berry curvature. In geometry, a surface can adiabatically transform into another surface under specific constraints. For example, a cube can be transformed into a sphere but not into a donut, because, to get a donut, we have to make a hole on its surface. The conditions are that for a transformation to be successful, we must keep track of the number of holes on both surfaces. We say then that two surfaces with the same holes number are topologically equivalent, or they have the same topological order, or they are in the same topological class (homotopy class). 42

2. Geometry of Quantum states: The adiabatic approximation

We define the Euler characteristic,  , as a topological index for a certain surface: 1  dS  2 1  g  , with  the “Gaussian curvature” of the surface (which is 2  actually the product of the inverse eigenradii of the curvature) integrated along the surface, resulting in an integer number 2 1  g  . g is called the genus of the surface, and it counts the number of holes on it. A sphere has the same Euler characteristic with a cube,   2 , because they both have genus g  0 . A mug has the same Euler characteristic with a torus,   0 , because they both have genus g  1 . Other examples involve polyhedral surfaces: All polyhedral surfaces has the same topological index and therefore belong to the same topological order. In these cases, Euler characteristic turns out to be equal to   V  E  F , V is the number of vertices, E is the number of edges and F is the number of faces. All polyhedral (with no holes) have   2 , and are topologically equivalent to a sphere:

The Euler characteristic has the same interpretation (as a topological index) with the closed surface integral of the Berry curvature (the first Chern number), appearing in physical systems. We then talk about the Gauss-Bonnet-Chern theorem. Both Gaussian and Berry curvature have the same origin and their quantization is exact. This means that small geometrical changes/deformations of the material‟s surface won‟t change its topological properties, as along as topological symmetries are preserved i.e. time reversal symmetry in topological insulators is not broken (see Chapter 1). All physical systems characterized by non-zero Chern number exhibit robust and coherent properties, which do not alter upon small changes in the material characteristics, temperature, impurities, crystal disorder etc.

2.4 Generalization of Hellmann-Feynman theorem with Berry terms In this section, we explore a different version of the familiar Hellmann- Feynman theorem (HF), which is extended by Berry gauge fields that emerge in parameter space during adiabatic changes of the Hamiltonian through a t-dependent vector parameter. This extension of the Hamiltonian has been first noted by Κ. Kyriakou, see [12] for an early version, although we are giving it in a slightly different form: it will turn out from the viewpoint we are following here that the motion of the vector 43

2. Geometry of Quantum states: The adiabatic approximation

parameter resembles the motion of a charge  in external magnetic and electric fields and is a useful tool for dealing with adiabatic problems without the need of expanding the wavefunction using time-dependent perturbation theory. For that, we consider a general Hamiltonian depending not only on parameters, but also depending explicitly on time, which gives rise to a gauge invariant Berry electric field in the mixed parameter space R , t (i.e. apart from the axes R1 , R2 , R3 we have an additional axis t).

2.4.1 General formulation 

Consider a Hamiltonian that depends on an orthogonal set of parameters Rt  at the time t. In addition, we allow the Hamiltonian to have explicit time dependence,





namely H  H R  t  , t and then we solve the time dependent Schrodinger equation:



 

H R t  , t  R t  , t



i



d  R t  , t dt



(2.14)

Since we have two variables that have to be treated on equal footing, we use the material derivative defined as:

d     R . R (2.15) dt t

 Projecting (2.14) onto  R t , t we have:





     d  Rt , t H Rt , t  Rt , t  i  Rt , t  Rt , t dt



 

 











(2.16)

 Now, dropping the notation R t , t and proceeding as in the Hellmann-Feynman  case, we partially differentiate (2.16) with respect to parameter R :





 R  H     R H    H  R   i  R 

d d   i   R  dt dt

(2.17) For the Schrodinger equation, the following relations hold (real and complex conjugate parts): H   i

d d  and  H  i  . Plugging them into (2.17) we get: dt dt

44

2. Geometry of Quantum states: The adiabatic approximation

  R H   i

d d   R   i   R  (2.18) dt dt

Finally, by making use of the previous material derivative we arrive at the result:       R H    R B B   R E   R, t , (2.19)

 



 with R being the rate of change of the parameter, B B being the Berry curvature defined through the time dependent wavefunction, E being the instantaneous energy  and  being an emergent electric field. These generalizations of Berry quantities are defined as:

  BB   R  AB (2.20)

       R, t  i   R    R   , (2.21) t t  

 

  AB is a generalization of the Berry connection, defined as AB  i   R  . It is 

easy to see that under a phase transformation    e i R,t   

both Berry electric

 and magnetic fields are unchanged. In the limit R  0 and H  H R only, we recover the usual Feynman-Hellman theorem. It is also interesting to notice that eq. (2.19) is the classical force in parameter space acting on a fictitious particle with  charge   and velocity dR / dt which moves under the influence of a Berry vector potential, and a gauge potential which will be discussed below.



2.5 The adiabatic limit Until now, the previous results are exact both for fast and slow changes of the Hamiltonian. In the adiabatic limit, Berry showed that the wavefunction may be written as:     e iaRt ,t  n Rt , t , (2.22)







with a Rt ,t  the phase containing both the dynamical term and the Berry phase term.  n Rt , t are the instantaneous eigenkets namely, they are defined by:





   H n R t , t  E R t , t n R t , t . (2.23)











45

2. Geometry of Quantum states: The adiabatic approximation

In this limit, eq. (2.19) has the same form[7]: 

 R H



      R B B   R E   R, t , (2.24)

 

but the Berry quantities are changed, determined only by the eigenvectors:  AB  i n  R n

, BB   R  AB (2.25)

    n n  R, t  i   R n   R n  (2.26) t t  

 

The Berry electric field can be written using potentials in the form:

 

 R, t   RVB 

AB , (2.27) t

 n (2.28). It is interesting to t note that the third Maxwell‟s relation (the analogue of Faraday‟s law) holds between Berry electric and magnetic field:

with the gauge scalar potential defined as: VB  i n

 R     R    RVB  

BB (2.29) t

Eq. (2.29) describes a continuity equation for the monopole current, if we define   R .BB   m (2.30) with  m the magnetic monopole density. Note the interesting fact that if the term R    RVB  is non-zero, it can be interpreted as a magnetic monopole current density J m  R    RVB  .

46

2. Geometry of Quantum states: The adiabatic approximation

2.6 Applications 2.6.1 Charge transport and Polarization change in Bloch crystals As a first example, we consider the adiabatic charge transport[7,8] in a band insulator  when the Hamiltonian has only explicit time dependence, namely, when R  0 . Equation (2.24) then reads:     R H    R E   R, t (2.32)

 

For this purpose, we consider a two dimensional Bloch crystal under the influence of external perturbation V t  such that V t  T   V t  (cyclic evolution) and the    parameter under consideration is the crystal momentum k , R  k . Note that because the external variation occurs only in time, the translational invariance is conserved and the eigenvectors are described according to Bloch‟s theorem. In the absence of an external electric field the crystal momentum does not acquire any time dependence and therefore it is considered as a constant. For a crystal, the left hand side of (2.32) is directly related to the electric current density:  S     k H k , t    J n k , (2.33) e

 



where S is the area of the crystal. Using this, we may write (2.32) by using (2.33) as follows:   e  e   Jn k    k E   k , t (2.34) S S



 

If the crystal is one dimensional (e.g. a chain of atoms), eq. (2.34) describes the adiabatic charge (electron) transport along the crystal which is quantized, because   then  k , t is just the Berry curvature in the mixed parameter space. In this case, eq.

 

(2.34) reads: J n k   

e E e  k , t  , (2.35) L k L

and the total current may be written as an integral in the Brillouin zone where the first term vanishes due to periodicity: J tot 

e 2

 B.Z . dkk , t   dt  e dt dI

dC

, (2.36)

where C is the particle number, and L is the length of the crystal (in one dimension). Upon integration of (2.36) with respect to time we have: 47

2. Geometry of Quantum states: The adiabatic approximation 

1 C dt B.Z .k , t dkdt (2.37) 2 0  which is an integer. For the two dimensional case however, we may allow in addition to the external perturbation, an homogeneous electric field running through the crystal. This additional electric field moves the crystal momentum in the Brillouin Zone according to the relation:  e  k   E el (2.38) 

Having the crystal momentum as a slowly moving parameter, eq. (2.35) is further modified by the transverse term:    e2  e  e   Jn k   E el  B B   k E   k , t , (2.39) S S S



 

and the total current is (the second term vanishes):  e2 J tot   h2  

  2 d k E  B B.Z . el B 

e

2 2 

  2 d k  k , t (2.40) B.Z .

 

Dividing the results into x and y components we get: Jxtot  

Jytot 

e 2 E el y

B. Z . d h2  

e 2 E el x h2 



B .Z . d

2

  n n n n  with  x k , t  i  t k x  k x t

 

2

kBB 

kBB 

e

B. Z . d 2 2 

e

2 2 

B.Z . d

2

2

 k x k , t

 

 k y k , t ,

 

 n n   n n  k , t  i   and  y t k y  k y t 

 

  

Now we have the key to calculate the change in electric polarization induced by the variation of the external potentials, defined by the relation: T

Pi   dtJitot (2.41) 0

For example, for the x component we have: Px  

e 2 E el y h2 

T

 0

B.Z . dtd

2

kBB 

e

T

2 2 0 

B.Z . dtd

2

 k x k , t

  48

2. Geometry of Quantum states: The adiabatic approximation

2.6.2 Spin in time dependent magnetic field This is the well-known Zeeman effect[7], i.e. electron‟s spin is coupled to an external  time varying magnetic field B t  . This problem can be solved analytically for any slowly-varying magnetic field B  t  , that returns to itself by the end of variation process. The Hamiltonian which describes the system is:   H t     B  .B t  ,

with  B  e / 2mc the Bohr magneton. The energy spectrum is E  B B t  with ξ=-1 for spin up and ξ=+1 for spin down. Because time dependence comes into play only through the magnetic field, (and so the magnetic field is considered as the slowly varied parameter) it turns out that the Berry curvature BB has the following form:

  B BB   , (2.42) 2B 3 which is a magnetic field generated by a magnetic monopole at the origin (where the two energy levels meet). Eq. (2.24) gives then:     B H    B BB   B E , (2.43)

 with   B H    B  , we arrive at the result (see equation 24 in ref. [10]):    B mc  B B   B 3     B 3   (2. 44) 2 B B e B B B 





 B

The result gives the zero and first order correction to the mean spin (or dipole moment) value without the need of using perturbation methods. The first term is a Coriolis force-like solution[11] accompanying the Berry‟s phase in 3D space, while the second term is the mean value when the parameter is time independent. The usefulness of our method is therefore demonstrated with an elementary example of quantum mechanics.

2.7 Quantization of magnetic charge – Dirac monopole A magnetic monopole[12] is an isolated magnet with only one pole (north, without a south one, or vice versa). It is supposed to curry a magnetic charge, similar to electric monopole, which carries an electric charge. The quantum theory of magnetic charge started with a paper by the physicist Paul A.M. Dirac, in 1931. In his paper, Dirac showed that if any magnetic monopoles exist in nature, then all electric charge in the 49

2. Geometry of Quantum states: The adiabatic approximation

universe must be quantized. This is not a surprise, because electric charge is indeed quantized, but this does not provide any proof for the existence of magnetic monopoles. Second Maxwell‟s law is modified in regions of space containing magnetic monopoles as: .B  4m with m the magnetic charge density. For a single, isolated, monopole we have m  qm  r  , with qm the magnetic charge magnitude and   r  the Dirac delta function. The magnetic field generated from a magnetic monopole is described by the equation (in spherical coordinates): B

qm rˆ (2.45) r2

similar to electric field generated by an electric, isolated charge. Dirac[13] consider a vector potential of the form

A

qm 1  cos   ˆ , (2.46) r sin 

where θ is the polar angle and φ is the azimuthal angle, both expressed in spherical coordinates. But this vector potential becomes pathological when reaching the south pole, θ=π, and the vector potential must therefore be described by two different functions, each of them defined in regions of θ which won‟t result in pathologies for A:

A1  A2 

qm 1  cos    ˆ ,   2 r sin 

qm  1  cos    ˆ ,   (2.47) 2 r sin 

At the overlap region, where    / 2 , the two vector potentials describe the same magnetic field, therefore they must be connected through a gauge transformation:

A1  A2   , with a gauge function   2qm Now, for a quantum particle with charge q that lives in the sphere, its wavefunctions must differ by a phase factor:

1   2e

2iqqm c

(2.48)

Both wavefunctions must be single valued with respect to azimuthal angle:

1   2   1   and  2   2    2   : cn 2qqm 2 , (2.49) with n being an integer.  2 n and qm  c 2q 50

2. Geometry of Quantum states: The adiabatic approximation

Note that these are just the values needed so that the magnetic flux that penetrates the sphere is quantized [12].

2.7.1 Our proposal for the quantization of magnetic charge in crystal momentum space Alternatively, let‟s start with Maxwell‟s second law for the Berry curvature, which, as showed before, is an effective magnetic field generated in the parameter space states that  k .BB  4 m ,

with m the magnetic charge density in parameter space. Using divergence theorem,

 d k .B 3

k

B



 B .dS  4  d k 3

B

m

 4 qm (2.50)

As stated before, the flux of Berry curvature across a closed surface in parameter space is always quantized (the Chern number):

 B .dS  2 n  q B

m



n (2.51) 2

On the other hand, the electric charge in parameter space equals to q  , and therefore using (2.50) and (2.51) we arrive at the result:

qm 

n (2.52) 2q

which indeed gives the usual quantization of magnetic charge in the parameter space, using the fictitious charge q  .

2.8 Non-Hermitian corrections to Ehrenfest and Hellmann-Feynman theorems We find that two of the most important theorems of Quantum Mechanics, the Ehrenfest theorem and the Hellmann-Feynman theorem, lack – in their standard form – important information: there are cases where non-Hermitian boundary contributions emerge. These contributions actually appear naturally, in order for the above theorems to be valid and applicable, and this occurs for physical quantities that are not represented by well-defined self-adjoint operators (such as the position operator in a periodic potential, or in general Aharonov-Bohm configurations, either in real or in any parameter space, in the sense of Berry‟s adiabatic and cyclic procedures). In this 51

2. Geometry of Quantum states: The adiabatic approximation

short note, we report modifications of these two theorems when such nonHermiticities appear, and we demonstrate how these modifications restore earlier but also recent Quantum Mechanical paradoxes (these are violations of the so-called Hypervirial theorem in Quantum Chemistry, and also a textbook error in the formulation of band theory in Solid State Physics that seems to have gone unnoticed). This restoration of paradoxes (which is actually the re-establishment of applicability of the Ehrenfest theorem even in multiply-connected spaces) always proceeds through the appearance of certain generalized currents, in a theoretical picture with interesting structure (where a generalized violated continuity equation shows up naturally).

2.8.1 Ehrenfest Theorem The total time derivative of the mean value of any operator that depends on position or momentum operator and has explicit time-dependence B  r, p, t  can be written as:

d d  B  (2.53) B  r, p, t    B  B      B dt dt t t t This leads to the well-known Ehrenfest theorem of quantum mechanics (usually called like this in its application for B = p , and giving the well-known velocity operator v 

i

H , r 

in its application for B = r ). Making use of the t-dependent

Schrodinger equation we may write  i  i   H  and   H† t t

for its complex conjugate. Substituting these into (2.53) we have

d B i i B  r, p, t      H  B    BH  dt t B i i i     H  B    HB     H , B   t

(2.54)

Now, if H were Hermitian (with respect to  and B ), we clearly see that the result d B i B  r, p, t        H , B   . In the more dt t general case, however, we can rewrite (2.54) as:

would be the familiar

d B i i   2  B     2 B   , (2.55) B  r, p, t       H , B    dt t 2m 

52

2. Geometry of Quantum states: The adiabatic approximation

e with  the kinematic momentum:   p  A  r  , with A  r  the vector potential, c minimally substituted in H , and:

i e 2e e2 2  p  . A  A. p  2 A c c c 2

2

Substituting into (2.55) we get: d B i i B  r, p, t       H , B   d 3r  2  * B   * 2 B     dt t 2m (2.56) e  d 3r  . A  * B  A.  * B    mc 

 

 





For a specific component of the vector operator Bl  r, p, t  the above equation reads: Bl d i Bl  r, p, t      dt t  

 H , Bl    dS .  * Bl   *  Bl    A * Bl   mc  2m  i

Bl i   t

e

 H , Bl    dS .J gen

(2.57), where the two volume integrals in (2.56) can be written as closed surface integrals (divergence theorem) on the boundary of a generalized current defined as: J gen 

i e  * Bl   *  Bl     A * Bl  (2.58)   2m mc

This current has a form very similar to the familiar quantum probability current, J prob 

i  * *   e A  2 , (2.59)      mc 2m 

which would correspond to the special case of Bl  1 (identity operator), and obeys the standard continuity equation: .J prob  p / t  0 with p the probability density,

p   *  . In the more general case, for any Bl, it can be proved that the above generalized current J gen obeys a generalized continuity equation, that is violated by a nonvanising inhomogeneous term, namely

.J gen 

pgen

 B i   *  l  H , Bl   , (2.60) t  t 

with pgen  * Bl  a generalized density. To prove this, we consider the integral form of eq. (2.60) which is eq. (2.57), and upon integration in a specific volume of all terms we get: 53

2. Geometry of Quantum states: The adiabatic approximation

Bl d i Bl  r, p, t       H , Bl    dS . J gen dt t (2.61)   i 3 * 3 * Bl 3 * 3   d r   Bl     d r    d r  H , Bl     d r. J gen t t

If this equality is true for any volume then we recover the differential form of generalized continuity equation, that is exactly eq. (2.60). Note here that, if Bl / t  0 and if p gen is time independent, i.e. Ψ is a single H-eigenstate, we have: . J gen 

i

 *  H , Bl   

 dS . J

gen



i

 H , Bl 

, (2.62)

and d B / dt  0 . This means that the time derivative of mean value of any time independent operator calculated in a single stationary state is always zero. A bit more generally, if Bl is an invariant operator,

Bl i    H , Bl  then t

  0 . This is the Liouville equation. It describes the flow of Bl  r, p, t  t through the surface (boundary of volume V where the system is restricted in). If  B i  Bl  r, p, t  is a conserved quantity, then the source term    *  l   H , Bl   is  t  . J gen 

zero, meaning that Bl i B i   H , Bl   0  l    H , Bl  , (2.63) t t

i.e Bl  r, p, t  must be an invariant operator. On the other hand, if the source term is nonzero,   0 , then the above continuity equation describes the rate of flow   0 of the quantity Bl  r, p, t  in the interior of the volume V.

2.8.2 Hellmann-Feynman theorem Eq. (2.57) can be further modified if operator Bl acts in a parameter space as a i.e. differential operator. If we assign Bl with the operator  R that acts in parameter space {R1,R2,…}, we get the Hellmann-Feynman theorem in a boundary-related generalized form: d i  R    R H   dS . J gen , (2.64) dt

54

2. Geometry of Quantum states: The adiabatic approximation



iEt

because  H , R   R H . And if we consider only one eigenstate,   e n , we d i have  R    R E and the Hellmann-Feynman theorem (eq. 2.64) becomes: dt R E  R H  i

with J gen 



 dS.J

gen

, (2.65)



i  e  * R    *  R    A * R  .  mc 2m 

A rigorous Mathematical Physics presentation (through discussion of domains of definitions of operators etc.) of this type of extra boundary contributions that can show up in the Hellmann-Feynman theorem has been given in ref. [14].

2.9 Examples: (a) Free particle Although it is rarely mentioned, one of the main consequences of the non-Hermitian boundary terms appears already in the simplest problem of quantum mechanics: the free particle (in a volume V with the standard periodic boundary conditions) whose Hamiltonian is: p2 H 2m

and eigenfuctions:   r   eik .r / V (box normalization). If we choose operator B  r, p, t  to be the position operator, B  r, p, t   r , which is clearly time

independent, eq. (2.57) gives: d i x  dt

with

 H , x

 i

H , x



i 2m

 dS .  x     x  , *

*

px / m  kx / m , and the second term must be evaluated on the

surfaces of the cube:  * x   *  x   1  V

2ikx  iˆ 1  V V

 dS .  2ik x  1 iˆ  2ik

which all together result in:

x

y

 dS . 2ikx  iˆ 

(2.66)

xjˆ  2ik z xkˆ   2ik x

k k d x  x  x  0 . (2.67) dt m m

55

2. Geometry of Quantum states: The adiabatic approximation

This is true for any of the components of r (and of course only for a single eigenstate). Note that if we had neglected the surface term in eq. (2.57), then

d x / dt would not be zero, violating the condition that all mean values of time independent operators calculated in a single state must also be time independent! (a paradox earlier noted in [15] and which is also essentially what has been noticed by Quantum Chemists (as a violation of the so-called Hypervirial theorem) [16]). It is also good to notice that, if we choose B  r, p, t   p the result is once again

d p / dt  0 but without the appearance of a non-Hermitian boundary term (here the reason being that the momentum operator is a good self-adjoint operator for these boundary conditions).

(b) General example for any gauge potential – Aharonov-Bohm configurations The fact that any mean value of a time-independent operator must not depend on time, can be generally proved for any real gauge (and vector) potential. Here we first  consider for simplicity the case A  0 , and examine the position operator in 1D (our method is valid for any time-independent operator, either differential or of other form) d i x  dt

L

i  d * d   *  H , x    2m  dx dx  0

p i  d * i  d * d  2 * d  x     x      x     x *    m 2m  dx dx  0 m 2m  dx dx  0 L

p

d

L

Now, p  i

 dx * dx

L

(2.68)

and by using integration by parts we conclude to:

0

p 

i 2

L L  d d  * 2 L     dx  *  dx   (2.69)   0 dx 0 dx  0 

Use again integration by parts to get the second derivative of Ψ with respect to x: p 

i 2

L L  d d *  d  d  d  d  *  2 L      x  *    dxx  *  dxx           0 dx dx  0 dx  dx  0 dx  dx    

i  2

L L   d 2 *  d  d  *  d 2  2 L     dxx    *       x   *    0 dx dx   0 0 dx 2 dx 2       (2.70)

Combining (2.68) and (2.70) we find that: 56

2. Geometry of Quantum states: The adiabatic approximation L  d 2 * d i d 2  x  dxx    *   (2.71) dt 2m 0 dx 2 dx 2  

Making use of the Schrodinger equation (for a real scalar potential) we can eliminate  in (2.71): d 2 2m   2  E  V ( x)  , 2 dx

to get: L

d i x    dxx    E  V ( x)  *  *  E  V ( x)    0 (2.72) dt 0 It should be noted that the above shows the necessity of including the non-Hermitian boundary terms in the case of a ring threaded by a static magnetic flux (i.e. an Aharonov-Bohm configuration [17]), so that the theorem is valid. This is in contrast to the standard literature on Aharonov-Bohm rings, where it has been stated (i.e. see [18] for a driven ring), that the Ehrenfest theorem is violated in multiply-connected spaces. The restoration of the above paradox can therefore also be seen as a re-establishment of the “practical applicability” of the Ehrenfest theorem in multiply-connected space. By following the above, the reader can actually find the exact form of the nonHermitian boundary term (or more generally of the above discussed generalized current) that heals the Ehrenfest theorem in the case of an Aharonov-Bohm ring (or, further, whenever the magnetic flux is even a time-dependent quantity).

(c) A note on Hellmann-Feynman theorem in the Bloch problem Up to now, by dropping the above mentioned boundary (surface) terms, HellmannFeynman theorem (for differentiations with respect to a static parameter k) had to be written in the form: dE dH  dk dk

(2.73)

Notice however that, by taking as example a Bloch electron, whose Hamiltonian (in 1D) is: H  p 2 / 2m  V ( x) , with eigenfunctions   eikx uk  x  , k is the crystal momentum and uk is the periodic cell function, (2.73) results in dE / dk  0 , namely, it predicts that the energy bands must not depend on crystal momentum k. This contradicts the fact that if one minimally substitutes crystal momentum k in the Hamiltonian, 57

2. Geometry of Quantum states: The adiabatic approximation

H

 p  k

2

2m

eq. (2.73) gives dE / dk 

 V ( x) , with eigenfunctions   uk  x  ,

 p  k/ m

 0 , the slope of the energy bands in a

crystal. What is really happening here is that dE / dk is always non zero, and can be analytically obtained in full generality without the need of minimal substitution, by directly using a modified Hellmann-Feynman theorem containing our non-Hermitian boundary terms (eq. 2.65): 2  * d  d   d  d *  dE dH    (2.74)   dk dk 2m  dx  dk  dk dx 

with H 

*

and

p2  V ( x) and   eikx uk  x  we have: 2m

2 d  d   d  d * du du du * 2 * * d u     u i  2 xk  ix u u  uu *  2 iku *  u        dx  dk  dk dx dk dxdk dk dx

dH  0 as it should be! Substituting then in eq. (2.74) we obtain: dk 2 L dE  2 u 2 xk  ix  u*u  uu *  (2.75)  0 dk 2m 

Now, making use of eq. (2.70) and Schrodinger‟s equation:

u  

2 2 2 2 2m  k  2m  k    E  V x  u  i 2 ku u *   E  V x  and       u * i 2ku * , 2  2  2 m 2 m    

we arrive at the correct result: L

2 p  dE  2 kx 2 i 2 k (2.76)  u  x  u*u  uu *    dk  m 2m m m 0

Here, an error in the Solid State Physics literature should be emphatically mentioned, that is naturally corrected by the above. I.e. on p.51 of textbook [19] it is claimed that

u= 0, the expectation value being with respect to the cell-periodic wavefunctions uk of the Bloch form ςk=eikx uk, this giving

ς= ħ k, which is not the same as the correct

ψ = m/ħ ∇kE that is also written later on the same page (essentially the usual group velocity that comes out of the standard Hellmann-Feynman theorem); one should note (and it is easy to verify) that the latter correct result comes out if in the first erroneous vanishing result one includes (adds) the non-Hermitian boundary contribution (when determining

u with an integral), something that we leave to the reader. 58

2. Geometry of Quantum states: The adiabatic approximation

(d) Linear combination of states d x / dt may not be zero only in the case of a linear combination of states as it can be easily proved using eq. (2.71): L  d 2 * d i d 2  x  dxx    *   (2.77) dt 2m 0 dx 2 dx 2  

If Ψ is a single eigenstate, then eq. (2.77) is zero, as shown before. But if now Ψ is a linear combination of states, i.e.

   Cn e

 iEnt

 n  x  , then eq. (2.77) becomes:

n

d i x  dt

 Cl * Cne

i

 El  En t

n ,l

L

 En  El   dxx l *(x) n ( x) , (2.78) 0

which is the correct result we obtain using elementary quantum mechanical methods. For example, consider the simple case of a particle in a quantum well (Q.W), with wavefunction

 n ( x) 

2 n x sin with L the length of Q.W. and n=1,2,3.., Eq. (2.77) then gives: L L d 4i  2 x  dt mL

2

 C Cne 2mL * l

n ,l

i

2

l

2

  nl 

 n2 t

 l 2  n 2  , (2.79)

with the constraint: l  n  odd . In this case, the extra boundary contribution is still important, and some nice closed patterns can be written, but it is a matter that we currently leave to the reader as well.

2.10 Contributions of boundary terms to Ehrenfest and HellmannFeynman theorems when the parameter has explicit time dependence It is interesting to recall how the Hellmann-Feynman theorem is further modified when parameters depend on time [6]. Additionally to the implicit time-dependence through the parameters, we also let H depend explicitly on time t, i.e.





H  H R  t  , t . Starting with eq. (2.64): d i  R    R H   dS . J gen dt

59

2. Geometry of Quantum states: The adiabatic approximation

d     R . R , the material derivative, which takes into account changes dt t with respect to time-dependent parameters, we have:

with now

    d d   R  i AB  i   R . R  AB  i AB  i R . R AB , (2.80) dt dt t  t 

with AB  i R . Using then nabla product rules, we get:     R . R AB   R  R . AB   R BB , (2.81)  

with BB   R  AB and 



R . AB  i  R . R   i 

with VB  i 

d  d E   i    VB   VB (2.82) dt t dt

  and E   H  . Combine (2.80), (2.81) and (2.82) to arrive t

at the result: 

 R H   R E    R BB  i

 

with  R, t   RVB 

 dS . J

gen

, (2.83)

AB the “Berry electric field” and BB   R  AB the Berry t

curvature, defined through potentials: AB is the Berry vector potential (the wellknown Berry connection) and VB is a “Berry scalar potential”. It is interesting that equation (2.83) can be interpreted as describing the Lorentz force (in parameterspace) acting on a particle of charge - which moves in the presence of scalar potentials E and VB , and a vector potential AB (although the contribution of nonHermitian boundary terms is generally still present and of separate importance). All quantities are defined through the full time dependent wavefunction, while, in the 

adiabatic limit R  0 , they reduce to AB  i   R   i n  R n and VB  0 (the standard quantities in Berry‟s seminal paper [2]).

60

2. Geometry of Quantum states: The adiabatic approximation

2.11 Invariant Operators derived from Hellmann-Feynman theorem when the operator has time dependence of the form B  B  t ,  / t  . d d B  r, p, t    B  , we observe that when dt dt the operator under consideration has time dependence, extra care must be taken when applying the differentiations:

Starting from Ehrenfest theorem:

d   d d d d B    B  , with   R . R the B  r, p, t    B   t dt dt dt dt t





material derivative, to take into account cases there is time dependence in some parameters. In this case, it is no longer valid the fact that

d  d dB B   B ; dt dt dt





and the commutator must be used:

d  d d  B    , B   B dt dt  dt 





Then, the Ehrenfest theorem becomes (in integral form): d i d  Bl  r, p, t     , Bl    dt  dt 

 H , Bl 

  dS . J gen 

i

d    H  i , Bl     dS . J gen dt  

(2.84) or, in differential form: . J gen 

d i *  d    H  i , Bl   with   * Bl  (2.85) dt dt  

It is interesting to note that the right term (the source term) of (2.85) is the generalization of Lewis Riesenfeld invariant operators theory[20]. If this term is zero, namely, the quantity  d 3r* Bl  is conserved, d d d d      (2.86)  H  i dt , Bl    0   H  i dt  Bl   Bl  H  i dt     H  i dt  Bl   0 0

then Bl  must be a solution of Schrödinger equation. This structure of generalized continuity equation with a source term that depends on the input operator Bl and its properties (which in turn reflect the non-Hermitian contributions) is an interesting mathematical structure that certainly deserves further study.

61

2. Geometry of Quantum states: The adiabatic approximation

Chapter References [1] T. Kato, “On the adiabatic theorem of quantum mechanics,” J. Phys. Soc. Jpn. 5 (1950) 435–439. [2] M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A 392 (1984) 45–57. [3] F. Wilczek and A. Zee, “Appearance of gauge structure in simple dynamical systems,” Phys. Rev. Lett. 52 (1984) 2111–2114. [4] Manfredo do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, 1976. [5] Wu Hung-Hsi The Historical Development of the Gauss-Bonnet Theorem, Science in China Series A: Mathematics, Vol. 51, 4, 777-784. [6] Kyriakos Kyriakou, Konstantinos Moulopoulos, Dynamical extension of Hellmann-Feynman theorem and application to nonadiabatic quantum processes in Topological and Correlated Matter, arxiv: 1506.08812-an updated version to be submitted. [7] R. Resta, Ferroelectrics 136, 51 (1992) [8] R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993) [9] D. Xiao, M.-C. Chang, and Q. Niu. “Berry phase effects on electronic properties.” Rev. Mod. Phys. 82, 1959–2007 (2010). [10] K.Yu. Bliokh, On spin evolution in a time-dependent magnetic field: postadiabatic corrections and geometric phases, arxiv.org/pdf/quant-ph/0702119 [11] M.V. Berry, Proc. R. Soc. A 392, 45 (1984); A. Shapere and F. Wilczek (eds), Geometric Phases in Physics (Singapore: World Scientific, 1989). [12] T. Wu and C. N. Yang, Phys. Rev. D 12, 3845 (1975) [13] P.A.M. Dirac, Quantised Singularities in the Electromagnetic Field, Proc. Roy. Soc. A 133, 60 (1931) [14] J. G. Esteve et al, Physics Letters A 374(6) Dec. 2009 (arXiv:0912.4153) [15] R. N. Hill, “A Paradox Involving the Quantum Mechanical Ehrenfest Theorem”, Am. J. Phys. 41, 736 (1973) [16] F. M. Fern ndez and E. A. Castro, “Lecture Notes in Chemistry: Hypervirial Theorems”, Springer-Verlag (1987) [17] Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in 62

2. Geometry of Quantum states: The adiabatic approximation

quantum theory”. Physical Review 115, 485–491 (1959) [18] P.-G. Luan and C.-S. Tang, “Charged particle motion in a time-dependent fluxdriven ring: an exactly solvable model”, J. Phys.: Condens. Matter 19, 176224 (2007) [19] D. W. Snoke, Solid State Physics-Essential Concepts, Addison-Wesley (2009) [20] Lewis H R Jr and Riesenfeld W B 1969 J. Math. Phys. 10 1458 [21] H. K. Moffatt, “The degree of knottedness of tangled vortex lines”, Journ. Fluid Mech. 35, 117 (1969)

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3. Electron systems in strong magnetic fields

Chapter 3 Electron systems in strong magnetic fields: Thermodynamic and hints of transport properties In this Chapter we report a systematic study of the energetics of electrons in an interface in a magnetic field with exact analytical calculations based on a Landau Level (LL) picture, by giving priority to the role played by the finite thickness of the Quantum Well (QW). The approach is physically transparent and subtly different in its line of reasoning from standard methods avoiding any semi-classical approximation. We find “internal” phase transitions (at partial LL filling) for magnetisation and susceptibility that are not captured by other approaches and that give rise to nontrivial violations of the standard de Haas-van Alphen periods, in a manner that reproduces the exact quantal astrophysical behaviours in the limit of full three-dimensional (3D) space. Upon inclusion of Zeeman splitting, additional features are also found, such as global energy minima originating from the interplay of QW, Zeeman and LL Physics, while a corresponding calculation in a Composite Fermion picture with Λ-Levels, leads to new predictions on magnetic properties of an interacting electron liquid. By pursuing the same line of reasoning for a topologically nontrivial system with a relativistic spectrum, we find evidence that similar effects might be operative in the dimensionality crossover of 3D strong topological insulators to 2D topological insulator quantum wells. In the Last Section, we also present a systematic study on the energetics of an Aharonov-Bohm 1D and 2D electron systems that move on a ring or on a nanocylider‟s surface, with microscopic radius R and the longitudinal direction being extended. Our results confirm the standard periodicities of energy and electric current with respect to flux quantum, and they are applicable to experimental setups for the design of quantum nanodevices.

64

3. Electron systems in strong magnetic fields

3.1 Introduction Recently, there has been a surge of interest in the new area of topological insulators [1,2] (and also Chapter 1 for a theoretical background), namely electronic systems characterised by a bulk insulating gap but also possessing topologically-protected gapless edge (or surface) states, i.e., dissipationless conducting surface modes, immune to nonmagnetic impurity scattering and geometrical defects. The simplest example of such a phase with broken time-reversal symmetry, can be found in a twodimensional (2D) electron gas under a strong perpendicular magnetic field in the Quantum Hall regime. Through a very general bulk-edge correspondence [3], it has been well established that the number of dissipationless edge states is equal to the integer that arises from the so-called TKNN invariant [4], or the 1st Chern number in a fibre bundle language [5], of the occupied energy bands. This is a bulk property related to the “vorticity” of the wavefunctions in the magnetic Brillouin zone. In a jellium model picture, the 1st Chern number or the number of edge states turns out to be equal to the number of completely filled Landau Levels (LLs) in the Integral Quantum Hall Effect (IQHE) regime. If one wanted to include the 3rd dimension, i.e., to take into account the thickness of the macroscopic quasi-2D sample (interface or film) with open (rigid) boundary conditions, then a treatment of the above mathematical (topological) properties would be a formidable task. In fact, it would spoil the beauty of the standard topological arguments normally applied to the 2D Brillouin zone. Here, we point to an alternative general procedure that is rigorous and based on physical rather than purely mathematical arguments and that seems to have not been discussed in the past. It is based on energy interplays in a one-electron (or oneComposite Fermion) picture, leading to the possibility and in fact, showing the existence of abrupt changes in the occupancy of transverse, i.e., thickness-related modes in the ground state. These occur at partial LL filling and are accompanied by associated changes in thermodynamic and also possibly in transport properties; changes that, as it turns out, happen to occur in an interesting, although in a certain sense, non-integrable fashion as the thickness is varied. The method we are presenting is a canonical ensemble approach (fixed number of particles), which is subtly different from standard canonical or grandcanonical approaches that at some point invoke semi-classical approximations and that usually have mathematical difficulty in dealing exactly with the zero-temperature limit, i.e., it does not anticipate or assume a Fermi sphere in the 3D zero-field limit as part of the quasi-2D calculation but naturally derives it in a direct and rigorous manner. The method is exact, involving no approximations whatsoever and it describes the zerotemperature case, although this is immediately generalisable if Fermi factors are included. What is most important is that it is physically transparent at every step of the procedure and hence, rather easy to use for other systems that are more involved or exotic. The method works directly in k-space by taking careful advantage of anisotropies in different directions by not using at all the density of states (DOS). The 65

3. Electron systems in strong magnetic fields

DOS is the key quantity in all other approaches through which, by reducing everything to the energy variable, basically masks the Physics (i.e., the intermediate physical steps) that take place in k-space and that depend on the geometry of each system. It is also not necessary to go through the rather difficult step of first finding the DOS by determining the exact energy spectrum. This is advantageous, especially if we want to have as much analytical control on our solution as possible. Moreover, a physical criterion (of “equilibrium”) applied to the occupation procedure of a strongly anisotropic system is shown from the results to be superior to the usual semi-classical treatments that lead to the standard “magnetic oscillations” [6]. Unlike those methods, the present approach leads to exact quantal violations of the de Haas-van Alphen (dHvA) periodicities in the quasi-2D interface or film, which become smooth quantal deviations (from the dHvA periodicity) in the 3D limit. The method can actually be useful in a wide range of applications because the precise role of thickness in various quasi-2D systems seems to be currently attracting considerable attention. By way of an example, mention should be made of bulk Quantum Hall Effect (QHE) measurements in a 3D topological insulator [7] where Shubnikov–de Haas oscillations in highly doped Bi2Se3 give evidence for layered transport of bulk carriers, in which the sample thickness plays an essential role on the quantisation of magnetotransport but also of the more exciting thickness-related issue of 2D to 3D dimensionality crossover in topological insulators (an issue that is actually briefly touched upon in this paper, as will be seen shortly). However, in the bulk of this work, we take a step back and present the method in the simplest possible but still nontrivial setting. We solve exactly thickness-related problems involving an electron gas system in the jellium model, both with and without a magnetic field in various dimensionalities, demonstrating that even in these simplest possible cases, the role of thickness is nontrivial and noteworthy. [The jellium model gives the luxury of dealing with simple LLs, where their number is automatically identified with the topological (Chern) number or the number of edge states (whenever the LLs are completely filled). This gives one the opportunity to identify possible abrupt changes in the Chern number (when LLs are abruptly depopulated – as will actually occur many times in this work) with possible interesting consequences on transport properties. However, these deserve a separate article, as this one focuses on thermodynamic consequences, i.e., violations of dHvA periods.] Furthermore, because the largest part of our analysis utilises a jellium model of electrons in extended states, mention should also be made of a 2D semimetal that has recently been observed in wide HgTe quantum wells (QWs) with a broad range of interesting properties [8] and with their thickness still being an important factor not yet seriously studied. Moreover, very recent works on the 5/2-Fractional Quantum Hall Effect (FQHE) [9–11] examine the stability of the effect in wide QWs against the variation of their thickness and find anomalous features. It is with this in mind that we have applied the same method by carrying out a thickness-adapted Composite Fermion calculation, as will be seen shortly. Mention could also be made of recently 66

3. Electron systems in strong magnetic fields

studied highly quantum-confined nanoscale membranes, the thickness of which is crucial for their (mostly optical) properties [12], as well as of the newly discovered almost free electron gases in oxide heterointerfaces [13]. Finally, returning to the one-body Physics of the recently discovered topological insulators, our approach and results might actually cast doubt on the completeness of recent findings on a simple oscillatory crossover from a 2D to a 3D topological insulator [14], where transitions between different z-modes (with z being the direction of the external magnetic field) may not have been treated entirely properly. This will be apparent from the present work – the point being that, in that work, energy comparisons are made under the assumption of a given (fixed) transverse mode, not taking into account the energetically favourable possibility of abrupt changes of such modes that might occur in nontrivial ways as the thickness is varied. As we will see in a preliminary study towards the end of this paper, although such transitions might occur at points located a little further than the Γ-point in the Brillouin zone, their distance from the Γ-point in k-space is actually quite small, such that these effects might be operative. We will actually see that they might occur inside the k-space region where the low-energy approximation that is widely used (namely, a modified Dirac equation) is valid and at points that are well within an estimated Fermi wavevector kf resulting from surface carriers. In order to present our analysis in the jellium model, first it is useful to remind the reader of systems that are a little more traditional, in the sense of being well-studied, than the above. For example, the standard sawtooth behaviour of the low-temperature magnetization of an electron gas in 2D interfaces and in the presence of an external perpendicular magnetic field is well-known both from experimental measurements [15], as well as from analytical calculations of the total energy of a noninteracting electron system with the use of a picture of LLs in a canonical ensemble approach (reviewed in Section 3.2). This sawtooth behaviour occurs as a function of the magnetic field. As a function of the inverse field, the “saw” has periodic steps, signifying the appearance of (or actually defining) the standard dHvA effect. In this article, we go further than these calculations by taking the issue of nonzero thickness of the interface seriously and by making a systematic study of its role on the ground state energetics of the interface, also by commenting on transport properties. We present extensions of the above type of analytical calculations to a quasi-2D interface with a finite-thickness QW in the z-direction, parallel to the magnetic field, by using rigid boundary conditions at the two edges of the QW, i.e., with an infinite potential barrier to represent the vacuum – similar to the “open boundary conditions” used in the area of 3D topological insulators. We also present independent analytical calculations, which are extensions of those that have already been carried out earlier in systems of astrophysical interest, for a fully 3D quantum system of noninteracting electrons in infinite space and in an external magnetic field with periodic boundary conditions parallel to the field, all at zero temperature (T = 0). Both systems, the quasi-2D interface and the full 3D space, seem to lead to previously unnoticed 67

3. Electron systems in strong magnetic fields

features in each system's magnetic response properties. For the interface, the crucial point is the single-particle energy competition between the LLs and the QW-levels for the different types of occupation-scenarios that are possible and allowed by the Pauli Exclusion Principle, when one attempts to determine the lowest total energy of the many-electron system. The basic physical reason is that each one-particle state is now characterised by three quantum numbers. There are then cases when the system energetically prefers to change (increase) a z-mode and then it can, or in fact it must, go back to lower quantum numbers of the 2D motion (in our case LLs) without violating Pauli‟s principle and in so doing, it can acquire a lower (in fact the lowest possible) total energy. It is shown that the manner in which occupancies (and transitions) occur, according to the above criteria, is an interesting and nontrivial exercise with the total energy probably not reducible to closed analytical forms immediately when an arbitrary field and an arbitrary thickness are given. One actually has to run the occupation scenarios starting from special values of parameters (for which the problem is easy) and then vary these parameters in some well-defined manner until they assume their values under consideration. When this exercise is carefully and properly solved, it defines a sequence of critical fields (or correspondingly of QW thicknesses) where “internal transitions” occur, in the sense that the highest LLs are only partially filled, which in turn leads to a number of new singular features in global magnetisation and in magnetic susceptibility. As a result, nontrivial quantal corrections to, or more appropriately, violations of the standard dHvA periodicities are found. In the independent calculation in full 3D infinite space, we determine the exact quantal behaviour of magnetisation, which in strong magnetic fields is found to deviate considerably from the standard semi-classical dHvA period but is also found to rapidly converge to this semi-classical periodicity as the magnetic field is reduced. The complete solution of this latter problem, derived here in closed form, also demonstrates some interesting analytical patterns in terms of the Hurwitz zeta functions that seem to have not been properly identified in earlier works. The mathematical problem of how to go analytically from the quasi-2D results to the results of the full 3D system (in the limit of infinite thickness) is also tackled; thus, providing a test and a proof of correctness and consistency of all the analytical expressions found here to describe the quasi-2D interface problem. Upon inclusion of Zeeman splitting, additional features are also highlighted, such as certain minima in total energy that originate from the interplay of QW, Zeeman and LL Physics in the full 3D problem, which might possibly be useful for the design of stable 3D quantum devices, i.e., in cases where the magnetic field can be self-consistently considered as self-generated. Furthermore, a corresponding calculation, now with the so-called ΛLevels in place of LLs in a Composite Fermion picture, in the approximation of noninteracting Composite Fermions, demonstrates the utility of our method, because it leads to new predictions on magnetic response properties of a fully-interacting electron liquid, possessing a certain form of universality, in which the finite thickness 68

3. Electron systems in strong magnetic fields

of the interface plays a major role, albeit different from earlier works such as [11]. These predictions should be compared with the much earlier reported mere monotonic reduction of FQHE gaps with thickness (see [16] for conventional FQHE systems – while for recent topologically nontrivial systems see [17]). In our results, they exhibit a richer and more delicate structure that possibly could be detectable with present day technology. In the bulk of this Chapter, particles are assumed nonrelativistic with a parabolic spectrum. A similar procedure for a model system with the relativistic energy spectrum of Graphene in the plane could be easily followed, although this is something that is not pursued here. Moreover, the method of energy-interplays presented in this work is immediately extendable to include Rashba or other types of spin-orbit coupling [18,19], although we will not consider this either in the present Chapter. However, towards the end of the Chapter, we do provide hints of relevance or of the applicability of the present method to analogous systems, namely systems with topologically nontrivial k-space behaviours, such as the dimensionality crossover from a 3D to a 2D topological insulator, i.e., systems with strong spin-orbit coupling and with low-energy properties described by a Dirac-type of equation.

3.2 Nonrelativistic electron gas in 2D in a perpendicular magnetic field As a precursor to the main results of this Chapter, we begin with the well-known problem of a system of many (N) noninteracting electrons, each with charge -e, effective mass m and spin s that are free to move in a 2D plane in the presence of an external homogeneous magnetic field B perpendicular to the plane, at temperature T = 0. For simplicity, let us first ignore the Zeeman splitting, i.e., we take the gyromagnetic ratio g* = 0 – however, note that we consider particles that do have spin (i.e., s =1/2); thus, providing a slightly more complete treatment than the standard (academic) one with spinless fermions. As is well known, this simple jellium model accounts for both the thermodynamic and transport properties of electrons, as these are observed in experiments on QHE systems in properties, such as magnetisation or Hall magnetoresistivities. However, we should state at the outset, that although these types of systems (interfaces or films) are not purely 2D, we can always reduce their thickness to achieve an effectively two-dimensional system (see Section 3.3 for the corresponding “critical thickness”, which depends on the areal density of electrons, as this is rigorously determined (at T = 0) by our analytical calculations). It is well known that in this 2D problem, the orbital motion of noninteracting electrons, which satisfy the nonrelativistic Schrodinger equation, is described by a Landau Level (LL) picture for the single-particle energy spectrum, namely  

1

 n  c  n   , (3.1) 2 

69

3. Electron systems in strong magnetic fields

where  c  eB / mc is the cyclotron frequency, e is the absolute value of charge of each electron and n (the LL index) is a non-negative integer that characterises all LLs. It is also well known that each LL has degeneracy 2Φ/  0 (accounting for the spin s = (1/2) of each electron – more generally, the prefactor being 2s + 1), where Φ is the total magnetic flux passing through the system and  0 = hc/e is the flux quantum. Each LL can then contain 2Φ/  0 electrons (due to Pauli‟s principle at T = 0) such that in the most general case, when there are π (a positive integer) LLs occupied by electrons (namely π = n+1, with n the LL index of the highest occupied level) the following inequality is satisfied 2  1

   N  2 , (3.2) 0 0

or equivalently, given that Φ = ΒS, with S being the total surface area of the sample:

1 1 n A 0  B  n A  0 , (3.3) 2 2  1 where N is the total number of particles and nA  N / S is their areal density. When the magnetic field varies in the above window, the electrons occupy π LLs, where the last occupied level of LL index π-1 is not necessarily completely filled up; a complete filling merely corresponds to equality in the right-hand side of (3.3). First, if π = 1, valid for B  1 2 n A 0 , all electrons are accommodated in the lowest LL and the total energy is simply:

EN

 c , 2

linear in B. For many LLs (π > 1), it is easy to sum over all occupied LLs to find the total energy of the system, namely  E2 0

 c n  1 2   N  2  1 

 2



n 0







  1   c   1  2 . (3.4) 0

After some algebra we can determine the total energy in units of 2D Fermi energy E f   2 k 2f 2m , k f  2n A , which has the following final form:  E  NE f 2   2    





 B   n A 0

2

  B   2   n A 0

     1  (3.5) 2  





One immediately notes that the energy varies quadratically with respect to B (for π > 1), such that one notes a linear behaviour of the magnetisation or a constant value of the magnetic susceptibility, quantities that are determined by derivatives of E with 70

3. Electron systems in strong magnetic fields

respect to B, as discussed further below. As already mentioned, for very strong B, i.e., for B  1 2 n A  0 (such that π = 1) E is given only by the last term in (3.5) and is linear in B, the magnetisation being therefore constant and having the value –NμB with μB the Bohr magneton, an “atomic value” of magnetic moment that is expected for almost nonoverlapping particles in the strong field limit (see more general discussion below). From application of the first law of thermodynamics at T = 0 one can determine the global magnetisation M and Susceptibility χ defined by: M 

E  and   B B

and these turn out to give:









  B M  N B  4  2    2   1  (3.6) 2 n A 0  





4 N B  2   . (3.7)  n A 0

It should be noted that χ is always non-negative for this 2D case; it is probably useful to state early on that when we later include a thickness for our interface, we will find cases (ranges of parameters) where χ will also assume negative values.

1.0

F

1.0 0.9

0.5

0.8 0.0

0.7 0.5

0.6 0.5

1.0

0.2

0.4

0.6

B

nA

0.8 0

1.0

0.2

0.4

0.6

B

nA

0.8

1.0

0

Fig. 3.1: Energy per electron (in units of 2D Fermi energy) and Magnetisation per electron (in units of μΒ)

71

3. Electron systems in strong magnetic fields

0

40

nA

50

30 20 10 0 0.2

0.4

0.6

B

nA

0.8

1.0

0

Fig. 3.2: Susceptibility per electron (in units of μΒ/nAΦ0) Therefore, in this manner, one obtains the well-known sharp sawtooth behaviour of magnetisation (in a system with a constant number of electrons) measured in low-T experiments [15]. If the above were plotted as a function of 1/B, then the above windows would be periodically repeated with a period 1 / B   2 / n A  0 , which is compatible with the dHvA period 2e / cA f (with A f  k 2f and k 2f  2n A ) (see, e.g., [6]). Also note that, for B  0 , the above energy correctly reproduces the 2D noninteracting result E / N  0.5 E f (i.e., in (3.5) take B  0 and    in such a way that the product Bπ is fixed).

3.2.1 Relation to transport properties – Hall conductivity It is useful to mention in passing a physical interpretation of the above thermodynamic results (at T = 0), which has a connection to transport properties and that in particular relates the above magnetisation discontinuities with diamagnetic currents. Indeed, the discontinuities in M can be associated with the abrupt change of chiral currents on the edges, despite the fact that the edges did not directly enter anywhere in the above formulation. This connection is through the simple relation of the magnetisation M with the diamagnetic electric currents I that flow around the edges (in opposite directions), namely M = IS/c (as one can immediately see by comparing   E / B with the Aharonov-Bohm formula I  cE /  , if I is assumed flowing along the edges such that the flux   BS can be viewed as an enclosed flux), in combination with the quantised values of the Hall conductance (    2  e 2 / h for spinfull electrons) and the fact that, during the transitions to a different LL, the current responds to a transverse potential that is equal to    c divided by e. Therefore, we expect to have (for the magnitudes of the various quantities involved) I  

 S with    c  M     2 N B , e ec

72

3. Electron systems in strong magnetic fields

where in the above, the values of B  n A  0 / 2  (where the transitions occur) have been used in the last step; therefore the above relation gives the correct magnitude of discontinuities 2 N B for the magnetisation that we see in Fig. 3.1, which occur whenever we have complete filling of π LLs. For completeness, we simply mention here that the above could have also been derived with the well-known Widom-Streda formula combined with a thermodynamic Maxwell relation, a more frequently followed procedure that gives M  N  / B (for the simultaneous discontinuities of M and μ), which turns out to be equivalent to the above relation but this diamagnetic current interpretation is preferable if we want to later generalise in a similar line of reasoning to the finite-thickness case (see corresponding discussion of transport in Section 3.4). The above also shows immediately how the discontinuities of M are directly related to the Hall conductance. One could determine ζH by measuring the simultaneous discontinuities of M and μ. This is a line that is actually going to be followed in the finite-thickness case of Section 3.4. The electron gas in full 3D space inside a homogeneous magnetic field would normally be the next example to consider and it will indeed be discussed in Section 3.7. This problem has mostly been treated in astrophysical applications but here, we want to place it within a framework interesting to fully 3D solid-state systems. Although it might be useful to present it at this point, in order to see the rather large differences from the above 2D case, e.g., the smooth deviations from the above dHvA periods, we choose to present it after discussion of the quasi-2D cases that follow. In this manner, we can study in detail the dimensionality crossover from 2D to 3D, thus, addressing issues regarding possible dHvA violations both in quasi-2D and in bulk 3D solids in a unifying manner.

3.3 Finite-thickness interface (without magnetic field) Let us now consider an interface with a finite (nonzero and non-infinite) thickness d but let us first begin with the simpler problem of a vanishing magnetic field. In this case, we will see that the standard Fermi circle or disk of 2D noninteracting electrons in the jellium model will now be replaced by a sequence of many Fermi circles of appropriate radii, each one connected to a particular QW-level associated with the zmotion; the procedure of determining the appropriate radii being not so trivial and rather tedious, as we shall see. Once again, we will work in the canonical ensemble with a fixed number N of electrons, such that the surface areal density n A is the control parameter, although at the end this can be relaxed. The results can recover those that would have been obtained if the control parameter were the volume density nV  nA / d (see later below) and especially so, in the limit d   .

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3. Electron systems in strong magnetic fields

Indeed, consider an interface (or film) that again extends in a macroscopically large area S in the x and y directions, whereas in the z-direction it is characterised by a width d, which we can initially consider as very small (of the order of nanometres, i.e., a few atomic layers thick). In the jellium model that we consider here, the Hamiltonian is effectively just a nonrelativistic kinetic energy term in 3D space, namely p2   E , (3.8) 2m  with p being the canonical momentum (we have obviously taken the simplest gauge  A  0 ). For a large system on a plane, it is natural to impose periodic boundary conditions in the x and y directions but the z axis can be treated like a 1D double quantum well with impenetrable walls at z = 0 and z = d (the simplest way to impose the spatial confinement). The eigenfunctions of (3.8) can then be written as simple product functions of the form

 x, y, z   sink z z e

ik x ik y y x e .

(3.9)

A quantum state is then characterised by the eigenvalues of the three Cartesian  components of canonical momentum p /  kx, ky, kz (or nx, ny, nz after quantisation, see below). The Pauli principle requires that each such orbital state (namely a triplet nx, ny, nz ) can be occupied at T = 0 by only two electrons (of opposite spins) and this is a very important criterion, which for strongly anisotropic systems such as this one, must be imposed in a careful manner, as we will see below and also in later Sections. The single-particle energy spectrum is

 n x , n y , n z   kx 

 2 k z2 2k 2   n z  where  n z   and k 2  k x2  k y2 , 2m 2m

2n y 2n x n ,ky  , k z  z , with nx , n y  0,1,2,.. and n z  1,2,3.. (3.10) Lx Ly d





i.e., kx and ky are quasicontinuous variables (because Lx, Ly   ), whereas kz is strongly quantised. For extremely small d, the variable kz is expected to take its lowest value (corresponding to nz = 1) for all electrons, which is the case that is usually discussed in the literature, where the particles are “frozen” at the lowest QW-level nz = 1, making the system effectively 2D. This is so, because of the enormous energy difference between the nz = 2 and nz = 1 levels (that goes as 1/d2) and therefore, because it is indeed energetically favourable to start filling states with increasing |kx| and |ky| (or equivalently |nx| and |ny|, starting from 0 and gradually occupying higher numbers in a symmetric manner, with nz always being 1), thus, forming the standard Fermi circle of 2D noninteracting electrons. However, the reader should note that for any fixed nonzero d, even at T = 0, the above-mentioned 2D character may be violated for sufficiently large density (to be quantified below). 74

3. Electron systems in strong magnetic fields

There might come a point (i.e., if the number of electrons to be accommodated in single-particle states is sufficiently large) when it is no longer favourable to continue increasing the Fermi circle and maintain nz = 1; it may be favourable for the remaining electrons to start jumping to the nz = 2 QW-level and then kx and ky can start taking values back at |nx| = |ny| = 0, i.e., start forming a new Fermi circle, now associated with the level nz = 2. We emphasise that this occurs without violating Pauli‟s principle, because in the triplet nx, ny, nz , which labels a single-particle state, nz has changed value, so that nx and ny can now acquire the same values as they had before this transition, starting again from 0. The transition to nz = 2 will of course occur whenever the “initial” Fermi circle (for nz = 1) becomes so large (with such a long radius kf1) that the single-particle energy 2 k 2f 1 / 2m will become equal with and from that point on, exceed the energy difference between the two QW-levels, or equivalently, whenever the following equality holds .

 2 k 2f 1 2m

  n z  1   n z  2  , (3.11)

The left-hand-side of (3.11) is the single-particle energy of an “extra” electron that we wish to place on the perimeter of the Fermi circle, previously formed by all other electrons that were in the QW-level nz = 1. The right-hand-side of (3.11) is the analogous single-particle energy if we were to put the “extra” electron at the QWlevel nz = 2 and start a new Fermi circle from the beginning, namely from zero radius. It is now important to note that (3.11) provides a sense of “equilibrium” in the occupation procedure. As stated, from that point on, a 2nd Fermi circle is being formed (corresponding to nz = 2) and what is more important, the above sense of “equilibrium” must be preserved during the entire occupation procedure that follows. If we still have an excess of electrons and we keep occupying available (empty) single-particle states, then the extra electrons must be placed back and forth in both QW-levels nz = 1 and nz = 2 in a way that the Fermi radius associated with nz = 1 and the one associated with nz = 2 will both keep increasing and will at every point (for every density) be related with each other through the “equilibrium” relation  2 k 2f 1 2m

  n z  1 

 2 k 2f 2 2m

  n z  2 , (3.12)

such that the occupation procedure is “fair”, guaranteeing that it will give the lowest possible total energy for the many-particle system. (3.12) demands that the extra electron to be placed anywhere at any moment of the occupation procedure must have the same single-particle energy in any of the possible occupational scenarios. If the equality of (3.12) were not satisfied and one side were larger than the other, it would mean that the procedure followed up to that point was not the optimal (energetically lowest) one, because we could always move electrons around in state-space to gain energy. [It can actually be shown variationally [20] that the above procedure is the 75

3. Electron systems in strong magnetic fields

lowest energetically.] The reader should notice that this “fairness” strategy is actually a generalisation of the standard symmetric manner of occupation scenarios that are followed in the usual construction of the 3D Fermi surface, where this “equilibrium” in the single-particle states occupation procedure is the usual isotropic filling in kspace that leads to the standard Fermi sphere; the above is a generalisation of this to a highly anisotropic system. This optimal partitioning for our anisotropic problem (in the above-described cases of one or two Fermi circles) is represented pictorially in Fig.3.3:

Fig. 3.3: Only one Fermi circle is created, (p = 1) when d < dcrit1 and two Fermi circles are created (p = 2) when dcrit1 1) 2 2 p 2 2 2 kf 12  2 . (3.26) 2m 2md 2md 2 2

By solving this equation with respect to the sample thickness d and by using (3.24), we find a series of critical values of thickness (for various values of p = 1,2,3,…), namely

dcrit ( p) 

p( p  1)(4 p  1) . (3.27) 12nA

That is, for values of thickness larger than (3.27), the system occupies p QW-levels, until the (p+1) Fermi circle starts over. This of course happens (as can be seen by just replacing p with p+1 in (3.27)) when d is equal to

dcrit ( p  1) 

p( p  1)(4 p  5) . (3.28) 12nA

Therefore, the conclusion is that when the thickness d varies in the following window

dcrit ( p)  d  dcrit ( p  1) , (3.29) the system occupies p (and no more than p) QW-levels. The above results (3.27), (3.28) and (3.29) reproduce the previous examples for p = 1 and 2: For 0  d  d crit 1  3 / 2nA

(p = 1), (3.30)

we have the case of Fig.3.3 (and the above (3.30) can be viewed as a criterion of 2dimensionality). For d crit 1  3 / 2nA  d  d crit  2   13 / 2nA

(p = 2) , (3.31)

79

3. Electron systems in strong magnetic fields

we have the second case of Fig.3.3, where two Fermi circles are present, etc. The final step is to determine in full generality, the total internal energy of the electron gas when d lies in the range (3.29). This is given by p 1 j 2 2 2  (3.32) E    NjEfj  Nj 2md 2  j 1  2

with Nj  nAjS the number of electrons at QW-level j and hence, with nAj given by (3.25) and with Efj 

2

k 2fj / 2m 

2

2 nAj / 2m , the corresponding 2D Fermi energy.

Therefore, we have E

 j 2 2 2  n Aj S  n Aj  2m j 1  d2 2

 S  , (3.33) 

p

which after carrying out the sums, turns out to be E

N



2

(2 nA)  1  ( p  1)(2 p  1)  2 p( p 2  1)(2 p  1)(8 p  11)     (3.34) 2m  2 p 12nAd 2 1440nA2 d 4 

that gives directly the total ground state energy per electron, when the thickness of the system lies between (3.27) and (3.28). This reproduces the earlier results (3.14) for p = 1 and (3.19) for p = 2. We note that, even in this rather trivial system, the role of thickness on the energetics is noteworthy. Once again we should stress that, compared with earlier work [21–23], (3.34) is exact and does not generally describe quantum oscillations with wavelengths that are governed by the extremal diameters of cross sections with an anticipated 3D Fermi surface; those being expected only for a large number of QW states, i.e., with very large p‟s involved. In contradistinction to earlier work, (3.34) is also valid for any small value of p. A final point that is of interest is to take the thickness of our system to infinity but keeping the volume density nV  nA / d constant. One expects that in the limit of infinite space, the above expression will converge to the well-known energy of noninteracting electrons in full 3D space, which is the standard result that is achieved through use of a macroscopically large cube. However, this has rather to be checked, because the standard problem that leads to the symmetric spherical Fermi surface utilises periodic boundary conditions in all Cartesian directions, whereas here we have infinite potentials (rigid boundary conditions) at two points of the z-axis. To examine if the above expectation is true, we choose to write the total energy in units of the 3D Fermi energy: Ef (3D ) 

2

(3 2 nV ) 3 . 2m 2

We then have from (3.34) 80

3. Electron systems in strong magnetic fields

E

1 1 2 2  8  3  nV 3 d  ( p  1)(2 p  1)  p( p  1)(2 p  1)(8 p  11)    .  Ef (3D)     2 5 N  9   2 p  12nV 3 d 2 1440nV 3 d 5 (3.35)

Substituting (3.27) into (3.35) we can plot the energy for large values of p, keeping the volume density nV constant (see Fig.3.3), from which it is readily seen that it indeed tends to the well-known energy of free electrons in full 3D space, namely E

N



3 Ef (3D ) . 5

To see this analytically, we need to make explicit use of the volume density. From (3.27) and (3.38), after setting nA = nV d and solving for d, we take the limits d   and p   such that the d window of values is shrunk to only a single value

d3 

p 3 (3.36) 3nV

and from (3.35) we then have E

1 1  p2  2 p5   8  3  nV 3 d   . (3.37)  Ef (3D)     2 5 N  9   2 p 6nV 3 d 2 90nV 3 d 5 

Substituting  d / p    / 3nV (due to (3.36)), we finally have 3

1 5 2 1 3 2 3 3 3 3   3nV   E  Ef (3D)  8   nV       3nV         , (3.38) 2  5  N  9   2  3nV  6nV 3    90nV 3     1

which transpires to be equal to 3Ef (3D ) / 5 . Therefore, we have given a full analytical treatment of the dimensionality crossover of nonrelativistic noninteracting electrons (in zero-field) from pure 2D to full 3D, passing through a sequence of quasi2D well configurations. The above results can be viewed as an extension of, or more appropriately, as an exact quantal correction to, the extremal free-electron crosssections picture, usually employed in this problem [21–23]. We can also note that with the above analytical solution, one can extend calculations to the derivation of other (thermodynamic) properties of the interface, such as pressure or compressibility, by taking proper derivatives with respect to volume (for constant N); however, this is something that we will not pursue here.

81

3. Electron systems in strong magnetic fields

Fig. 3.4: Energy (in units of 3D Fermi energy) as a function of z-levels quantum number. Note that for approximately more than 300 z-axis occupied levels, the total energy tends rapidly to E N  3 5 Ef (3D) .

3.4 Finite-thickness interface in a perpendicular magnetic field In the previous example of the finite-d interface, the single-particle energies were quantised only in the z-direction and they were quasicontinuous in the xy plane. Later in Section 3.7, where the full 3D problem in a magnetic field is considered, we will find single-particle energies that will only be quantised in the xy plane (Landau levels) and will be quasicontinuous in the z-direction. In all these cases, we have quasicontinuity in at least one direction, such that the above discussed “equilibrium conditions” can continuously be satisfied, giving at every point, i.e., for every density, the optimal partition or arrangement of our Fermionic particles in single-particle states. One might wonder how the above method could be used if the single-particle energy is strongly quantised in all directions. This is actually the case of our main interest, namely when we consider a finite-thickness interface inside a perpendicular magnetic field. In such a case, the previous equilibrium condition is not satisfied in a continuous way, as there are no quasicontinuous Fermi circles (of Section 3.3) or Fermi line segments (of Section 3.7). We now do not quite have equilibrium equalities at every density, as in the other two problems but we rather have inequalities that change directions in a discrete manner with variation of density, which actually determine the lowest-energy occupation scenario. However, we will still have distinct points of transition (into different occupation scenarios) whenever certain equalities are again satisfied, as we will see. Specifically, it will transpire that to determine these equalities, requires a close and careful study and that there is no simple analytical solution that can be written directly for a generic B and d, even though we are dealing with noninteracting electrons at T = 0, i.e., the energy cannot be written directly in closed form for an arbitrary field and thickness – one has to actually run the occupation procedure carefully for all “previous” values of B and d starting from easy limiting cases, unlike the other two problems. The interplay between the strong quantisation in the xy plane and the simultaneous strong quantisation in the z-axis leads to rather unpredictable patterns under combined 82

3. Electron systems in strong magnetic fields

variations of B and d, when one simply occupies one-electron states in a manner that maintains the lowest possible total energy. The single-electron spectrum is now given by  

1

 (n, nz )   c  n    εnz , (3.39) 2 

where n is again the Landau level index (n = 0,1,2…) and the QW-levels are again represented by 2

kz 2 ε nz  , where 2m

kz 

 nz d

,

nz =1,2,3,… (3.40)

Let us first see a simple example of the above-mentioned competitions that are expressed with inequalities. If d is extremely small (to be further quantified below), then for a given B (not very strong – such that there are more than one LLs needed (see below)), it is energetically favourable for the electrons to be placed in several distinct LLs consecutively, starting from the lowest and moving upwards in energy until all the electrons of the system are accommodated and to keep the system “frozen” in the nz  1 QW-level. In such a case, the problem is essentially equivalent to the 2D problem of Section 3.2, apart from an extra constant term in the energy (i.e., common to all electrons) due to the QW confinement. However, if the thickness d starts increasing, then there might come a point (in density) when an extra electron would energetically prefer to be placed in nz  2 and start to occupy from the beginning a lower LL, which is already occupied by other electrons that correspond to nz  1 without violating Pauli‟s principle (note that, apart from n and nz, the 3rd integer l is already implicitly used, counting the degenerate states for each combined pair (n, nz ); therefore, it does not need to be mentioned in any special way). The simplest nontrivial case is when two lowest LLs, i.e., n = 0 and n = 1, are originally involved (for nz  1 ) and then, upon an increase of d, the above transition to nz = 2 and back to n = 0 only takes place; this transition will happen when

c 2

 ε nz  2 

3 c  ε nz  1 . (3.41) 2

This is in the spirit of “equilibrium” that was used earlier in (3.12), although here, it occurs in steps for discrete values of parameters. Once again, the extra particle that is about to be accommodated according to various possible occupation scenarios, has a single-particle energy that must be the same in all of them; otherwise, the process would not be fair and it would lead to higher total energy [20]. (3.41) leads to a critical value of thickness d where the transition occurs for a given B, namely

83

3. Electron systems in strong magnetic fields

dcrit 

3 . (3.42) 4B

The above was only the simplest example in order to stress the essential point and to motivate what follows, the general case involving an arbitrary number of LLs and QWs still needs to be worked out. One then wonders what one can say in full generality for the correct partition (in combined n and nz states) for arbitrary values of B and d for this problem. In the general case, there is “asymmetry” in the manner with which we need to treat d-B variations, in order to have good control on all possible cases and better understanding of the patterns that occur. If, for example, one follows the route of having fixed d and varying B, analogous to what is done in the 2D case (Section 3.2) but for a finite d (the experimentally relevant route, of a given interface), then the problem is rather difficult to analyse systematically, producing results that sometimes look “surprising”, i.e., new transitions appear in the interior of certain windows of B-values (windows with ends that are consistent with dHvA variations), the origin and location of these “internal transitions” not being easily identifiable. The point is that variation of B for fixed d changes not only the energetic distance between LLs but also their degeneracies and this interplay, together with the competition with the energetic distance between QW-levels, leads to a multitude of cases to be investigated that do not appear easily subdued to a systematic control. However, it transpires that the opposite route of temporarily keeping B fixed and varying d and then changing B in a particular way and repeating the procedure of the variation of d, offers a much better control in the theoretical treatment; this is basically because degeneracies of each LL are fixed and we need only to focus on competitions between LL-QW energetic distances. Although the results are of course equivalent with both methods, what we called “surprising results” of the 1st route will find a better understanding through the 2nd route, both in terms of origin and location. In the following, we will pursue the 2nd route for theoretical convenience but in the final figures, we will also show results as would appear from the 1st method and later, we will also provide 2D figures that show the full results under combined variations of B and d; the ordering then of what is kept fixed not being important. Let us start being more quantitative and in accordance with the mathematically 2D problem of Section 3.2, let us first assume that the number of electrons lies in the following window: N 2

 , (3.43) 

such that only a single LL is involved, although now combined with a QW-level (see below). Always treating N as fixed (so that nA  N / S is fixed as well), (3.43) is equivalent to

84

3. Electron systems in strong magnetic fields

B

1 nA , (3.44) 2

where it should again be noted that the effective areal density nA  N / S is related to the volume density nV through nA  nV d and   hc / e is the flux quantum. Now, each quantum state is again characterised by three quantum numbers, namely {n, l, nz } with the positive integer l counting the degenerate states inside an LL (or more appropriately, inside a combined (n, nz )-pair) and taking 2Φ/Φν values, such that each combined pair (n, nz ) can contain up to 2Φ/Φν electrons (according to Pauli‟s Exclusion Principle). Then it is easy to see that, when (3.44) is satisfied, the electron system will occupy only one combined-pair, n  0, nz  1 , while l runs from 1 up to N, which here, is less than the LL degeneracy 2Φ/Φν and this will give a total energy E  N ε{n  0, nz  1} (3.45)

with ε{n  0, nz  1} the single-particle energy (3.39) with n = 0 and nz = 1. We can write this energy in terms of 2D Fermi energy (corresponding to the absence of magnetic field), as

E  B  Ef  N  nA

kf      , where Ef  2  2m  2nAd   2

2

2

2 nA . (3.46) 2m

That is, if B satisfies (3.44) then, for every value of thickness d, electrons occupy only the states with the lowest possible quantum numbers n and nz (see Fig. 3.5 – note that in this and all following figures we simply compare single-particle energy differences, by always placing at the same level the beginning of each energy difference that needs to be compared. In such a way, the comparison is visually obvious; therefore, the placement of the levels does not have an absolute meaning in energy and only the differences matter).

Fig. 3.5: Occupied states for every d

85

3. Electron systems in strong magnetic fields

Let us now start lowering B. The next window of B-values, a natural choice if we follow the 2D paradigm of Section 3.2, is 4

  N 2  

or

1 1 nA  B  nA . (3.47) 4 2

Now the usual occupation scenario would normally be the one in which the extra N2Φ/Φν electrons will need to be placed in the next LL (the one with n = 1) as in Section 3.2 but this is not necessarily true. It might be energetically favourable for some electrons to occupy another QW-level with respect to Pauli‟s principle, because of the extra degree of freedom provided by nz . We can see immediately the possible options; the two appropriate possibilities are {n  1, nz  1} (increase n by 1) or {n  0, nz  2} (increase nz by 1 and go back to the lowest LL). However, which one

is the correct one and under what conditions? The answer is that this will be determined by the thickness of the sample. Let us try to find the critical thickness at which the two possibilities lead to the same single-particle energy: ε{n  1, nz  1} = ε{n  0, nz  2} , (3.48)

which is (3.41) that we saw earlier as a motivating example, or equivalently

 c  3 2 2 / 2md 2 , (3.49) which in turn leads to dcrit  3 /4 B (the same as (3.42)) . Note that the critical thickness depends on the value of the magnetic field, as long as this field lies inside the window of (3.47). It is easy to see that for values of thickness lower than (3.49), it is the states {n  1, nz  1} (always meaning for all 0< l Bcrit1

Fig. 3.44: Two Fermi segments are created when BBcrit2 (see (5.10))

If we now drop B to a value slightly lower than Bcrit1, the lowest LL cannot accommodate all the electrons and the next LL (for n = 1) will have to be used. We then start having a second Fermi line segment forming (extending from -kf2 to +kf2) associated with the n = 1 LL, while we simultaneously also have a first Fermi segment (with a kf1, always associated with the n = 0 LL) that now increases in size as we keep placing more electrons. The actual manner in which we now place the remaining electrons in the two LLs is back and forth in both of them, in a way that the “Fermi height” of the segment associated with n = 0 and the one associated with n = 1 will both keep increasing and will at every point (for every density) be related with each other through the “equilibrium relation”:

c  2

2

2 2 k 2f 1 3 c kf 2 , (3.91)   2m 2 2m

where kf1   2lB2 n 1 and kf 2   2lB2 n 2 (coming out from an argument exactly like the one in (3.86) but now with partial densities). For any given volume density nV , (3.91) will determine the proper (energetically favourable) partition ( n 1 , n2 ) of the total density between the two LLs involved. Again, (3.91) reflects the fact that the extra electron at every point of the occupational procedure must have the same singleparticle energy for either of the two options (or scenarios). If (3.91) were not satisfied and one side were larger than the other was, it would indicate that the occupational procedure followed up to that point energetically was not the lowest. Compare the above “equilibrium condition” with the one that was implemented in (3.17) of Section 3.3, or (3.48) of Section 3.4, when only two QW levels were occupied. Note that, although the cases are different, they are along a similar line of reasoning. 115

3. Electron systems in strong magnetic fields

From (3.91) and the expressions for kf1 and kf 2 , we obtain the optimal density partition in the two LLs (by also utilising nV  n1  n 2 ), the final result being: nV B3  (1  3 )  2 Bcrit1   , (3.92) nV B3  n 2  (1  3 ) 2 Bcrit1  n1 

where Bcrit1 is given by (3.89). The above partition of density is valid only for B < Bcrit1. With regard to the lowest value of B allowed, i.e., the complete range of Bvalues where (3.92) is valid, see further below. Note how the full three-dimensionality and the extra presence of the magnetic field have modified the earlier found partition (3.17). The total energy in the above case will be determined by: 2 2 kf1  c 1 E  N1  N1  N 2 3 c  1 N 2

2

3

2m

2

2  E Ef   B   2  onV 3     4  N (3 2 ) 2 3   nV 2 3   48  B    

3

2

2

k 2f 2 2m

  B    16  2    n 3 o    V 

4

  . (3.93)  

The lower value of the range of B can then be determined by considering the next case, namely, when a 3rd Fermi segment (of LL index n = 2) is about to form, for which we have the equilibrium condition equivalently, 3 σc / 2 

2

c / 2  2 k 2f 1 / 2m  5 c / 2

or

k f22 / 2m  5 σc/2 and turns out to be 1

Bcrit 2  (3  2 2) 3 Bcrit1 . (3.94)

Following these two examples, to proceed further with the most general case requires greater mathematical sophistication. In the most general case, in every ith LL, electrons build a 1D Fermi segment that defines a Fermi wavevector kfi , where the index i (defined by i = n+1, n is a LL index) runs over all occupied LLs and has positive integer values. When the magnetic field is a constant B, let us say that we know that the system occupies in general k LLs (k  1) and creates k 1D Fermi segments in the z-axis (and then i runs from 1 to k). The associated kfi s must be determined as in the example shown above, namely: 2 2 L Ni  1   k :occupied  2

kfi

 dk  kf

i

  2lB2 n i , (3.95)

 kfi

where ni  Ni / V is the partial volume density corresponding to the ith LL (i = 1,2,3….k). A similar line of reasoning as that of Section 2 must then be followed. The 116

3. Electron systems in strong magnetic fields

last electrons on the ends of any of the k 1D Fermi segments must have equal singleparticle energies, i.e., in the spirit of Section 2, „equilibrium‟ is satisfied; otherwise, we would have transitions and rearrangements between the states, such that equilibrium is recovered, to assure that the energetically optimal occupational procedure has been followed. The appropriate mathematical expression for the equilibrium is then: 2

k 2f 1

2m



c 2

2



k 2f 2

2m



2 2 kf k 3 c 1  ...    c (k  ) . (3.96) 2 2m 2

From the above condition and with the use of (3.95), we can determine in the general case (i.e., for any k) the proper partition of all 1D densities in each LL:

ni2  n12   i  1

B3

16 

 3 0

, (3.97)

k

while it also holds that

nV   ni , (3.98)

and index i runs from 1 to k and nV

i 1

denotes the total (global) volume density of electrons. This is a system of k equations with k unknown variables, which can be solved analytically. We will return to this solution soon. Let us first think of the appropriate values of magnetic field B that force the system to occupy exactly k LLs; from the equilibrium condition (3.96) we can find a critical value of B as a function of n1 . When B is exactly equal to this critical value, electrons start the occupation of k+1th LL. However, this is rather easy to describe; it occurs when the 1D Fermi segment at the k+1th LL is just about to be formed, namely: 2

k 2f 1

2m

 k c , (3.98)

such that B is just 3 Bcrit (k ) 

1   3 2    0 n1 , (3.99) k  16 

by following steps similar to the ones followed to derive (3.89) – but note that now n1 also depends on B. The same line of reasoning gives the other critical value of B, which makes electrons start the occupation of the kth LL: 3 Bcrit (k  1) 

1   3 2    0 n1 (3.100) k  1  16 

and all the previous conditions are correct only in the case that the magnetic field varies in the following window:

117

3. Electron systems in strong magnetic fields

Bcrit (k 1)  B  Bcrit (k ) , (3.101) this being true for k > 1; for k = 1, we only have B  Bcrit (1) . Here, we remind the reader that the first linear density n1 also depends on the magnetic field and it must be calculated analytically. Now, writing (3.97) in a more convenient form, we obtain: 2

ni  nV

3  n1  Bcrit 1    i  1 3  B  nV  3 Bcrit 1 3 B

, (3.102)

  3 2 3 where we set Bcrit 1    0 nV that was found to be the first critical value of B (see  16  (3.89)). It is also convenient to define a quantity (a filling factor-like quantity): 2

B3  n  a1  crit3 1  1  . (3.103) B  nV  It is then easy to observe that when B  Bcrit (k ) (see (3.99)), a1  k and when B  Bcrit (k  1)

(see (5.17)), a1  k  1 , so it must hold that:

Bcrit (k 1)  B  Bcrit (k )  k 1  a1  k . (3.104) When B lies on a critical value, then a1 is an integer (k or k-1 accordingly), otherwise it must be a fractional (more generally irrational) real number. Then (3.97) becomes:

ni  nV

a1  i  1 3 Bcrit 1 B3

, (3.105)

i.e., the coefficient of nV is just the percentage of density that corresponds to the (i)th LL. Now, by using (3.98) we have: 3 k Bcrit 1  a1  i  1 , or  B3 i 1

3 Bcrit 1  a1  a1  1  a1  2  ...  a1  (k  1) . (3.106) 3 B

118

3. Electron systems in strong magnetic fields

Unfortunately, it does not seem possible to solve the above equation with respect to

a1 . However, one observes that (3.106) can be written with the use of generalised 

Riemann or Hurwitz zeta functions (defined by   s, a    1/  i  a  ), as follows: s

i 0

3 Bcrit 1  i    1 2 ,  a1      1 2 , k  a1   , (3.107) 3 B

where i is the imaginary unit and k the number of occupied LLs and k  1  a1  k ,

0  k  a1  1 . Therefore, the difference between Hurwitz zeta functions must be a pure complex number:

Re    1 2 , a1   Re    1 2 , k  a1  a1 (3.108) and it is also true that Im    1 2 , k  a1   0 , because k  a1  0 (3.109). Finally, we find that

   1 2 , a1      1 2 , k  a1   i Im    1 2 , a1  (3.110) 3 Bcrit 1 . (3.111) B3

Im    2 , a1   1

This is the key to the solution of this problem; only the imaginary part of the Hurwitz zeta functions has physical meaning. With the help of (3.107), (3.110) and (3.111), we can then write down analytical expressions for the critical values of B that do not depend on n1 : Bcrit (k ) 

Bcrit (k  1) 



Bcrit1

Im    12 , k 



Bcrit1

 

2

(3.112) 3



Im    12 ,   k  1 

2

. (3.113) 3

As a test of consistency, we can check that the above reproduce the earlier results of (3.89) and (3.94) For k = 1, then: Bcrit (1) 

Bcrit (2) 



Bcrit1

 Im   

1

2

Bcrit1

, 1

Im    1 2 , 2 





2

2

 3

Bcrit1  Bcrit1 , which is (5.5) and for k = 2 we have 1

 3



1

1 2



2/3



Bcrit1  3  2 2



1

3

Bcrit1 , which is (3.94).

119

3. Electron systems in strong magnetic fields

It is also interesting to check what the differences of neighbouring inverse Bcrits are and relate their behavioural pattern to the standard period of the de Haas-van Alphen effect. It is true that in weak magnetic fields and hence, large values of k, the system starts behaving semi-classically (then the segment sizes will come from cuts of Landau tubes inside a Fermi sphere) and then we expect an oscillating period similar to that of the dHvA effect. Having calculated the critical values of B analytically, we have the ability to check the period directly, without any approximations. Indeed, the semi-classical dHvA period is: 1

B

 1 

4

 3  2

2

3

4   1   9   0.76314 . (3.114) 2 Bcrit1 n 3  Bcrit1 3

The difference of inverse B that we have found is (from (5.29) and (5.30)):

 

1 1  1   B Bcrit (k ) Bcrit (k  1)

 Im    

1

2 ,  k 

   Im    2

3

1



2 ,   k  1 

Bcrit1

2

3

.

(3.115) Note that when k = 1, then  1/ B   1/ Bcrit1 , which deviates from (3.114) by about 31%, while if k = 2, then  1/ B   0.7996 / Bcrit1 , which deviates from (3.114) by only 5%. Now, comparing (3.114) with (3.115), leads to the conclusion that the following must be proven (for large k):

 Im  

1

2

, k 

   Im   2

3

1

2



,   k  1 

1

2

3

4    . (3.116) 9 3

Using the well-known relations (which are true, because k is an integer):



k



k 1

Im    1 2 , k    j and Im    1 2 ,   k  1    j , (3.117) j 1

j 1

then the following must be proven 2

2

3 3  k   k 1  4 2 j  j           when k  1 . (3.118) 9 3  j 1   j 1  3

1

3

2

For this, let us think momentarily in a slightly different manner. Instead of calculating

 1/ B  , we can calculate  1 / B 

3

2

and then relate it to  1/ B  . Using (3.115) we

obtain:

120

3. Electron systems in strong magnetic fields

B

 1

3

2



k 3

Bcrit2 1

. (3.119)

Equivalently, we can write  1/ B  as:

B

B

2  1   1 3

3

2

Bcrit (k ) 

2 1 k 3 Bcrit1 Im    1 ,  k  2





1

3

(3.120)

and now comes the approximation. For large k (weak magnetic fields), we must expand the term Im    1 2 , k  /

 k

3

around k =  , to see that it is almost equal

to 2/3, which is indeed true: Im    1 2 , k 

 k

3

2 1 1     3 2k  k 

3/2

11/2  1 1  1    2  1           . 2 4  2  24k 1920k  k    3 

(3.121) Therefore, the result is:

 

1

2

3 23 3 1 2 1 , (3.122)  1      B 3  2  Bcrit1  3  Bcrit1

as anticipated (see (3.114)). The conclusion is that in magnetic fields that are not extremely strong (i.e. for many LLs occupied), the system rapidly converges to the semi-classical behaviour. However, this semi-classical dHvA period is violated at exceedingly strong magnetic fields. Unfortunately, the magnetic fields necessary in order to observe these extreme quantum effects are very large and therefore, we cannot see them in the laboratory. However, we can effectively reduce them by lowering the value of electronic number density. For example, consider (3.89), which gives the first critical value of B (the largest of all critical values). Nowadays, we might achieve magnetic fields up to 40 Tesla, so: 1

nmax

3

2 2  16   40T  24 3      2.17 10 m .      

This can be considered as the maximum number density of charge carriers that a material must have in order for our extreme quantum results (reflected in the dHvA violations) to be seen experimentally. The above density is four orders of magnitude smaller than typical metallic densities. The final important step for this section is to calculate the total energy and magnetisation. The energy is just a sum over all occupied LLs and z-axis levels:

121

3. Electron systems in strong magnetic fields 2 k 1  kf j21  1 E    N j 1  ( j  1 2 )  N j 1  , (3.123)  3 2m  j 0 

where kf j21   4lB4 n2j 1 (from (3.105)), which leads to: k 1  N j 1 1 2 4lB4   E     jN j 1   N j 1n 2j 1  . (3.124) 2 3 2m j 0  

Using N j 1  n j 1V and substituting nj with its equal from (5.13), we find: 3 3 2   N j 1 1 2 4lB4 3 n3j 1  Bcrit Bcrit 1 n1 1 E     N j 1 3 2   Vn j 1  V 3  . (3.125)  B n 2 3 2m B n 2  j 0  k 1

 4lB4

2

Now, observing that

2m

 

3 Bcrit 1 , (3.126) the energy becomes 3 2 Bn

3 k 1   N j 1 2 B3 n2 B3 n j 1  E     N j 1 crit3 1 12   V crit3 1 2  . (3.127)  B n 2 3 B n  j 0 

The first term gives k 1



 N j 1

j 0

3 3 2 2 Bcrit Bcrit 1 n1 1 n1  N   N  a1 , (3.128) B3 n2 B3 n2

the second term gives



k 1

N j 0

j 1

2

N

 2

(3.129)

and the third term gives 3 k 1 n  B3  Bcrit 2 2 j 1 1 V 3  2   N  3  3 B j 0 n 3  Bcrit1  3

k 1

Now, the sum

a  j j 0

3

2

a  j j 0

1

3

2 k 1

a  j j 0

1

3

2

. (3.130)

is just a difference of two zeta functions of order -3/2:

1

k 1

1

2

 i    3 2 , a1      3 2 , k  a1   . (3.131)

Energy must be a real quantity; therefore, it must hold that:

   3 2 , a1      3 2 , k  a1   i Im    3 2 , a1  (3.132) 122

3. Electron systems in strong magnetic fields k 1

a  j

and

3

1

j 0

2

  Im    3 2 , a1  . (3.133) Finally, the energy per electron is:

1   2 3   2 B 3 , a E   1 a    . (3.134) Im       2  1 1 3 N 3  Bcrit  2  1   

Once again, it does not seem possible to solve (3.111) with respect to a1 (there is no analytic expression for the inverse function of the imaginary part of the Hurwitz zeta functions). However, this is not quite necessary, because we can solve (3.111) numerically and then determine the values of (3.134) for every B. If up to this point all our calculations are correct, the derivative of energy with respect to a1 must vanish, i.e., energy is indeed minimal and the correct density distributions are given by (3.105). Although tedious, it is straightforward to check this expectation and indeed, we have:

 N 

 E

a1

 2  B 3  1 2   k 1    B 3  1 2 k 1  3  1 2 2    c 1   3  a  j   1  a  j      3   1  1  c B  3  Bcrit1  a1  j 0    j 0     crit1     B3  12  B3  12   c 1   3   crit3 1    0   Bcrit1   B    

At the critical values of B (the ones expressed by (5.29)), the energy has a simple analytic form, namely:

E

N



 eBcrit1  1 mc  Im    1 , k  2 





 3    1  k  2 Im    2 , k   , (3.135) 2 3  2 3 Im    12 , k    

Note the amusing fact that

eBcrit1 / mc   2 / 3 where Ef (3D ) 

2

2/3

2

(3 2 nV ) 2/3 2/3   2 / 3 Ef (3D) , 2m

(3 2 nV ) 2/3 / 2m , the 3D Fermi energy of electrons when no

magnetic field is applied on the cube. In the limit B  0 (or k   ), (3.135) can be shown to tend to  3 / 5 Ef (3D) . This can also be seen from Fig. 3.44 below. Finally, by taking the derivative of (3.134), we can also determine analytically the magnetisation per electron using the relation: M  E / B . However, we will need to know the derivative of a1 with respect to B; this can be calculated from (3.107) and the result is

123

3. Electron systems in strong magnetic fields

da1  dB i 

M

3B  2 Bcrit2 3B  2 Bcrit2   12 , a1     12 , k  a1   Im   12 , a1  5

3

5

3

10 3      1  2a1  x 2 Im    3 2 ,  a1  (3.136) N 3  

and for the magnetic susceptibility, the corresponding procedure gives: X

N



 

9 x Bcrit1 

52

 1  5 x 2 Im    3 2 , a1  , (3.137) Im   12 , a1   1

where   e / 2mc is the Bohr magneton and x  B / Bcrit1 . The above solves exactly the problem of noninteracting electrons in a uniform magnetic field in full 3D space, by applying a procedure (of equilibrium relations) that is in a similar line of reasoning as in earlier sections, which is actually, the central line of approach that has been introduced in this article, to be used as a common tool for quite disparate problems (see also next section). It should also be noted that the above results are the limiting behaviours of the previous quasi-2D interface, when thickness becomes exceedingly large (it can be rigorously shown, for example, how the first critical field (3.89) arises from the rather involved analytical patterns of Section 3.4 in a complete analytical manner, demonstrating the consistency of our results). Earlier works that follow different methods either do not give results for the total energy [27], or they mostly deal with a relativistic system [28], both of which are considerably involved in mathematics and do not quite reflect the basic Physics of the problem (i.e., the basic physical processes that are involved in the formation of the proper Fermi segments). Below, the reader can find plots of all thermodynamic properties as functions of B. We should note again the continuity of energy and magnetisation but with the latter having cusps, leading to discontinuities and a highly nonlinear behaviour of susceptibility, something that we did not witness in the quasi-2D results of Section 3.4, where susceptibility was always piecewise constant.

124

3. Electron systems in strong magnetic fields

0.2

M/N M N (μμBΒ)

0.88 0.86 0.84 0.82 0.80

0.0

0.2

0.4

0.6

0.8

0.78 0

1

2

BB

3

0.0

4

0.5

1.0

B B

Βcrit1 (Bcrit1)

1.5

2.0

Βcrit1 (Bcrit1)

40

XN μ Β Βcrit1 (μB/nAΦ0 X/N )

e Βcrit1

N E/NE (heBcrit1/mc) mc

0.90

20

0

20

40 0.0

Fig. 3.45: Energy (in units of

0.5

B B

1.0

1.5

(Bcrit1) Βcrit1

2.0

eBcrit1 / mc ), Magnetisation (in units of   ) and

Susceptibility (in units of  / Bcrit1 ) as functions of B. Susceptibility, apart from being discontinuous, is highly nonlinear (compared with the quasi-2D cases of Section 4). A final point must be made concerning Zeeman coupling. Analytical solutions involving imaginary parts of the Hurwitz zeta functions could also be obtained if the Zeeman term is included in the above calculations. In such a case, the energy spectrum (3.85) is modified as follows: 2 2 g * m* kz ε n , kz  (n  1  )  c*  2 4 m 2m*

* with g the gyromagnetic ratio, m* the effective mass and  c*  eB / m*c the effective cyclotron frequency. Although the problem is also completely solvable, we * choose to only report the observation that, for sufficiently large g , we find a

pronounced minimum in total energy as a function of B, i.e., when the gyromagnetic * ratio is g  1.5 we obtain the behaviour shown in Fig. 5.4. Such behaviours originate

from the interplay of QW, Zeeman and LL Physics in the full 3D problem and have not been reported earlier; as already noted in the Introduction, such minima might be important for the design of stable 3D quantum devices, i.e., in cases where the magnetic field can be self-consistently considered as self-generated.

125

3. Electron systems in strong magnetic fields

Fig. 3.46: Energy, as function of B, for g *  1.5 , for the first two windows of B. This minimum might be important in fabrication of small quantum devices.

3.8 Relevance and applicability to the dimensionality crossover in Topological Insulators We have seen with an exact analytical solution and through a detailed analysis of energy interplay that the finite thickness is not as innocent as widely believed or implied. Its presence does not merely provide just another variable and just another label to the wavefunctions and energy spectra. Basically, this is because the Pauli principle can be circumvented momentarily at every step; these steps forming a sequence that leads to interesting and rather unpredictable behaviours. The method that we have followed for determining ground state thermodynamic magnetic quantities, such as the magnetisation, is not only exact but is also based on physically transparent arguments (on energy interplay and comparisons at the single-particle level, without the need of using the density of states). As a reward for this more physical approach, we have found, even for the above conventional systems that special values of thickness induce certain “internal transitions” (i.e., occurring at partial LL filling) that violate the standard de Haas-van Alphen periodicity; transitions that apparently have not been captured by other approaches. However, as an equally important reward, we should stress that because of its simplicity and universality in its line of reasoning, the same method could also be applied to other more involved systems of current interest, such as 3D strong topological insulators (such as Bi2Se3) and its dimensionality crossover to 2D topological insulators (such as HgTe/CdTe wells). To show this, we now briefly turn our attention to the well-known effective four-band model by Zhang et al. [29] that describes the low-energy behaviour of Bi2Se3. Such systems are described by a modified Dirac equation rather than the Schrodinger equation. This leads to very different wavefunctions (with nontrivial topological properties) and energy spectra, where the role of thickness is coupled to the 2D motion; however, the line of reasoning that we have developed and the general method that we have followed can still be applied in a similar manner. All one needs is essentially the one-particle spectrum, which incorporates the effect of thickness, even though this effect might be strongly coupled to the 2D degrees of freedom. Indeed, even for the coupled problem, one could determine the single-particle energy 126

3. Electron systems in strong magnetic fields

for the lowest value of a thickness-related quantum number, then determine the same for the next higher value of this quantum number and then study the comparison between the two energies – looking for cases of crossover between the two that might occur not too far from the Γ-point (kx = ky = 0). If there are also sufficient charge carriers that give a kF that is further away than the crossover point in k-space, this would be a strong indication that effects like the ones discussed above might also be present in these systems as well. We will carry out a quick calculation in the above spirit in the following but only in the thin-film limit, i.e., we will now have massive Dirac Fermions, which is even more relevant to our method because recently, it has been found [30] that for thin films there is a gap opening and the Fermi level does not fall in the gap; hence, surface carriers are present in the electron band with an estimated areal density of ~ 5 × 1016 m-2. In this thin-film limit, we will indeed find theoretical evidence of a clear crossover close to the Γ-point, inside the region of k-space where the Dirac equation is valid and with an estimated kf that is further away – something that shows that for these more exotic systems a more careful study of effects like the ones presented in the present work is probably needed. One can start with the effective model that describes the bulk states near the Γ-point for bulk Bi2Se3 [29]. The Hamiltonian is given by:

 

 

H k  

M k   iA  1 z k I 44    0   A2 k 

 

iA1 z

0

 

M k

A2 k

A2 k

M k

0

iA1 z

 

A2 k   0   . (3.138)  iA1 z   M k  

 

In a basis p1z ,  , p 2z ,  , p1z ,  , p 2z ,  where +(-) stands for even (odd) parity with

 

 

  k  C  D1 2z  D2 k 2 , M k  M  B1 2z  B2 k 2 , k  k x  ik y , k 2  k x2  k y2 with the model parameters having values:

M  0.28eV ,



A1  2.2eV  ,



A2  4.1eV  ,





B1  10eV  , 2



B2  56.6eV  2 ,



C  0.0068eV , D1  1.3eV 2 , D2  19.6eV  2 and with a 4-component trial wavefunction:

    e z (3.139)

127

3. Electron systems in strong magnetic fields

(3.138) has been diagonalised [31] giving ια as functions of E and k [see (5) of [31]]. By inverting them, we obtain E = E(λα,k) and by focusing on the electron band, we obtain:

 

Eel  2 k  0.0068  19.6k 2 

12.830522

2

 0.0784  14.886k 2  3203.56k 4 

7.522

2



11172.4k 2 22

2

 10024

(3.140) Although this problem must be treated numerically for a self-consistent determination of E and ιs, we can immediately check the thin-film limit, where it is found [31] that ι = i nz π/d. By plugging this into (3.140), we obtain the single-particle spectrum for the electron band as a function of k for each nz, namely:

 

Eel  2 k  0.0068  19.6k 2 

12.8305nz2 7.5n 2 11172.4k 2 nz2 n4  0.0784  14.886k 2  3203.56k 4  2 z   100 z4 2 2 d d d d

(3.141) Then, by using a value of d = 10 nm and plotting (3.141) for nz  1 and nz  2 , we indeed find a crossover close to the Γ-point, as shown in Figs. 6.1–6.3.

0.20

E E (meV) 2 eV

E (meV) E 2 eV

0.22

0.22

0.24

0.26

0.23 0.24 0.25 0.26 0.27

0.28 0.28

0.00

0.01

0.02

0.03

0.04

0.05

−𝟏 0 1

kk (ĂA )

Fig. 3.47:

  for

Eel  2 k

0.06

0.00

0.01

0.02

0.03

0.04

0.05

−𝟏1

k k (ĂA ) 0

nz  1 .

Fig. 3.48:

 

Eel  2 k

for

nz  2 .

128

0.06

3. Electron systems in strong magnetic fields

EE (meV) 2 eV

0.20

0.22

0.24

0.26

0.28 0.00

0.01

0.02

0.03

0.04

0.05

0.06

−𝟏 0 1

kk (ĂA ) Fig. 3.49:

  for

Eel  2 k

nz  1 and nz  2

shown together

Moreover, note that, in analogy to Section 3.3, the occupational procedure is similar. For example, a Fermi wavevector for this band is now kF   nA (with nA being the mean surface areal density) for a certain spin configuration. Then by using the 

estimate of density given above, a value of k F  0.04  1 is obtained, which is further on the right of the crossover point in Fig. 3.49, indicating that we have sufficient carriers that might exploit the crossover for internal transitions of the general type studied in this paper. Independently, let us try to examine the first critical value of d where the first transition occurs; however, in this quasi-2D topological insulator, we must now have

Eel 2  k  kF , nz  1  Eel 2  k  0, nz  2  , (3.142) or equivalently 12.8305 7.5 11172.4k F2 1 2 4 0.0068  19.6k   0.0784  14.886k F  3203.56k F  2   100 4  2 2 d d d d 2 F

12.8305  4 7.5  4 16  0.0784   100 4 . (3.143) 2 2 d d d This equation determines the first critical value of d in which the two-dimensional topological insulator starts becoming three-dimensional. By plugging in the estimated 0.0068 

k F above, the solution of (3.143) gives d = 3.86 nm; something that indicates that our tentative value of d (of 10 nm) is indeed in a region where interesting effects might be expected and that, generally speaking, strengthens the necessity for more careful treatment of this system.

129

3. Electron systems in strong magnetic fields

3.9 Thermodynamic properties of Aharonov-Bohm electronic systems We now complete our calculations by studying a 2D electron system that moves in nanorings and nanocylinders in the presence of an Aharonov-Bohm (AB) flux tube penetrating the empty spaces in the interior of the circular area. This system differs dramatically from the previous cases; in the sense that now the magnetic field is completely localized inside the AB tube and cannot reach the particles. But as we know, an AB flux tube may have measurable consequences on the thermodynamics of our system. Both single-electron spectrum and total energy (and also electric current) depend on the ratio  /  0 where Φ is the magnetic flux and  0 is the flux quantum. The most interesting aspect of this work is that all properties depend only on the fractional part of the ratio  /  0 , indicating the standard periodicity with respect to flux quantum.

3.9a Nanoring threaded by a magnetic flux When a magnetic flux Φ is inserted in the empty space of a nanoring of radius R the eigenenergies of the electrons are influenced according to the Aharonov-Bohm effect (AB) as follows: 2

   n  n  , (a.1) 2  2mR  0  2

with n  0, 1, 2.. ,  0  hc / e the flux quantum, and e the electron charge. All thermodynamic properties e.g. total energy, persistent currents, depend only on the fractional part of ratio y   /  0 , (denoted by y from now on) because if this ratio is an integer, the AB effect disappears. The fractional part is a periodic function upon shifting      0 , therefore all thermodynamic properties are also periodic with period exactly one flux quantum. This is exactly the case for small particle number N, while for large numbers the period becomes exactly half flux quantum   0 / 2  as we will show below. To prove the previous statement, we consider a nanoring of radius R threaded by an AB flux Φ in which an electron gas consisting of N particles moves at zero temperature (T=0). The single electron spectrum (a.1), given that the ratio y>0 can be decomposed into its integer and fractional parts, namely y  Int  y   Frac  y  can be written as: n 

2

2mR

2

 n  Int  y   Frac  y  

2

, (a.2)

130

3. Electron systems in strong magnetic fields

where n  Int  y  is just another integer, let‟s call it l  n  Int  y   0, 1, 2... such that (a.2) becomes: n 

2

2mR

2

 l  Frac  y 

2

(a.3)

Now, the ground state clearly depends on the fractional part of y. For example, if Frac  y   1 / 2 then the ground state is located at l  0 while if Frac  y   1 / 2 then the ground state alters to l  1 . This means that two energetic sequences emerge depending on the value of Frac  y  : For Frac  y   1 / 2 the sequence of increasing energy is

l  0, 1, 1, 2, 2... and

for

Frac  y   1 / 2

the sequence becomes

l  1,0, 2, 1, 3, 2... and so on. If the electrons do not interact through their spin

degree of freedom then each state may contain no more than two particles, due to Pauli Exclusion Principle. Also, if the electron number is countable, namely very small, then we must distinguish the following 4 cases regarding N: 1) N  2k , k odd integer, 2) N  2k , k even integer, 3) N  2k  1 , k odd integer, 4) N  2k  1 , k even integer for each of the Frac  y  cases. As a first example we consider the case N  2k with k being an odd integer and Frac  y   1 / 2 . Then, the energetic occupations are shown schematically below:

k= 3

k= 1 l= 1

l= 1

l= 0

l= 0

l= -1

l= -1

131

3. Electron systems in strong magnetic fields

k= 5 l= 2 l= 1 l= 0 l= -1 l= -2 Fig. a1: Occupations for

N  2k ,

k odd integer and

Frac  y   1 / 2

Then, the total energy of the system is a sum over all occupied states:

E

 k 1     2 

2

mR

2

  l  Frac  y 

2



 k 1     2 



2

96mR

2

N  N 2  4  N

2

2

2mR

2

 Frac  y 

2

, (a.4)

2

2 E  N 2  4  Frac  y  2  2  N 96mR 2mR

with the Fermi energy located at l   k  1 / 2 . Note that in the thermodynamic limit N   , R   such that the number density n  N / 2 R remains a constant, the

second term vanishes, and the total energy per electron becomes 2 2  N 2  1 2   n  1 2 k F2 E N2   , (a.5)      N 96mR 2 2m  48R 2  3 2m  2  3 2m 2

where kF   n / 2 the Fermi wavevector in 1D for spinfull electrons. The fact that total energy is independent of the flux in the macroscopic limit can also be verified through density of states, which is a series of delta functions: g E 





  E 

n 



2

2mR

2

n  y

2

  (a.6) 

In the limit R   we may approximate the sum as an integral with respect to continuous variable k  n / R , namely

132

3. Electron systems in strong magnetic fields 



n 



  R  dk

so (a.6) can be written as: 2   2 2 2   y  k   g  E   R  dk  E  k   R dk  E         2m  R   2m     

Substituting u 

2

k2 we have: 2m g E 

2 Rm 2m



 0

du

 E  u u



2 Rm 2mE



 du  E  u   0

2 Rm 2mE

, (a.7)

which is nothing else but the 1D density of states in the thermodynamic limit for noninteracting electrons. Note that the magnetic flux does not appear in the final result, and this indicates the independence of the total energy upon the magnetic flux in that limit, as expected. In fact, as it is easily seen from (a.4), it is not strictly necessary to reach the thermodynamic limit. For a fix radius, increasing electrons number it is clear that the first term will dominate, and the second term vanishes. Below we plot total energy as a function of y. The case Frac  y   1 / 2 will be

𝟐 𝟐 E/NE (𝒉 ) 2 2mR2 N /𝟐𝒎𝑹

included in all graphs.

2.25

2.20

2.15

2.10

2.05

2.00 0.0

0.5

1.0

1.5

2.0

y

Fig. a2: Energy per electron as function of y for N  2k , k odd integer. The period is a flux quantum.

The other case, N  2k with k even and Frac  y   1 / 2 :

133

3. Electron systems in strong magnetic fields

k= 0

k= 2 l= 1

l= 1

l= 0

l= 0

l= -1

l= -1 k= 4 l= 2 l= 1 l= 0 l= -1 l= -2

Fig. a3: Occupations for N  2k , k even integer and

E

2

 k     Frac  y    2 mR  2  mR 2

2

2 E   N 2mR 2

2

k   1 2 

Frac  y   1 / 2

  m  Frac  y 

2

k   1 2 

  N  4 2 N    Frac  y   Frac  y   1  (a.8)  8  48 

134

3. Electron systems in strong magnetic fields

2 2 mR2 𝟐 E 𝟐 /𝟐𝒎𝑹 E/N (𝒉 )

2.00

1.95

1.90

1.85

1.80

1.75 0.0

0.5

1.0

1.5

2.0

y

Fig. a4: Energy per electron as function of y for N  10 . Period is a flux quantum.

The cases where N is even are not interesting, because the period of oscillations is always a flux quantum. The period remains a flux quantum for any particle number because the Fermi level is always completed with two electrons. On the other hand, the case where N is odd is more interesting because there is one unpaired electron at the Fermi level. As the Fermi level increases (namely, when N increases) the period becomes a half flux quantum as we shall see below. Following the same reasoning as above, for the case N  2k  1 with k odd, we find:

E

2

96mR

N  1 N  1 N  3   N  1 2 

2

2

32mR 2

2



 N  1 Frac

4mR 2

 y 

N 2 Frac 2  y  2 2mR

(a.9)

𝟐 E/NE(𝒉𝟐 /𝟐𝒎𝑹 ) 2 2mR2

𝟐 E/NE(𝒉𝟐 2/𝟐𝒎𝑹 ) 2mR2

8.0

7.9

7.8

7.7

7.6

7.5

0.0

0.5

1.0

1.5

y

2.0

49 026

49 024

49 022

49 020

0.0

0.5

1.0

1.5

2.0

y

a

b

Fig. a5: a) Energy as a function of ratio y for N=7. b) For N almost 133 and above, period is a half flux quantum.

135

3. Electron systems in strong magnetic fields

And for N  2k  1 with k even, we find:

3 2 E  N  1 N  5 N  3   N  1 96mR 2 32mR 2 2 2 N 1   Frac y Frac y   Frac  y       2mR 2 2  4mR 2 2

𝟐 E/N E(𝒉𝟐2/𝟐𝒎𝑹 ) 2mR2

𝟐 E/N E(𝒉𝟐2/𝟐𝒎𝑹 ) 2mR2

2

47.2

47.0

46.8

46.6

46.4

0.0

0.5

1.0

1.5

58 414

58 412

58 410

58 408

0.0

2.0

(a.10)

0.5

1.0

1.5

2.0

y

y

a

b

Fig. a6: a) Energy as a function of ratio y for N=9. b) For N almost 141 and above, period is a half flux quantum.

3.9b Nanocylinder threaded by a magnetic flux Now we consider the electronic gas to be confined on a cylinder of small radius R (let the folded direction denoted by x) but macroscopic in the y direction, like a nanotube. The presence of the magnetic flux combined with periodic boundary conditions leads to the following single electron energy spectrum:



2

2mR 2

 nx  y 

2 2



k y2

2m



2

2mR 2

 l  Frac  y  

2 2



k y2

2m

(b.1)

with l  nx  Int  y   0, 1, 2... and y   /  0 . Because energy levels in x-direction are strictly quantized, while those in y direction are continuous, we expect the appearance of 1D Fermi segments for each x-axis level. In the end, taking the thermodynamic limit N   , R   we expect that the presence of flux won‟t alter the results just like in the case of the nanoring. Once again, we deal with the quantum problem of N spinfull and noninteracting electrons (N is fixed) at zero temperature (T=0) moving freely on the cylindrical surface. The most interesting thing about the 136

3. Electron systems in strong magnetic fields

energy spectrum is the fact that quantum number k y is considered as a continuous variable, while quantum number l is strictly discretized. This means that electrons form a one dimensional Fermi segment upon occupying energy levels at y - axis. We make the supposition that the parameters R, Φ, are such that the system occupies q xaxis levels, with q being an even positive integer (q=0,2,4,6…). The second case with q being an odd positive integer will be studied later. We also fix the condition Frac  y   1 / 2 while obtaining the results, and therefore the energetic sequence at x-

axis becomes l  0, 1, 1, 2, 2... The x-axis state labeled with l  0 has the lowest energy. When the following relation is satisfied: 2

k 2f  l  0  2m

with

k 2f  l  0 

the



2

2mR 2

 Frac  y 

Fermi

2

2

 q     Frac  y   (b.2) 2  2mR  2  2

wavevector

defined

through

the

relation

k f  l  0   n l  0 / 2 with n  l  0   n(0) the linear (1D) number density of electrons

at x-axis, namely, when the total energy of the last electron at l=0 equals to the total energy of l=-q/2 for zero Fermi segment then the system occupies no more and no less than q levels at x-axis. Solve (b.2) with respect to radius to find: R

q 2  4qFrac  y 

 n 0

(b.3)

And respectively, for the system to occupy exactly q+1 levels the following relation must be satisfied: 2

k 2f  m  0  2m



2

2mR 2

 Frac  y 

2

2 q 2  4qFrac  y  q   (b4)   Frac  y    R  2mR 2  2  n 0  2

We therefore conclude to the following window of values for the radius R: q 2  4qFrac  y 

 n 0

R

q 2  4qFrac  y 

 n 0

, (b.5)

needed for the system to occupy exactly q x-axis levels. It is important to notice here that for Frac  y   0 the above window shrinks to a single point, indicating that the usual degeneracy is recovered. Note that parameters q,y are considered constants, but the number density n(0) is a function of (R,y). Another condition we must take into account is that the chemical potentials of each x-axis level must be identical, so that the whole system is in an equilibrium state: 137

3. Electron systems in strong magnetic fields 2

k 2f  l  0 



2 2

 Frac  y 

2 2



k 2f  l  1

2m 2mR 2m 2 2 2 k f  l  1 2   1  Frac  y   ..  2  2m 2mR

2



2

 1  Frac  y 

2mR 2 2 k f  l  q / 2  2m

2



 q     Frac  y   2  2mR  2  2

2

Solving the above equations with respect to 1D densities we find an analytical expression for the number density of each x-axis level: n i   n  0  2

2 q q 4  i  Frac  y   Frac 2  y  , i   ,...,  1 (b.6) 2    2 2  R 2

The total 1D density n has to be a constant: 2 2 q / 2 1   Rn  0   2 n   n i       i  Frac  y   Frac  y  (b.7)   R 2 i  q / 2 i  q / 2   2

q / 2 1

We define x   Rn / 2 and the quantity a0   Rn  0  / 2 such that (b.7) becomes: x

q / 2 1

a0 2   i  Frac  y   Frac 2  y  (b.8)



2

i  q / 2

Using (b.5), the quantity a0 is bounded in the range q 2  4qFrac  y  2

 a0 

q 2  4qFrac  y  2

(b.9)

q 2

Note that when Frac  y   0 , then a0  . Finally, using once again (b.5) we conclude to the following window of values of x (which is proportional to the radius of the cylinder) in which the system occupies q x-axis levels:



   q  2i  

q / 2 1

i  q / 2

 q  2i  

4

q / 2 1   Frac  y   x   i  q / 2 

  q  2i 

 q  2i   

4

  Frac  y  (b.10) 

The total energy of the system is: E

2 2 1 N i E i  N i i  Frac  y          f  2  2mR i  q /2  3  q /2 1

After carrying out the math we conclude to: 138

3. Electron systems in strong magnetic fields 3  2 1 q / 2 1  E 1  E f 1D    3   a02  i 2  2iFrac  y  2  2  a02  Frac 2  y   , (b.11) N x  3 x i  q / 2 

with chemical potential located at 

1   q   Frac  y     n  q / 2    2  2m  2  2mR  2  2

2

2

2

We should point out in this point that the total energy of the system is written in units of 1D Fermi enegy in the absence of AB flux, to emphasize on the distinct differences between the two cases. Below are the graphs of total energy and electric current (in units of e F / h ) as functions of the ratio y:

1.1

1.5

2e h Ef

I/N (eεF/h)

0.9

Εf

0.8 0.7 0.6

ΙΝ

ΕΝ

E/N (εF)

1.0

0.5

1.0 0.5 0.0 0.5 1.0

0.4

1.5 0.0

0.5

1.0

1.5

2.0

0.0

0.2

0.4

y

Φ Φο

a

y

0.6

0.8

1.0

y

b

Fig. b1: a) Total energy (in units of 1D Fermi energy as a function of y for x=0.5. b) Persistent current as function of y for x=0.5. The usual periodicity is recovered in the macroscopic limit.

Note that both energy and the electric current (derived from an induction argument I  cdE / d  ) are periodic functions of the ratio y, as expected. Also, observing the graphb b) we can clearly see the standard saw-tooth behavior of the global current (which is related directly to global magnetization) that appears in 2D systems in the interior of magnetic field. This is no surprise; in reality an AB system with an inaccessible magnetic flux and a magnetic system have both similar properties, owing to enclosed fluxes in the electronic orbital motions.

139

3. Electron systems in strong magnetic fields

3.10 Conclusions An exact solution providing analytical expressions for the magnetic thermodynamic functions of an interface or film in a perpendicular magnetic film (with rigid walls) has been presented, in a picture of noninteracting electrons. Interactions were later taken into account by following the same method for the Landau Λ-levels in a picture of noninteracting Composite Fermions. The method used, different from standard density of states methods and grandcanonical and semi-classical approaches, is exact but also physically transparent at every step; hence, providing the possibility of application to more involved systems, such as 3D topological insulators, in which the thickness-related modes are strongly coupled to the planar motion. Even for conventional systems, it has been found that the finite thickness is not as innocent as widely believed or implied. Its presence does not merely provide just another variable and just another label to the wavefunctions and energy spectra. Basically, this is because the Pauli principle can be circumvented momentarily at every step; these steps forming a sequence that leads to interesting and rather unpredictable behaviours. The finite thickness has been found here to induce certain “internal transitions” (at partial Landau Level filling) of magnetisation that are not captured by earlier approaches and that violate the standard de Haas-van Alphen periodicities. The correctness of all these results has been tested against an independent exact analytical solution of the full 3D problem, which apparently, also leads to certain behaviours that have not been reported earlier. For topologically nontrivial systems, evidence that such effects might also be operative in the dimensionality crossover between 3D and 2D topological insulator wells has also been given. This suggests that the versatile method presented here needs to be carefully applied to such systems, a task that can be carried out numerically if the analytical patterns are too involved.

140

3. Electron systems in strong magnetic fields

Chapter References [1] M.Z. Hasan and C.L. Kane, Rev. Mod. Phys. 82(4), 3045 (2010) [2] X.L. Qi and S.C. Zhang, Rev. Mod. Phys. 83(4), 1057 (2011) [3] Y. Hatsugai, Phys. Rev. Lett. 71, 3697 (1993) [4] D.J. Thouless, M. Kohmoto, P. Nightingale and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982) [5] M. Nakahara, Geometry, Topology and Physics, 2nd Edn. Taylor & Francis (2003) [6] D. Shoenberg, Magnetic Oscillations in Metals, Cambridge University Press (1984) [7] H. Cao, J. Tian, I. Miotkowski, T. Shen, J. Hu, S. Qiao and Y.P. Chen, Phys. Rev. Lett. 108, 216803 (2012) [8] Z.D. Kvon, E.B. Olshanetsky, D.A. Kozlov, E. Novik, N.N. Mikhailov and S.A. Dvoretsky, Fiz. Nizk. Temp. 37, 258 (2011) [Low Temp. Phys. 37, 202 (2011)] [9] J. Nuebler, B. Friess, V. Umansky, B. Rosenow, M. Heiblum, K. von Klitzing and J. Smet, Phys. Rev. Lett. 108, 046804 (2012) [10] Y. Liu, D. Kamburov, M. Shayegan, L.N. Pfeiffer, K.W. West and K.W. Baldwin, Phys. Rev. Lett. 107, 176805 (2011) [11] M.R. Peterson, T. Jolicoeur and S. Das Sarma, Phys. Rev. B 78, 155308 (2008) [12] K. Takei, H. Fang, B. Kumar, R. Kapadia, Q. Gao, M. Madsen, H.S. Kim, C.H. Liu, E. Plis, S. Krishna, H.A. Bechtel, J. Guo and A. Javey, Nano Lett., 11 (11), 5008 (2011) [13] A. Ohtomo and H.Y. Hwang, Nature 427 (6973), 423 (2004) [14] C.X. Liu, H. Zhang, B. Yan, X.L. Qi, T. Frauenheim, X. Dai, Z. Fang and S.C. Zhang, Phys. Rev. B 81, 041307(R) (2010) [15] M.A. Wilde, M.P. Schwarz, C. Heyn, D. Heitmann, D. Grundler, D. Reuter and A.D. Wiecket, Phys. Rev. B 73, 125325 (2006) [16] T. Chakraborty and P. Pietilainen, The Quantum Hall Effects, 2nd Edn. Springer (1995) [17] V.M. Apalkov and T. Chakraborty, Phys. Rev. Lett., 107, 186801 (2011) [18] S.K.F. Islam and T.K. Ghosh, Journ. Phys.: Condens. Mat. 24, 035302 (2012) [19] X.F. Wang and P. Vasilopoulos, Phys. Rev. B 67, 085313 (2003) [20] K. Moulopoulos and M. Aspromalli (unpublished); the variational proof is actually contained in an early Master‟s Thesis of M. Aspromalli (“Energy studies of quantal electrons inside a magnetic field: from astrophysical systems to semiconducting heterostructures”, University of Cyprus (2004) (in Greek)) 141

3. Electron systems in strong magnetic fields

[21] M.C. Tringides, M. Jalochowski and E. Bauer, Phys. Today 60(4), 50 (2007) [22] W.A. Atkinson and A.J. Slavin, Am. J. Phys. 76(12), 1099 (2008) [23] V.D. Dymnikov, Fiz. Tverd. Tela 53, 847 (2011) [Phys. Solid State 53, 901 (2011)] [24] S. Das Sarma and A. Pinczuk (Eds.), Perspectives in Quantum Hall Effects, Wiley-VCH (1997) [25] J. Jain, Composite Fermions, Cambridge University Press (2007) [26] D. Lai, Rev. Mod. Phys. 73, 629 (2001) [27] Y.B. Suh, Ann. Phys. 94, 243 (1975) [28] C.O. Dib and O. Espinosa, Nucl. Phys. B 612, 492 (2001) [29] H. Zhang, C.X. Liu, X.L. Qi, X. Dai, Z. Fang and S.C. Zhang, Nat. Phys. 5, 438 (2009) [30] A.A. Taskin, S. Sasaki, K. Segawa and Y. Ando, Phys. Rev. Lett. 109, 066803 (2012) [31] W.Y. Shan, H.Z. Lu and S.Q. Shen, New Journ. Phys. 12, 043048 (2010)

142

4. Quantum phase variations in response to proper changes of vector potentials

Chapter 4 Quantum phase variations in response to proper changes of vector potentials Due to the importance of gauge invariance in all fields of Physics and motivated by an article written almost three decades ago that warns against a naive handling of gauge transformations in the Landau Level problem (a quantum electron moving in a spatially uniform magnetic field – a key problem in the area of the Integer Quantum Hall Effect), we point out a proper use of the generators of dynamical symmetries combined with gauge transformation methods to easily obtain exact analytical solutions for all Landau Level-wavefunctions in arbitrary gauge. Our method is different from the old argument and provides solutions in an easier manner and in a broader set of geometries and gauges. We then go further by showing that a similar methodology can be made to apply to the more difficult case of a spatially nonuniform magnetic field (where closed analytical results are rare), in which case the various generators (pseudomomentum and pseudo-angular momentum) appear as line integrals of the inhomogeneous magnetic field; we give closed analytical solutions for all cases, and we show how the old and rather forgotten Bawin & Burnel gauge shows up naturally as a “reference gauge” in all solutions.

143

4. Quantum phase variations in response to proper changes of vector potentials

4.1 Introduction and motivation It is well-known from Weyl‟s early work [2] (but also from independent proposals by Schrodinger, Fock and London [3]), that the structure of the time-dependent Schrodinger equation remains the same upon change of potentials (through A2  A1   and V2  V1   / ct ) if at the same time the wavefunction Ψ1( r ,t) (solution of the time-dependent Schrodinger equation for the set of potentials ( A1 ,V1)) is replaced by Ψ2( r ,t)=Ψ1( r ,t) − , with Λ generally being Λ( r ,t). Hence, the formal solution (i.e. before any imposition of boundary conditions) of this equation for the set of potentials (A2, V2) is the above Ψ2 . This formal connection between two systems (in which the same particle of charge -e moves in the above two different sets of potentials) may be seen as a mapping between the two problems, and this has been exploited in the past [4,5] for advancing new solutions of the t-dependent Aharonov-Bohm effect, both magnetic and electric [6], an area that seems to currently be growing rapidly [7]. In a similar manner, if we look at the t-independent Schrodinger equation, we can see that a similar formal mapping is also valid for the stationary state solutions a well, i.e. we now simply have Ψ2( r )=Ψ1( r ) − (now only under the change A2  A1   , with Λ being time-independent), and with the understanding that the corresponding solutions Ψ1 and Ψ2 will both belong to the same energy. Long time ago Swenson [1] criticized the above result, pointing out that there may be subtleties associated with cases of degeneracy; indeed, if we have many wavefunctions corresponding to the same energy, then care should be taken with respect to their grouping together (by using as a label the eigenvalues of some other constant of motion, apart from the Hamiltonian, e.g. the generators of the dynamical symmetry that is responsible for these degeneracies – such as the pseudomomentum for the magnetic translations, see below). In particular for the Landau problem, Swenson demonstrated that it is erroneous to apply the above simple result to the well-known wavefunction-solutions of the static Schrodinger equation in the two standard Landau gauges ( A1 =-yB iˆ and A2 =xB ˆj , with B the modulus of the uniform magnetic field and iˆ, ˆj unit vectors along x,y directions). He actually showed that the mapping between the two problems is more complicated (i.e. involving superposition of different Ψ1‟s that corresponded to different values of a canonical momentum but all to the same energy as the one of Ψ2), and he then drew the general conclusion that in cases of degeneracy, if one writes the simple mapping (with the single extra phase factor), then this produces a solution that is not necessarily a single stationary state of the 2nd system. Here we want to point out that if one is careful to map wavefunctions that correspond not only to the same energy but simultaneously to the same value of another constant of motion, such as a given component of pseudomomentum (a component which is chosen to be the same in both gauges), then the simple mapping is valid again. In Swenson‟s case, it happened that the Ψ1 and Ψ2 used, that are the standard solutions of this problem in the usual two Landau gauges, did not correspond to the same component of pseudomomentum (as will be reminded below), hence the arising of a more complicated result. The use of the same value of another constant of motion would simplify the problem, and this simplification can be exploited, as we will see, 144

4. Quantum phase variations in response to proper changes of vector potentials

in order to immediately write down analytical solutions for any gauge, to answer immediately a question on symmetric gauge posed at the end of Swenson‟s article, and finally, to go further by drawing general conclusions in the problem of nonuniform magnetic fields where analytical results are rather rare.

4.2 The Landau - Level system A key quantity in the Landau problem (for a quantum particle of charge –e), although a bit forgotten through the years, is what had earlier been called the pseudomomentum, defined as e e A  r  B (4.1) c c where B  Bz and A is the vector potential defined by  A  B (4.2), which A apart from (4.2) we intentionally leave unspecified. For a classical system it is immediately obvious that the vector quantity K is indeed a constant of motion: from Newton‟s equation of motion with the Lorentz force, namely K  p





d  / dt  d p  eA / c / dt  ev  B / c ,

by

simply

writing

v  dr / dt

we

immediately obtain dK / dt  0 . For a quantum system like the one we are interested in, it is also straightforward to see that this vector operator quantity K is a constant of motion: a direct calculation gives  H , K   0 (and it takes a little more to show that K is also the generator of translations inside a uniform magnetic field, in the sense that the magnetic translation operator that translates by a classical vector a is eiK .a / ). [This pseudomomentum is rarely mentioned nowadays, having yielded its place to the so-called guiding center operators – see later below about the precise relation between the two.] Our target is to find common eigenstates between H and one component of K for any vector potential that satisfies (4.2) [note that different components of K do not commute]. Let us first choose the x-component of K , that is e e K x  px  Ax  yB and solve its eigenvalue equation, namely c c K x   x, y   k x   x, y   i

 e e  yB  Ax  x, y    k x  (4.3) x c c

where k x is the continuous eigenvalue of K x . The solution of the above differential equation is k

 x  eyB  x  c  

i

  x, y   Ce



e

ie x  dx ' Ax  x ', y  c0

i

f  y   Ce

ie x

kx x

e

  dx ' Ax  x ', y  i eyBx c e c0 f

 y (4.4)

145

4. Quantum phase variations in response to proper changes of vector potentials

with f  y  a function of y, to be determined. This wavefunction must also satisfy the static Schrodinger equation H    with Hamiltonian that contains a minimal substitution (namely the kinematic momentum   p  eA / c in place of the canonical momentum p ), which upon expansion of Π2 gives

p2 e e e2 H  A. p  i . A  A2 2 2m mc 2mc 2mc for an arbitrary vector potential A that satisfies eq. (4.2). This leads, after several algebraic manipulations, to the equation that f  y  must satisfy, namely 2 2 2 f e f  1  eyB  1 e 2 e Ay  0, y     iA 0, y  k   A 0, y  i      f y x y     2m y 2 mc y  2m  c  2m  c  2mc y  2

 Ef

. (4.5) 

ie y  dy ' Ay  0, y ' c0

Finally, upon change of variable according to f  y   e Y  y  all terms containing components of vector potential cancel out and we obtain a very simple equation that Y(y) must satisfy, namely

 2Y 1  eyB     kx   Y  EY , (4.6) 2 2m y 2m  c  2

2

which is a one-dimensional shifted harmonic oscillator with the well-known solutions, Hermite polynomials times a Gaussian, and with the standard harmonic oscillator energies. The total solution can then be written (from (4.4)) as

  x, y   Ce

k  i  x  eyB  x c  



e

ie x  dx ' Ax  x ', y  c0

i

f  y   Ce

i e i  e ce

kx x

kx x

ie  x



y

   dx ' Ax  x ', y   dy ' Ay  0, y '   i eyBx c 0  0 Y y c e e   

Y  y

,



 y Yo 

(4.7) 2

 y  Yo  2lB2 Hn  and Yo  ck x / eB =kxlB2 being the well-known e  lB  guiding center operator eigenvalue (and with lB denoting the so-called magnetic length). [We remind the reader that, in general, the guiding center vector operator R0 (defined through the same combination of positions and momenta that give the point (x0,y0) of the center of the classical circular motion) is related to the pseudomomentum K through K  eB  R0 / c .] It is directly seen that the phase with Y  y 

146

4. Quantum phase variations in response to proper changes of vector potentials

factor Λ appearing in (4.7) contains line integrals of the vector potential and a flux of magnetic field B  Bzˆ , namely x

y

x y

x

y

0

0

0 0

0

0

  x, y   yBx   dx ' Ax  x ', y    dy ' Ay  0, y '    dxdyB   dx ' Ax  x ', y    dy ' Ay  0, y ' , (4.8) and one notes that its gradient is   x, y   Byiˆ  A  x, y  . It is therefore clearly seen that (4.7) is the solution of Schrodinger equation that maps the Landau gauge A   Byiˆ (whose solution is the 2nd and 3rd factor of the last part of (1.7)) to an arbitrary vector potential A  x, y  in the sense discussed in the Introduction (indeed there is a gauge transformation that connects the two gauges, namely A  x, y    Byiˆ   as already said above). Note also that the form (4.8) involves a path (in the line integral of A ) that connects the point (0,0) to the point (x,y) in a continuous way (and in place of (0,0) we could have more generally had an arbitrary (x0,y0) – see later below), and apart from this line integral it also involves an additional flux contribution that has the general form of the results in [4,5]. Similarly, if we had chosen to find the simultaneous eigenfunctions of H and K y we would end up with the result: y x  xy  k y y i e    dxdyB  dx ' Ax  x ',0   dy ' Ay  x, y '    i c  0 0 0 0 

  x, y   Ce

e

xy

x

y

00

0

0

X

 x ,

with the phase factor   x, y     dxdyB   dx ' Ax  x ', 0    dy ' Ay  x, y ' and with   x, y    Bxjˆ  A  x, y  which maps the problem of the second Landau gauge

A  Bxjˆ to an arbitrary vector potential A  x, y  . So far all the results are for an

arbitrary vector potential A . If we now choose a specific gauge for A , let us say a Landau gauge, namely A1  xBjˆ the solution (4.7) reads: k



A1

 x, y   Ce

 x  eyB  x  c  

i 



e

ie x  dx ' Ax  x ', y  c0

i

f  y   Ce

kx x

e

i eyBx c Y  y

(4.9)

And if we choose another Landau gauge, i.e. A2   yBiˆ equation (4.7) gives: k

 A2  x, y   Ce

 x  eyB  x c  

i 



e

ie x  dx ' Ax  x ', y  c0

i

 Ce

i

f  y   Ce

kx x

kx x

ieB  x



   dx 'y  i eyBx c 0   c e e Y  y

(4.10)

Y  y

which, incidentally, is the standard textbook solution in this latter gauge.

147

4. Quantum phase variations in response to proper changes of vector potentials

Note that (4.9) and (4.10) only differ by a certain phase factor, namely 

A2

i eyBx A  x, y   e c  1

 x, y 

(4.11) (and this phase indeed contains the correct Λ for

a gauge change from A1 to A2 , namely   A2  A1 ); hence we observe that different wavefunctions for different gauges may differ by a single phase factor (Weyl‟s result) as long as the same component (of the constant of motion) is chosen to be diagonalized with the Hamiltonian. This contradicts Swenson‟s [1] results on gauge changes, where he states that a wavefunction that belongs to a specific gauge must be written as a linear combination of wavefunctions in another gauge. Swenson‟s result is true only in case that in the two gauges, two different operators (components of the constant of motion) are used to be simultaneously diagonalized with the Hamiltonian. And we point out that even in this Swenson‟s case, the correct result is actually not the one given in eq. [18] of ref. [1], but k y2 



Uk

k y2 

y2

, ky1

 2  x, y  dk  e

i

e xyB c

y2

m ,k y 2

1  x, y  m ,k x1

with the matrix elements U (ignored in [1]) being

Uk

y2

,kx1

ˆ m, kx1   m, ky2 U 1 2

i e xyΒ   x, y e c . 1  x, y  dxdy   2 x  y  m, k

x  y 

m, k y 2

x1

which finally comes out to be

Uk

y2

,kx1



2

i ˆ m, kx1  i mlB  1  e m, ky2 U   1 2 

eB y x   c



with (x0,y0) being an arbitrary point for the lower limit of integrations – see more general discussion on this after the inhomogeneous field cases in the final section. (The corresponding rather long calculation that gives the above general correction of Swenson‟s result has been worked out in detail as a part of ref. [8].)

4.3 Homogeneous magnetic field – Polar coordinates Similar results can be obtained in polar coordinates, where the conserved quantity turns out to be what could be called a pseudo-angular momentum eB 2 Lz   r     r (4.12) with  the kinematic momentum   p  eA / c in two z 2c dimensions (see [8]). This pseudo-angular momentum turns out to be the generator of rotations around the z-axis. It is straightforward to prove that  H , Lz   0 , hence this quantity remains a constant of motion and we may choose one of its components to be simultaneously diagonalised with H [note that the two components of  do not commute with each other]. Note also that the pseudo-angular momentum is defined

148

4. Quantum phase variations in response to proper changes of vector potentials

with an arbitrary vector potential and it is gauge invariant [8]. It can be shown, after considerable algebra, that the correct wavefunction may be written as r  ie  2    rd ' A  r , '   dr ' Ar  r ,0   eBr  i i c 0  0

  r ,    Ce

e

2 c

e



R

r 

(4.13)

where λ is the eigenvalue of the pseudo-angular momentum operator and R  r  a function of r that satisfies the following differential equation 2

1   R  1  eBr 2   r      R  ER (4.14)   2m r r  r  2mr 2  2c  2

which is the Laguerre equation expressed in polar coordinates. Note that the following commutation relation between the pseudomomentum and pseudo-angular momentum holds  K , Lz   i  

 zˆ  K  (4.15)

Using the Landau gauge, A1  xBjˆ  A  rB cos 2  and Ar  rB cos  sin  the solution (4.13) gives



A1

i  i

 r ,    Ce e

eBr 2 sin2 4 c R  r  (4.16)

A2   yBiˆ  A  rB cos 2  A  rB sin 2  ,

While, in the other Landau gauge

Ar  rB cos  sin  , and eq. (4.13) gives 

A2

i  i

 r ,    Ce e

eBr 2 sin2 4 c R  r  . (4.17)

We can clearly see that (4.16) and (4.17) are again connected to each other through a simple gauge transformation that conserves the pseudo-angular momentum, with the Br 2 sin 2 (that indeed takes us from A1 to A2). correct gauge function    Bxy   2 It is also easy to see that the same method applied for the mapping from one of the Landau gauges to the symmetric gauge A3  B  r / 2 (which in Cartesian coordinates is A3   yBiˆ / 2  xBjˆ / 2 ) will immediately give a proper answer to the question posed at the end of Swenson‟s article, something that we leave to the reader.

149

4. Quantum phase variations in response to proper changes of vector potentials

4.4 Inhomogeneous magnetic field In classical mechanics the motion in a inhomogeneous magnetic field B  r  is described by Newton‟s second law m

 dv e     v  Br  , (4.18) dt c

   where m is the mass, v is the velocity and e the charge of the particle. B r  is an r -dependent static magnetic field. The energy of the particle is a conserved quantity, irrespective of the spatial structure of the magnetic field, namely E





1 2 dE dv e mv   mv .   v . v  B  r   0 , (4.19) 2 dt dt c

because the particle accelerates in a direction that is always perpendicular to its velocity. However, not only the energy is conserved; under certain circumstances there are other quantities that are also constants of the motion, like a generalized type of pseudo-angular momentum, defined now by the relation



Lz  r  



z



e B  r  r .dr  (4.20) c C

in polar coordinates, or the pseudomomentum in Cartesian coordinates, now defined through K 

e dr  B  r  , c

(4.21)

  all these quantities being defined in a 2D plane perpendicular to B r  which is oriented parallel to the z axis. In general, the previous constants are expressed as line integrals and therefore they cannot quite represent a well-defined constant of motion. But if the integrand is a curl-free quantity, i.e. when it happens that  B B     B  r  r   B  r    r  r  B  r    r  B  r     x y  zˆ , x  0  y

or,

in

polar

coordinates

   B   r Br   zˆ  becomes zero (i.e. if the field has cylindrical symmetry), then Lz becomes a welldefined constant of motion, that can be written as



Lz  r  



r

z



e B  r ' r ' dr ' (4.22) c 0

The above can be proven directly using the commutator 150

4. Quantum phase variations in response to proper changes of vector potentials

r  e   H , L  H , x   y   H ,  z    B  r ' r ' dr ' =0, y x c 0 

with H 

2  2x  y . This allows us to search for common eigenstates for H , Lz . Using  2m 2m

then eq. (4.20) expressed in polar coordinates we have  e e  rA   B  r ' r ' dr ' (4.23)  c c0 r

Lz  i

i 

and with the ansatz   r ,    Ce

e

ie r  r ' dr ' B  r '  c0



e

ie   rd ' A  r , ' c0

f r 

we can solve

the eigenvalue equation Lz   r ,     r ,  , with ι being an eigenvalue of L z . Substituting this ansatz in the static Schrodinger equation with Hamiltonian

H

p2 e e e2  A. p  i . A  A2 , 2m mc 2mc 2mc 2

and after a number of algebraic manipulations we finally obtain the following result (see Appendix A for the corresponding derivation) 

  X r 2mr r  r 2

1   2 2 mr 

2

  er     r ' dr ' B  r '   X  EX   c0  

, (4.24)

which is a generalized Laguerre equation. The full wavefunction can then be written as: i 

  r ,    Ce

e

ie r  r ' dr ' B  r '  c0



e

r  ie    rd ' A  r , '   dr ' Ar  r ',0   c 0  0  X

 r  (4.25)

This is a pure quantum solution of the time-independent Schrodinger equation for a general radially-varying magnetic field in a general gauge A . Note that the appearing phase factor again constitutes of a magnetic flux plus a line integral (along     a path) of the vector potential A . If Br   B , namely, when the magnetic field is homogeneous, eq. (4.24) becomes 

  X r 2mr r  r 2

1   2  2mr

2

 eBr 2      X  EX 2c  

(4.26)

which is the same with eq. (4.14) that we obtained in last section for a uniform B, with solution (eq. (4.25))

151

4. Quantum phase variations in response to proper changes of vector potentials

  r,   C

i 

e e

ieBr 2  ie  rd ' A  r , ' r dr ' A  r ',0  r   0  X r , (4.27) 2 c e c  0  

where X r  are Laguerre polynomials. In addition, note that when the vector potential has only azimuthal coordinate, namely r

1 dr ' r ' B  r '  r 0

A 

then

the

pseudo-angular

momentum

reads

 e e    dr ' r ' B  r '   B  r ' r ' dr '  i , which is equal to the canonical  c 0 c0  r

Lz  i

and Ar  0 (4.28)

r

angular momentum; the wavefunction (4.27) then becomes simply   r,   C

i 

e

X r 

(4.29)

which must be single-valued upon azimuthal trips by 2π, and therefore the angular momentum remains quantized (λ is an integer multiple of ħ, as expected). As for the second constant of motion, the generalized pseudomomentum K 

e dr  B (4.30) c

which is again path-dependent, it is convenient to solve the problem using magnetic   fields that depend only on a single spatial variable, for example B  B  x  , in which case we obtain e K x   x  B  x  y (4.31) c x e K y   y   B  x '  dx ' (4.32) c0

Direct algebraic manipulations give  H , K y   0 , while for the x coordinate we get

H , K x   

i e  B B  xy y  x   0 (4.33)  2mc  x x 

If the magnetic field were solely dependent on y, B=B(y), then we would have H , K x   0 and H , K y  0 . Taking this latter case as an example, we can therefore





choose H and K x and try to find their common eigenfunctions, which turn out to be the following

152

4. Quantum phase variations in response to proper changes of vector potentials

k

i 

  x, y   Ce 

x

x

e y B y ' dy '  x  ie  dx ' Ax  x ', y   c c 0  e 0 f ie  x

i

 Ce

 y 

y

y kx x i ex B  y ' dy '    dx ' Ax  x ', y   dy ' Ay  0, y '  c 0  c 0 0 Y e 

e

 y (4.34)

with Y satisfying 2

y   2Y 1  e   k  B y ' dy '     Y  EY (4.35) x 2m y 2 2m  c 0  2

and with   kx   the pseudomomentum Kx -eigenvalues. Note that the lower limit of the integrals appearing in these equations can arbitrarily be any constant number (an initial point) x 0 , y 0  , the final results being x

k x i x i

  x , y   Ce

e



y

ie e x  x0  y    dx ' Ax  x ', y   dy ' Ay  x0 , y '  B  y ' dy ' c x  c y0 y0 Y e 0

i

 Ce

kx x

e

 y

i e   x , y  c Y  y

(4.36)

with   x, y  

x y

x

x0 y0

x0

y

  B  y ' dx ' dy '   dx ' A  x ', y    dy ' A  x , y ' . x

y

0

Note again the

y0

basic structure of a magnetic flux (a nonlocal influence of B-fields on the observationpoint (x,y)) plus a line integral of potentials along a smooth connected path. This result in simply-connected space must be a single-valued phase function, and with Y satisfying 2

y   2Y 1  e   k  B y ' dy '  x  Y  EY .   2  2m y 2m  c y0  2

Eq. (4.36) represents a complete solution of the static Schrodinger equation that has a well-defined x-component of the pseudomomentum. One notes that  is indeed our arbitrary A minus the Bawin & Burnel gauge [9], hence the above Λ takes us from the Bawin-Burnel gauge (viewed as a “universal reference gauge”) to any arbitrary gauge A that one wishes. We should point out that considerations similar to the ones of last section (regarding wavefunctions defined in different gauges) apply here as well. For example, in the gauge Ax  0 and Ay  xB  y  the phase factor in eq. (4.36) becomes x y

y

y

x0 y0

y0

y0

  x, y     B  y '  dx ' dy '   dy ' x0 B  y '   x  dy ' B  y '  (4.37)

153

4. Quantum phase variations in response to proper changes of vector potentials

and (4.36) reads i

  x, y   Ce

y kx x i e x  dy ' B y ' c y0

Y  y  , (4.38)

e

y

while in the gauge Ax    dy ' B  y '  and Ay  0 (4.36) gives y0

i

  x, y   Ce

kx x

Y  y  (4.39)

Note that, once again, (4.38) and (4.39) only differ by the correct (single) phase factor (that is straightforward to see that it connects the two gauges). We see therefore that if one is sufficiently careful, the cautionary remark of Swenson does not apply, even in cases of inhomogeneous magnetic fields. Finally, before ending this article, we find it useful to include a couple of comments that we find interesting for further study (related to the role of the point (x0,y0)), one on the probability flux vector (quantum mechanical current) in connection to the Hellmann-Feynman theorem, and one on Berry‟s phases with respect to adiabatic and cyclic variations of the point (x0,y0).

4.5a Probability current The local quantum probability current density (or probability flux vector) is defined as    i e 2   *   *   J loc  Re  *    A  (4.40)   m  2m mc 

that, for this particular case, has components y  2 i   * 1 e 2  e *  J locx  y         A   k  B y ' dy '   Y  y   x x    2m  x x  mc m  c y0  (4.41)

J locy 

 i   *   e i    2    *    Ay   Y  y  Y *  y   Y *  y  Y  y    2m  y y  mc 2m  y y  (4.42)

Applying the well-known Hellmann-Feynman theorem [10] with respect to parameter y0 we get y  H eB  y0   e   k x   B  y ' dy '  (4.43)  y0 mc  c y0 

154

4. Quantum phase variations in response to proper changes of vector potentials

and therefore E H  y0 y0

 E eB Y0     dyJ locx  y  (4.44) y0 c 

We see that the eigenenergies depend explicitly on y0 (which is actually not a surprise, although changing y0 is equivalent to changing the vector potential by a

constant, see ref. [11]) except when B  y0   0 (or, for B uniform, whenever 

 dyJ  y   0 is satisfied). locx



4.5b Berry’s phase Using the general gauge-wavefunction (4.36) i

  x, y   Ce with   x, y  

kx x

i e  x, y  c Y  y

e

x y

x

y

x0 y0

x0

y0



 B  y ' dx ' dy '   dx ' Ax  x ', y    dy ' Ay  x0 , y '

we can interpret the initial point  x0 , y0  as a slowly varying parameter (or, equivalently, we have a parameter-dependent vector potential). The Berry‟s phase [12] picked up by the particle‟s wavefunction during cyclic adiabatic changes of  x0 , y0  is

  i  dR.   R  , with R   x0 , y0  (4.45) Performing the necessary calculations we have i  e   i   Ce y0 c y0

and

with

kx x

e

i e   x , y  c

Y (4.46) y0

 e   i  , (4.47) x0 c x0

    B  y0  x  x0   Ay  x0 , y0  and   Ax  x0 , y0  (4.48) y0 x0

i  e  i  B  y0  x  x0   Ay  x0 , y0    Ce y0 c

kx x

e

i e   x , y  c

Y (4.49) y0 155

4. Quantum phase variations in response to proper changes of vector potentials

 e  i Ax  x0 , y0   (4.50) x0 c

Choosing B  y0   0 (i.e. the initial point is outside the magnetic field), we have

Y / y0  0 and the Berry‟s phase can be written as   i  dR.   R   

e e e dx0 Ax  x0 , y0    dy0 Ay  x0 , y0     c c c

 dR.A , (4.51)

which is equal to the Aharonov-Bohm phase [6], picked up in the parameter-based cyclic loop. The above is valid for the flat 2D problem. If however the above system has periodic boundary conditions, it is “compactified” and it can equivalently be folded into a cylinder; then the Berry‟s phase picked up by trips around the empty space is an interesting story, to be told elsewhere: apart from the Aharonov-Bohm phase, the full Berry‟s phase also contains a term containing the global electric current [13].

4.6 Inclusion of a homogeneous Electric field When a homogeneous electric field is added to the system, eq. (4.1) becomes an invariant operator: K  p

e e AB  r  B  eEt (4.52) c c

and obeys the following relation (see Chapter 2) K i    H , K  (4.53) t K with  eE . In this case, the y-component of K and H do not have the same t eigenfunctions anymore, but the two eigenfunctions differ by a time dependent phase factor. If we choose the E-field to point in the x-direction, we can use the eigenfunction of K y to find the eigenfunction of the Hamiltonian, which is:

i

  x, y   e

ky y

y x   i e  xyB   dx ' Ax  x ',0  dy ' Ay  x, y 'ctV  xctEx   c  0 0 

e

g  x, t 

(4.54)

with 2  2 g  1  eBx  dg   k   exE  (4.55) xg   y  2 2m x  2m  c  dt  2

156

4. Quantum phase variations in response to proper changes of vector potentials

The usefulness of this result is now shown: We know that for the problem of a particle that moves in homogeneous magnetic and electric fields the choice of vector potential is important. For example, if the electric field is at x-direction, and the magnetic field is at z-direction and described by the Landau gauge Ay  Bx , the Hamiltonian is H





2 1 p  eBxjˆ / c  exEx (4.56) 2m

In this case, one can write down the Hamiltonian in the more compact form 2

 p y eB   mc 2 Ex2 cp y mc 2  x  eE   Ex (4.57)    x 2 2  2 e B mc 2 B B    And, by observing that p y is a constant of motion, the eigenenergies are p 2 e2 B 2 H x  2m 2mc 2

mc 2 Ex2 cp y    (n  1/ 2)   Ex (4.58) 2B2 B and wavefunctions in the x-direction are the Hermite polynomials. One then wonders what exactly the solution in the other Landau gauge is (that describes the same magnetic field) Ax   By 2 1 H p  eByiˆ / c  exE x (4.59) 2m Now one can not factorize the potential term into a perfect square! The solution of this equation is then given by (4.54) and the eigenenergies are given by (4.58). Equation (4.52) can be also used in the special case of spatial inhomogeneities of the magnetic and electric fields, but this needs better clarification and we leave it for the future.





4.7 Conclusions and Discussion We have demonstrated that, when working with Landau gauges (or in general with any other gauge), the wavefunction that corresponds to one gauge can be transformed to a wavefunction that corresponds to another gauge by a simple (single-phase) gauge transformation, such as the standard one of Weyl, as long as the same component of pseudomomentum (or pseudo-angular momentum) is simultaneously diagonalized with H in either of the two gauges (before and after the transformation). Alternatively, if one chooses to solve the problem using different components of K or Lz , then the two wavefunctions will not be connected by a single-phase relation, but through a linear combination on all quantum numbers that lie within a single degeneracysubspace (i.e. correspond to the same energy), and this is what occurred in Swenson‟s warning article [1]. Similar results hold even in the more difficult case of inhomogeneous magnetic fields that Swenson did not consider, and in such cases we additionally calculated certain Berry‟s phases (for the flat 2D Landau problem) upon small variations of the location of the initial point (that can be taken as the origin of coordinates), something that is equivalent to adiabatically changing the vector potential of the problem. [This looks like a rather innocent gauge transformation; 157

4. Quantum phase variations in response to proper changes of vector potentials

note, however, that if we had periodic boundary conditions, then the system would be compactified, i.e. it would be equivalent to one that is being folded along the periodicity direction; this change of initial point would then no longer be so innocent – it would correspond to a singular gauge transformation with subtle physical consequences, the simplest one being a gauge proximity effect [11].] In cases of spatially nonuniform fields in general, when both K and Lz are expressed as line integrals and therefore depend on the path connecting the initial and final points, extra care must be taken in the sense that the integrand quantities must be curl free. Independence from the path of integration then means that these quantities are welldefined constants of motion. We demonstrated in detail how utilization of such constants of motion can quickly lead to general solutions of the Schrodinger equation in certain nonuniform-field cases, making apparent how a careful use of gauge transformation techniques (combined with proper use of the generators of dynamical symmetries) can be a powerful tool for the quick solution of difficult problems of this type. Finally, it is useful to point out some alternative formulation [14] that has the advantage of making gauge-invariance explicit without the use of wavefunctions, but with the use of the probability flow. We saw in the present work that the quantum numbers (Landau level indices and degeneracy quantum number), or equivalently the constants of motion, are gauge-invariant physical quantities, and this was exploited here in terms of wave functions by using the pseudo-momentum operator. We should point out, however, that the hydrodynamic formulation of quantum mechanics, first presented by Madelung [15], where the basic object is by definition no more the wave function but the combination of the mass density (typically proportional to the modulus square of the wave function) and a flow velocity (related to the phase gradient), allows one to write down from the very beginning a system of equations from which we can derive afterwards the constants of motion/quantum numbers without resorting to any gauge, thus making their gauge-invariant nature explicit. Such a method was developed in ref. [14], and although it did not address the case of a non-uniform magnetic field, it is clear that this Madelung decomposition can again be very useful/convenient to discuss gauge issues and to identify the relevant constants of motion for this more complicated problem, depending on the spatial dependence of the inhomogeneous magnetic field. This has not been tried up to now and it deserves serious consideration.

Appendix A – Derivation of eq. (4.24) We start with the minimally substituted Hamiltonian p2 e e e2  A. p  i . A  A2 2 2m mc 2mc 2mc 2 i eA. e e2  2  i . A  A2 2 2m mc 2mc 2mc H

with  

 1  1     1 2 1  2 1 2 rˆ  ˆ and 2  r       r r  r r  r  r 2  2 r r r 2 r 2  2 158

4. Quantum phase variations in response to proper changes of vector potentials

We then calculate the necessary derivatives, namely i 



 ie ie   rB  r     d '  rA  r ,  '    Ce r c c0 r

e

ie r  r ' dr ' B  r '   c0



e

ie   rd ' A  r , ' c0

f r

(A.1)

and 



 2  ie  ie  ie 2 ie    rB  r     rB  r     d ' 2  rA  r ,  '     d '  rA  r ,  '    2 c r c r c0 c0 r r r r ie C rB  r   c

e e



C

i 

ie r  r ' dr ' B  r '   c0

ie  d '  rA  r ,  '   c 0 r

i 

e e

C

i 

e e



e

ie   rd ' A  r , '  c0

ie r  r ' dr ' B  r '   c0

ie r  r ' dr ' B  r '   c0



e



e

f r

ie   rd ' A  r , '  c0

ie   rd ' A  r , '  c0

f r

2 f r 2

(A.2)   Using then B  r   1    rA  r ,  '    Ar  r ,  ' , we find r  r

rB  r  

 '

  Ar  r ,  '    rA  r ,  '    ' r



and

 2 2 Ar  r ,  '  2 rA  r ,  '  rB  r   r r  ' r

and substituting this in (A.1) and (A.2) and in derivatives with respect to φ we obtain i 

 ie    Ar  r ,    Ar  r , 0     Ce r c

e

ie r  r ' dr ' B  r '  c0



e

ie   rd ' A  r , ' c0

f , (A.3) r

2 ie    ie      Ar  r ,    Ar  r , 0      Ar  r ,    Ar  r , 0   2 c  r r c r r  ie r ie r  r ' dr ' B  r '    ie  rd ' A  r , '   r ' dr ' B  r '    ie  rd ' A  r , ' 2   i i c c c c ie f  f 0 0 C rB  r   e 0 C e 0 c r r 2 ie r r ' dr ' B  r '   ie  rd ' A  r , '  i  c  c ie  f 0 C  d '  rA  r ,  '   e 0 c0 r r

e e

e e

e e

, (A.4)

159

4. Quantum phase variations in response to proper changes of vector potentials     ie r ie  i   r ' dr ' B  r '  rA  r ,    , (A.5)   c0 c    ie r   2 ie  ie  r A  r ,     i   r ' dr ' B  r '   rA  r ,    2 c  c0 c     2

  ie r  ie  ie  r A  r ,     i   r ' dr ' B  r '   rA  r ,     c  c0 c  

(A.6)

Finally, we substitute all the above derivatives in the Schrodinger equation to get

i 

ie r  r ' dr ' B  r '   c0



ie   rd ' A  r , ' c0

ie    ie  e f  Ar  r ,    Ar  r , 0     i Ce e e  Ar  r ,    Ar  r , 0   2mc  r r 2 mc  r mc r  ie r ie r  r ' dr ' B  r '    ie  rd ' A  r , '   r ' dr ' B  r '    ie  rd ' A  r , '    i i c c c c ie f ie  f C rB  r   e e 0 e 0 C d '  rA  r ,  ' e e 0 e 0 2mc r 2mc 0 r r



2

2m

i 

Ce

e

ie r  r ' dr ' B  r '   c0

e

ie    rd ' A  r , ' c0

i 

2 2 f ie  A r ,   A r , 0   Ce       r r 2mr r 2 2mrc

e

ie r  r ' dr ' B  r '   c0



e

2

ie   rd ' A  r , '  c0

2   ie r  ie  ie e2  A  r ,     i  r ' dr ' B r '  rA r ,    Ar  r ,    Ar  r ,    Ar  r , 0            2mrc  c0 c 2mr 2  mc 2    ie r  e ie e e2 i A  r ,   i  . A  A2     r ' dr ' B  r '   rA  r ,      i 2 mcr c c 2 mc 2 mc   0

(A.7)

that can be simplified as  2   ie Ar  r , 0   e A2  r , 0   ie A  r , 0   1 r r  2mc r 2mrc 2mc 2 2mr 2 



  X  r 2mr r  r

2

  f  ie A  r , 0  f r  mc r 

2 2 f f   f , (A.8) 2m r 2 2mr r 2

which, upon transforming f  r  as f  r   e 2

  er     r ' dr ' B  r '     c0  

1   2  2mr



ie r  dr ' Ar  r ',0  c0

X r 

leads in turn to

2

  er     r ' dr ' B  r '   X  EX   c0  

. (A.9)

This is equation (4.24) of the main text. 160

f r

4. Quantum phase variations in response to proper changes of vector potentials

Chapter References [1] R. J. Swenson, “The correct relation between wavefunctions in two gauges”, Amer. J. Phys. 57, 381-2 (1989) [2] H. Weyl, Z. Phys. 56, 330 (1929) [3] See the historical review of L. O‟Raifeartaigh & N. Straumann, Rev. Mod. Phys. 72, 1 (2000) [4] K. Moulopoulos, “Nonlocal phases of local quantum mechanical wavefunctions in static and time-dependent Aharonov-Bohm experiments”, Journ. Phys. A , 43 (35): Art. No. 354019 (2010) [5] K. Moulopoulos, “Beyond the Dirac phase factor: Dynamical Quantum PhaseNonlocalities in the Schrödinger Picture”, Journ. Mod. Phys. 2: 1250-1271 (2011) [6] Y. Aharonov and D. Bohm, “Significance of Electromagnetic Potentials in the Quantum Theory”, Phys. Rev. 115, 485 (1959) [7] D. Singleton and E. Vagenas, “The covariant, time-dependent Aharonov-Bohm Effect”, Phys. Lett. B 723, 241 (2013); J. MacDougall and D. Singleton, “Stokes' theorem, gauge symmetry and the time-dependent Aharonov-Bohm effect”, J. Math. Phys. 55, 042101 (2014); R.Y. Chiao, X.H. Deng, K.M. Sundqvist, N.A. Inan, G.A. Munoz, D.A. Singleton, B.S. Kang, L.A. Martinez, “Observability of the scalar Aharonov-Bohm effect inside a 3D Faraday cage with time-varying exterior charges and masses”, arXiv:1411.3627; M. Bright and D. Singleton, “The time-dependent non-Abelian Aharonov-Bohm effect”, Phys. Rev. D 91, 085010 (2015); J. Macdougall, D. Singleton, and E. C. Vagenas, “Revisiting the Marton, Simpson, and Suddeth experimental confirmation of the Aharonov-Bohm effect”, Phys. Lett. A379 1689 (2015); M. Bright, D. Singleton, A. Yoshida, “Aharonov-Bohm phase for an electromagnetic wave background”, Eur. Phys. J. C 75, 446 (2015); D. Singleton, J. Ulbricht, “The time-dependent Aharonov–Casher effect”, Physics Letters B 753, 91 (2016); S.A.H. Mansoori, B. Mirza, “Non-Abelian Aharonov–Bohm effect with the timedependent gauge fields”, Physics Letters B 755, 88 (2016); K. Ma, J.-H. Wang, H.-X. Yang, “Time-dependent Aharonov-Bohm effect on the noncommutative space”, arXiv:1604.02110 [8] K. Kyriakou, to be submitted (2016). The constant of motion (even for a manybody system) termed pseudo-angular momentum has been noted and fully worked out in this paper [9] M. Bawin and A. Burnell, “Single-valuedness of wavefunctions from global gauge-invariance in two-dimensional quantum mechanics”, Journ. Phys. A 18, 2123 (1985) [10] H. Hellmann, Einfhrung in die Quantenchemie. Deuticke, Vienna (1937); R. Feynman, Phys.Rev 56, 340 (1939) [11] K. Moulopoulos, “Topological Proximity Effect: A Gauge Influence from Distant Fields on Planar Quantum-Coherent Systems”, Int. Journ. Theor. Phys. 54: 1908-1925 (2015) [12] M. V. Berry, "Quantal Phase Factors Accompanying Adiabatic Changes", Proc. R. Soc. A 392, 45 (1984) [13] G. Konstantinou & K. Moulopoulos, in preparation; for a related earlier result on a ring, see K. Moulopoulos & M. Constantinou, “Two interacting charged particles in 161

4. Quantum phase variations in response to proper changes of vector potentials

an Aharonov-Bohm ring: Bound state transitions, symmetry breaking, persistent currents, and Berry's Phase, Phys. Rev. B 70, Art. No. 235327 (2004); Erratum: Phys. Rev. B 76, Art. No. 039902 (2007). On how this Berry‟s phase appears in a interacting mixture of different masses, see K. Kyriakou & K. Moulopoulos, “Arbitrary mixture of two charged interacting particles in a magnetic Aharonov-Bohm ring: persistent currents and Berry's phases”, Journ. Phys. A 43, Art. No. 354018 (2010) [14] T. Champel and S. Florens, “Quantum transport properties of two-dimensional electron gases under high magnetic fields”, Phys. Rev. B 75, 245326 (2007) [15] E. Madelung, Z. Phys. 40, 322 (1926)

162

5. High electric field influence on QHE systems

Chapter 5 High electric field influence on QHE systems In this final Chapter we investigate potential consequences of an exceedingly strong electric field (E field) on the ground state energetics and transport properties of a 2D spinless electron gas in a perpendicular magnetic field (a Quantum Hall configuration). We find fractional filling factors for certain values of E and B fields that are not resulting from interactions or impurities, but are a pure consequence of a strong enough in-plane E field. Additionally, we determine analytically the ground state energy, and response properties such as magnetization and polarization as functions of the electromagnetic field in the strong E field limit. The exact same method is followed in relativistic systems (Graphene) where we find irrational values of Hall conductivity for a relatively strong enough E field, such that L.Ls overlap in some region of single particle energy. The conditions under which our results are relevant (experimentally observable) are also discussed at the end of this Chapter.

163

5. High electric field influence on QHE systems

5.1 Conventional system - Low Electric field strength Consider an ideal nonrelativistic two-dimensional spinless electron gas in a  perpendicular and homogeneous magnetic field B directed along the positive z-axis. The dimensions of the plane are taken to be macroscopically large Lx  Ly . In   addition to B there is also an in-plane, homogeneous electric field E pointing in the y-direction that is very weak, i.e. it doesn‟t cause any overlap of different Landau Levels (L.Ls). In this manner, the Fermi energy can always be situated in the interior of an energy gap (i.e. for certain areal density), causing, as is well-known, the universality of quantization of Hall conductance   . If the electric field is further increased from its first critical value (to be determined below) then the gap closes and the system is highly nonlinear, the general consensus being that this results to destruction of the quantization of   , although from what we will see below several refined behavioral patterns remain (in this strong electric field case), that can even lead to the survival or even a different type of integer quantization in the nonlinear regime. This strong electric field case will be separately studied in the next section. We choose to work in the Landau gauge Ax   By , Ay  Az  0 along with a scalar potential V  eEy (we take the charge of the electron to be –e) resulting in the following Hamiltonian: 2 1   e  H  p  A   eEy , 2m  c 





with energy spectrum [1] ε  ω n  1 2  mVD2 / 2  eEY0 (5.1), with ω  eB / mc   being the cyclotron frequency, n  0,1, 2... the L.L. index, V D  cE  B / B 2  cE / B

cp x mc 2 E  the modulus of the drift velocity and Y0  (5.2) the guiding center eB eB 2 operator eigenvalue in the y-direction (restricted to the area  Ly / 2  Y0  Ly / 2 ). Now, in specifying the electric field‟s strength E, if we first want to avoid any overlap among different L.Ls, the single particle energy ε (n  1, Y0  L y / 2) must be lower than (or equal to) ε(n, Y0  Ly / 2) , a criterion which leads to the following inequality: E  ω / eLy (5.3). Therefore, this case of no-overlap involves a limitation of drift velocity‟s values that depends on both field strengths and is equivalent to either of two criteria: (i) mVD Ly  (the angular momentum of an electron in one 2 2 2 edge with respect to a point in the other edge is  ), or (ii) mVD / 2  / 2mLy (the drift kinetic energy is  a confinement energy along the y-direction (due to the uncertainty principle)).

164

5. High electric field influence on QHE systems

Fig. 5.1: A schematic representation of the energy states (L.Ls) when the electric field is weak E  ω / eLy : there is no inter-L.L. overlap. All L.Ls can be filled independently, and energy gaps between L.Ls can result in quantization of the Hall conductivity.

In this case all L.Ls can be filled independently: according to the least energy principle, at zero temperature T=0, all electrons occupy states labeled by small quantum numbers  n, l  , (with n the above mentioned Landau Level index, a non-zero positive integer, and with l another integer specifying the eigenvalue of p x due to periodic boundary conditions along the x-direction, the eigenvalue being px  hl / Lx ), starting from L.L n=0 and varying all possible values of l, and then successively following n=1, n=2 and so on. Each L.L (indexed by quantum number n) may host up to  /  0 spinless electrons (which is the total number of distinct values of l in the thermodynamic limit) with   BL x L y being the magnetic flux across the 2D plane and  0  hc / e the flux quantum, in order to be consistent with Pauli principle. This means that without loss of generality we always have, let us say, π (=1,2,3..) L.Ls that are occupied for fixed particle number N and for a certain value of B, and due to the minimal energy criterion, we must have (π-1) fully occupied L.Ls and the last L.L. either partially or fully occupied. Of course, if N is fixed, the entire occupational procedure is uniquely determined only by the magnetic field strength B and it can be described naturally by the inequalities

  1 

0

N

  0

1



nA 0  B 

1 n A  0 , (5.4)  1

with n A  N / L x L y the 2D areal density of electrons. The last L.L. is occupied by

N    1 /  0 electrons allowing us to determine analytically the maximum Y0 MAX of the last electron located at L.L. index value of n=π-1: 165

5. High electric field influence on QHE systems

Y0 MAX 

chN 1   Ly     , (5.5) eBLx 2 

which gives the expected results that when N   /  0 , Y0 MAX  Ly / 2 and when

N     1  /  0 , Y0 MAX   Ly / 2 .

5.1.1 Thermodynamic properties The total internal energy of the system at T=0 is a sum over all occupied quantum numbers n and Y0 :





1   E   ω n  1  mVD2  eEY0  (5.6) 2 2   n Y0 In the macroscopic (continuum) limit L y   we may approximate the sum with respect to Y0 with an integral: Ly

Ly

BL x f Y0   0

2

 Y0  

Ly 2

 dY f Y  , with f Y0  2

0



0

an arbitrary function of Y0 . (5.7)

Ly

2

We then have

BL x E Φ0

π2

Ly





1   BL x dY0 ω n  1  mVD2  eEY0     2 2   Φ0 n 0 L y  2

2

 dY ωπ  12   2 mV

Y0 MAX



0



Ly

1

2 D

  eEY0  

2

and after a number of algebraic manipulations we reach a closed analytical expression for the total minimal energy per electron, namely  n A 0   mc 2 E 2 eB  E e2 B2 1 B 1   1      1  eELy          2 2 N 2  2n A  0 mc  2 4mc n A 2B   2B  (5.8)

valid in the magnetic field range n A  0 /   B  n A  0 /   1 . Or, in units of the Fermi energy  F  2 4 nA / 2m (the one defined for 2D spinless electron gas in the absence of E and B),

166

5. High electric field influence on QHE systems

B 2     1 E B  1  Ly enA 0 E 1     eELy        2 2 N F 2 0 n A nA 0  2 2 B F 2  e 2 E 2  02 nA2 eELy B      1  4 F2 nA B 2 2 F nA 0

Now, let B be equal to B 

eELy

F

 y

(5.9)

1 nA 0 , with   1  k   , so that from (5.3) we have k

1 . Rewriting then (5.8) in terms of y and x  B / nA 0 we obtain k

 1    x E 1 y 1 y 1  yx     x       y         1  2 2 N F 2 2  2 x 4 nA Ly x 2 2   2

2 In Fig. 5.2 see the graphs of energy and magnetization per electron, for nA Ly  1 (a good value so that the internal structure of these quantities are shown in sufficient detail).

Fig. 5.2: Energy and Magnetization per electron as functions of the magnetic field B. Using the second thermodynamic law we can also determine analytically the equilibrium magnetization and polarization per electron, which turn out to be  nA 0     1  mc 2 E 2  E / N  M e2 B e 1      1  eEL      y     2 2 3 N B 2 mc nA 2n A  0  B mc  2  2B

 E / N  n   P 1   mc 2 E Ly    1   e2    eLy  A 0          B N E 2  B2 n A c  2h   2B 

(5.10)

These thermodynamic expressions demonstrate the effect of the electric field which is non-linear (with respect to the variable E), even for the case of relatively weak electric field. Now we proceed by examining some limits: when all L.Ls are fully occupied, namely when N   /  0 , then we obtain

Ly     M e m 3 2     E  3   E (5.11) N 2mc n A c  2  n Ac 167

5. High electric field influence on QHE systems

P m   2  2 E (5.12) N nA

e 2 with    the Hall conductivity (see below). It is interesting to note a h characteristic half of the Hall conductance in the coefficient that connects the socalled magnetoelectric effects with the fields that cause them (that in a problem with chiral properties, usually in Topological Insulator materials, correspond to an extra magnetization caused by a parallel electric field (as in (5.11) above) and an extra polarization caused by an extra magnetic field (see [2] and references therein); here we note such a trend even in a Quantum Hall system (which is not unexpected, and is actually justified based on general Physics arguments [2]). One can actually see such a trend in the polarization as well, for a general B, in (5.10), where in the last linear term we can again see ζH/2 appearing. [It should be added that all these behaviors originate from the final term of (5.9) that describes an EB–coupling, something that we will also see later for the relativistic case.] In addition to all this, note also that for E  ω / eLy the gap closes, and (5.8) becomes E 1 2 , (5.13)   F E  0, B  0  N 2 2mL2y 2 with  F E  0, B  0   4n A / 2m being the expected Fermi energy (in the absence

of fields) – and when all L.Ls are fully occupied, with B  nA 0 /  , and with any E (now less than its first critical value) (5.8) becomes

E 1 mc 2 E 2   F  E  0, B  0   N 2 2B2 i.e. the total energy per electron reduces to 2D energy of free electron gas plus a drift kinetic term. These last simple results are not quite unexpected and may be justified with proper semiclassical considerations.

5.1.2 Hall Conductivity The Hall conductivity is defined as    enA c / B . The usual plot representing the QHE [3] shows   as a function of B, and in this we have plateau formation for certain values of B. For B varying in the range defined by the inequaltities (5.4)

1



nA 0  B 

1 n A  0 , (5.14)  1

168

5. High electric field influence on QHE systems

then   is quantized in units of e /h with  being an integer, which counts the number of fully occupied L.Ls. Note that each plateau has maximum width 2

1 1 1 n A 0  n A 0  n A  0 (5.15)  1     1 In the following we show that, in the very strong E-field case, fractional filling factors may also occur, by variation of electron number N with E and B fixed. [In that case, the analogous width of the corresponding plateau is expected to be reduced, although a serious consideration of plateau-observability should include a study in the presence of disorder, which is beyond the scope of the present article.]

5.2 High Electric field Strength When the electric field exceeds its first critical value, E   / eL y , inter L.L. overlap occurs. As E gets stronger, more and more L.Ls overlap and degenerate states that belong to different values of the quantum number n appear (in the previous case of a weak electric case, the standard Landau degeneracy had been completely lifted). Energy gap closes, and Fermi energy is always located on a single quantum state, with a significant number of available nearby states. Although Fermi energy is no longer in an energy gap, it will make jumps from one L.L to another by varying the magnetic field (or the particle number N). There are some critical values of B (the transition points) where a jump occurs, at which the Hall conductivity takes fractional values, as we shall see. We now assume a general strength E field determined by E  z  / eLy , (5.16) with z a continuous number that describes “overlap percentage”: j  z  j  1 with j  0,1,2.. . When j  0 , or z  1 , overlap vanishes and L.Ls can be filled

independently, with the usual energy gap restored. This is just the case discussed in previous section. Eq. (5.16) can actually be derived for a certain electric field that obeys the following relation:

 n  j, Y0  L y / 2   n  0, Y0  L y / 2   n  j  1, Y0  L y / 2, i.e. the electric field has a strength such that the single particle energy of the last electron Y0  L y / 2  in L.L. n  0 is greater than single particle energy of the first particle Y0   L y / 2  in L.L. n  j and lower than the single particle energy of the first particle in L.L. n  j  1 . Then,

 j  1 / 2  eELy / 2   / 2  eELy / 2   j  3 / 2  eELy / 2

169

5. High electric field influence on QHE systems

that concludes in j

   or E  z with j  z  j  1 ,  E   j  1 , eL y eL y eL y ,

which is indeed (5.16).

Fig. 5.3: An example of different L.L mixture with j=5 and z=5.1 a possible position of Fermi Level (ρ=6), with only one full L.L. occupied and five L.Ls partially occupied. Ground state demands that the chemical potential of each distinct L.L (highest single-electron energy) must be equal, resulting in a purely horizontal Fermi energy.

Now we procced to occupancies: For a constant E and B field (i.e. a constant energetic configuration) varying electron number N results in variation of Fermi energy and a corresponding variation of occupied L.Ls. The goal here is to examine under what N-variations the Fermi level remains at a single (the topmost) L.L level. To achieve this, we fix Fermi level at the highest occupied L.L, n=π-1,   1 , (see Fig. 5.3 above) with energy

ε F  ω  1/ 2  mVD2 / 2  eEYm , (5.17) with Ym the guiding center position of the last electron (with highest single particle energy) located at n=π-1 (see the isolated dot in Fig. 5.3). In this way, we ensure that variations of N will result in moving Ym in the interval YL  Ym  YR , with YL   L y / 2 , the left edge of n=π-1 L.L. and YR is a critical guiding center above

which L.L. n=π is occupied. This can be determined by equating the Fermi energy to the single particle energy ε n   , Y0   L y / 2  :

ω  1 / 2  mVD2 / 2  eEYR  ω  1 / 2  mVD2 / 2  eELy / 2 , resulting in: YR  Ly / 2  ω / eE , or, using eq. (2.1): YR  Ly / 2  Ly / z . Any number N that takes YR above this critical value will result in a nonzero occupation of

170

5. High electric field influence on QHE systems

n=π L.L. Therefore, to ensure that exactly π L.Ls are occupied, Ym must vary in the following window:

 Ly / 2  Ym  Ly / 2  Ly / z , (5.18) guaranteeing that exactly π L.Ls are occupied. Naturally this equation becomes pathological when z is smaller than unity – the weak electric field case – YR becomes infinite when z  0 . This of course happens because the above equation is not valid in this case, where no inter-L.L overlaps occur. Limitations must therefore be imposed here. The smallest value that z can take, must be above 1, with z=1 defining the nonoverlap limit. If z  1 the above relation just becomes:

 Ly / 2  Ym  Ly / 2 (5.19) as earlier. Having now defined the Fermi level‟s exact location analytically, we proceed with our method by counting how many L.Ls are fully occupied and how many are partially occupied. An L.L. is completely filled with electrons only when the Fermi energy is greater than the single electron energy located at Y0  L y / 2 in each L.L. Using this information, we may write down the number of fully occupied L.Ls iF  as a sum of theta functions over all L.Ls:    Y 1   L    iF Ym ,  , z     F    n, y       1  n  z m   for z  1 , (5.20)  L 2 n0   2  n0   y 

where  x is a step function,  x  1 if x  0 and  x  0 otherwise. Variations of

Ym (i.e. for fixed E and B) according to eq. (5.20) result in variations in iF as follows:

i FL  , z   i F Ym ,  , z   i FR  , z  , (5.21)       Ym YL

Ym YR





n 0

n 0

with i FL  , z       1  z  n  and i FR  , z       z  n are the numbers of completely filled L.Ls calculated at the edges of eq. (5.18). Let us see now some examples:

171

5. High electric field influence on QHE systems

Fig. 5.4

Fig. 5.5 

In Fig. 5.4, z is equal to unity, z  1 and   2 , then i FL     n    0  1 , and n 0



i FR    1  n   1   0  2 . In Fig. 5.5, z  2.2 and   3 , i FL  0 and i FR  1 . It n 0

L R is interesting to note that for any value of the variable z, iF and iF always differ by

one, which means that varying N or equivalently, Ym , in the interval defined by (5.18), one full L.L. is added to the system. This resembles the E=0 case, where variations of Ym in the same interval (but for z  1 ) the Fermi energy passes through all available states in the last L.L. until it reaches the next L.L., where an extra full L.L. increases the number iF by 1 (see Fig. 5.6 below). [This is actually consistent with the well-known corresponding result in the case we fold the system (in the x– direction) into a cylinder, that is a key result in the Laughlin argument [4] that gives the integral quantization of the Hall conductivity from general gauge arguments – or equivalently to the charge-pumping picture of Thouless [5] or a more general property of the so-called spectral flow [6] in topologically nontrivial systems, such as topological insulators [7].] It is interesting to see that when z reaches its lowest L possible value, z  1 , i F    1 , meaning that the maximum number of partially occupied L.Ls is always 1. Needless to say that z remains a constant only when E and B fields are constant too, (or in a special case that both fields vary in the exactly same rate) and the only variable is the electron number. This happens because of eq. (5.16), E  z / eL y , which relates B to E through the variable z. For a constant electric

172

5. High electric field influence on QHE systems

field, varying B will result in a variation of z such that E remains a constant. The same principle holds for keeping B constant and varying E. Generally, from definition of iFR and iFL we have that: iFR    Int[ z ]   and

iFL    Int[ z] 1   , where   1 if z is an integer and   0 otherwise. These equalities hold for   Int[ z ]   . If   Int[ z ]   or lower, then iFR  0 and iFL  0 .

Fig. 5.6: For z=0 case (identical to z  1 case), with no E-field, and i F   . Left figure shows π=3 occupied L.Ls, with two of them fully occupied and the last one partially occupied. An addition of electrons will stimulate Fermi energy to pass through all available states in its right side until right end of L.L. n=2, where a full L.L. will be added to iF .

i FL    1

R

Now, about partially filled L.Ls, these must intersect at a certain Y0 Fermi energy. Optimal energy requires that states (per L.L.) starting with Y0  L y / 2 until the intersection with the Fermi energy must be filled, while states with higher Y0 s must be empty. These intersections can be easily found at the following points: Y0 l  





1  1  ε F  ω i F  l  1  mV D2  , with l  0,1,2...  1  iF , (5.22)  2 eE  2 

where   i F is the number of partially occupied L.Ls.

5.2.1 Number of states under Fermi energy Each L.L contains  /  0 available quantum states. Each state corresponds to a single spinless electron, in accordance to Pauli Exclusion Principle. The number of states under Fermi energy is a sum of i F full states and a sum term that counts all states in the partially occupied L.Ls from  L y / 2 until the intersection with Fermi energy:

173

5. High electric field influence on QHE systems

# states  i F

  1iF  L x B F BL x  1  mV D2 L x B         iF  l     (5.23) 0 2cE  2 2 Ehc 2 0  l  0  Ehc

This number, in a canonical ensemble, is exactly equal to the constant particle number N: N  iF

 ε BLx  L E eBLx Ly  eB 2 Lx     iF   F  mc x     iF  (5.24) 2  0 h 2B ch 2 4 mE c  E ch 

or, from the definition of Fermi energy:  1 eB 2 Lx BLx eBLx Ly   N  iF     iF     1  i  eY    F m  2 0 ch ch 2   4 E mc

with iF  iF  F , E, B also a function of  F . This equation determines particle number N when  F is also known. Unfortunately, it is rather difficult to solve directly with respect to  F , but it can be easily determined numerically. Further simplification of (5.24) can be made using the window of values of  F , given by eq. (5.17) and (5.18): Ly  Ly Ly  1  mVD2 1  mVD2    , (5.25)        eE   F         eE   2 2 2 2 2 2 z    

which results in the following window of values for N: R B 2 Lx B 2 eLx  L   R    iF   L L R R e    iF     1  iF   iF  N     iF     1  iF    iF  z  Emc 2 4 0 4 Emc 2  0   (5.26)

with i F  i F  , E , B  and i F  i F  , E , B  . The above relation defines windows of values for B (if it is considered as a variable) for constant E field, and constant electron number N. Equivalently, one may prepare an experiment, where electron number is not conserved (i.e. an electric circuit) and keep E and B fixed. Note that the above relation shrinks to eq. (5.4) when z  1 : L

L

R



R

   N    1 (5.27) 0 0

5.2.2 Thermodynamics We now proceed to ground state energy calculation, which is a sum over all single electron states, until reaching the Fermi energy: 174

5. High electric field influence on QHE systems

BL E x Φ0

iF 1 Ly /2





1   BL dY0  ω n  1  mVD2  eEY0   x   2 2   Φ0 n  0  Ly /2

 1iF Y0  l  l 0





1   dY0  ω iF  l  1  mVD2  eEY0  2 2    Ly /2

 

The left term describes the energy due to fully occupied L.Ls, while the right term describes the energy of partially occupied L.Ls. After a number of algebraic manipulations, and using eq. (5.22) we conclude to the following result for the total energy per electron:

   iF  mc 2 E 2 Ly EnAch  e2 B 2 e B  iF    eELy     iF     2 N 4 mc 2 nA 2cm 2    iF  B  2    iF   2 B   iF  e2 e2 B3  Ly EB  (   iF  1)(   iF )    iF  1  48 nA m 2c 3 ELy  2    iF   hcnA (5.28) that consists of a sum of several terms which are not all symmetric (with respect to E and B). The last term vanishes in weak E field limit ( iF   1 ). Note the interesting fact that the presence of the coupling term EB requires fully occupied L.Ls, i.e. iF  0 . This is a probably expected pattern resulting from axionic considerations in an electromagnetic field where the coefficient of coupling term involves fully occupied L.Ls (for the weak E field case, the corresponding coefficient is related to the Hall conductivity, and it was briefly mentioned in the last section, originating from the EB – coupling term). When iF  0 , i.e. when all L.Ls are partially occupied (   Int[ z ]   ), we have (from (5.28)):

L EnAch eELy mc 2 E 2  e B e2 B3   y     (  2  1) 2 2 3 N 2cm 2 B 2 2B 48 nAm c ELy and from (5.26): B 2 eLx B 2 eLx      1  N      1    2 2 Emc 4 4 Emc 0 z

Or, substituting E  z / eL y we get

  2z 0

   1  N 

  2z 0

   1

(5.29)

i.e. there are no quadratic terms with respect to B appearing in the total energy, and there are no coupling terms either. Also, when the Fermi energy travels from one edge of n=π-1 to the other, it passes exactly through  / z 0   /  0 states. On the other 175

5. High electric field influence on QHE systems

hand, when   Int[ z ]   , i.e. iF  0 , the Fermi energy passes through exactly

 /  0 states. Now, we want to plot eq. (5.28) by keeping E and B fixed and vary N in the interval given by (5.26). Using E as in (5.16) we rewrite (5.26) and (5.28) as N 0 N  0    iF  zN 2  02   i     i  z    F F 2 2 2 e B Lx Ly      iF    iF   4 mc 2 z  iF 1   (   iF  1)(   iF )    iF  1   iF 12 z  1 1   iFL     1  iFL   iFL  N 0     iFR     1  iFR   iFR  2z  2z

and

8

e2 B2 Lx Ly 4 mc2

15

6 4 2

10

5

E

E

e2 B2 Lx Ly 4 mc2

2  2 0  Nz  2 

0

0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

0

1

2

y

3

4

y

Fig. 5.7: Total energy (in units of e2 B 2 Lx Ly / 4 mc 2 ) as function of particle number

15 10 5

E

e2 B2 Lx Ly 4 mc2

(in units of Φ/Φ0, y=NΦ0/Φ) for z=3 (Left), z=1 (Right, low E field limit).

0 0

1

2

3

4

y

Fig 5.8: Total energy (in units of e2 B 2 Lx Ly / 4 mc 2 ) as function of particle number (in units of Φ/Φ0, y=NΦ0/Φ) for z=6.1.

176

5. High electric field influence on QHE systems

A few remarks are in order about these figures: It is clear that when the E field is exceedingly strong, in contrast to the case of a weak E field, there appears a global minimum with respect to particle number variations. This minimum is a consequence of the E field effect on the thermodynamic properties; as E gets stronger, more and more states will gain negative energy, (see for example Fig. 5.3). As these states get occupied by electrons, the total energy will become negative at first, then will rise up to positive values, because states with positive energy will begin to fill up. The result of this competition is this minimum, which can be fully controlled by examining the corresponding z-value. The greater z gets, the more negative-energy states will be occupied, and the minimum will move to greater particle numbers (or larger particle densities). In comparison with the low E field case, where the positive states are more, in this case the total energy may be even lower. We now plot the total energy as a function of variable z, for a constant E field with y  eELy /  F (a suitable measure for the E field in units of 2D Fermi energy in the E=0, B=0 case) and B  mcLy E / z from eq. (5.16):  y2 y 1 z y    iF   2 iF      iF    N F 2z 2z    iF  2 2    iF 

  iF  y 2  z2      2  Ly 4 nA  2    iF   z

1 y2 (   iF  1)(   iF )    iF  1 24 z 3

where z lies in the following window: 1 1 1 1 1   iFL     1  iFL   iFL   2    iFR     1  iFR   iFR 2  2z z y 2z z

0.64 0.22

F

F

0.66 0.68

0.20

EN

EN

0.70 0.18

0.72 0.74

0.16

0.76 0.14

0.78 2.0

2.5

3.0

3.5

z

4.0

4.5

5.0

5.5

3

4

5

6

7

8

9

z

Fig. 5.9: Total energy per particle (in units of Fermi energy in the absence of E and B) as function of z (analogous to inverse B) for y=3 (Left), y=6.3 (Right, low E field limit). We should point out here that although the energy is periodic function with respect to inverse B (this is the de Haas-van Alphen effect [8] – see how this can be derived at 177

5. High electric field influence on QHE systems

T=0 in various cases from rather elementary considerations in ref. [9]), the windows of values of z (or 1/B) seem to be influenced by the presence of the electric field, namely  1/ B   / mcLy E , or, by using y  eELy /  F we find that

 1/ B   e / ymc F  1/ ynA0

(5.30)

which

deviates

from

the

standard

semiclassical periodicities  1/ B   1/ nA0 for a 2D system [9]. In addition, it should be pointed out that the analytical correctness of our results is witnessed by the fact that the total energy turns out to be a smooth (continuous and differentiable) function of z for every z (i.e. the positions of the windows match with the corresponding expressions) – this smoothness of the figures (in their joining through different window-values) strictly testifying for the analytical correctness of the overall expressions that we derived for the total energy.

5.2.3 Hall conductivity The Hall conductivity is defined as:    enA c / B (5.31) and is usually plotted as a function of magnetic field B. For convenience we will make here use of eq. (5.26) and examine   as a function of electron number N instead of B. Starting therefore from R B 2 Lx B 2 eLx  L   R    iF   L L R R e    iF     1  iF   iF  N     iF     1  iF    iF  z  Emc 2 4 0 4 Emc 2  0  

(5.32) we will show that by varying N, fractional filling factors appear, with no interactions and no impurities taken into account. E and B fields are considered as constants (and therefore so is z) throughout all N (and π) variations. Substituting eq. (5.16) into (5.32) we may eliminate E:  1  1     iFL    iFL  1  iFL   N    iFR    iFR  1  iFR      0  2 z 0  2 z   R Because for all numbers  and iF the two edges of (2.17) differ by

 , we choose 0

to calculate   only at the special values of electron number N given by the maximum edge of (5.32):

N

 0







1 R R R  2 z   iF   iF  1  iF  ,  

and   is then:

178

5. High electric field influence on QHE systems

 







 enA c e 2  1      i FR   i FR  1  i FR  with i FR  , z       z  n B h  2z  n 0

Let‟s examine now some cases regarding z-values:

z is an integer:

e 2 1) z=1  i   :    h 2 2 2 2 e2  1  e 1 e 3 e 6 e 10 R    1  , , , ..... 2) z>π:  i F  0 :    h  2 z  h z h z h z h z e2 1 R z  1  1. 3) z=π:  i F  1 :     h 2  R F

4) z0 and

 Lx / 2



2 BLy 0

X0

Lx / 2



dX 0 for n=0 L.L. (5.46)

 Lx / 2

we can write:

   FULL   PART , (5.47) where  FULL is the energy of the fully occupied L.Ls and  PART is the energy of the last partially occupied L.L.

 FULL 

4 BLy 0

 2

Lx /2

n 1

 Lx /2

 

dX 0

2n eBuF

1   2 

1

4 2 eBuF   0 1   2 14  



4

4 BLy eE   2 0

  n 1

dX 0 X 0

 Lx /2

(5.48)

 2



Lx /2

n

n 0

To calculate  PART , it is necessary to determine the last electron‟s guiding center position in L.L n=π-1.

184

5. High electric field influence on QHE systems

X 0  lB2

2    1lB  2 l (5.49)  1 2 4 Ly 1    

The index l, appearing in k y , has a starting value l0 that needs to be determined by the condition X 0   Lx / 2 , (left edge):

Ly   2 0 2 lB



2    1 

1    2

1

4

 l0 (5.50)

 2  But the last L.L hosts  N   2   3  / 4 states for the remaining electrons to be 0   placed inside, so we conclude that the last electron in n    1 L.L. has a guiding

center position: X 0 MAX  lB2

2  N       (5.51) Ly  4 0 0 

For consistency reasons we check eq. (5.51) for certain values of particle number, i.e. when N   2   3 2 / 0 then X 0   Lx / 2 and when N   2   1 2 / 0 we have

X 0  Lx / 2 . So far we are correct. The energy of the electrons located at the Fermi L.L. EPART is:  PART 



4 BLy

4 BLy 0

X 0 MAX



2    1 eBuF

1   

0

dX 0

 Lx /2

2

1

4

2    1 eBuF

1    2

1

 eE

4

 X 0 MAX  Lx / 2  eE

4 BLy 0

X 0 MAX



dX 0 X 0

 Lx /2

4 BLy  X  0  2

2 0 MAX



L   4

(5.52)

2 x

Substituting eq. (5.51) in eq. (5.52) we get:

 PART  

2    1 eBuF   N  2  2   3   1 0  1   2  4 

(5.53)

2 BLy 2  2 hN E 1  eEN    1 Lx  e 2 E Lx    2    8BLy h 2  2

185

5. High electric field influence on QHE systems

From eqs (5.48) and (5.53) we obtain the final result for the total energy per electron of the system:

TOT F  1 N 1   2  4 

3    2   2 B 4    2 n   2   3    nA 0   n 1 

  B     1   2     nA 0 

nA 0 eELx 2 BLx  eE    1 Lx  eE 8B nA 0

1

    1   

2

1  2    2   2  (5.54),

 2

with

 n 1

3

  2  1  n     ,   1    , where   1/ 2,   1 is the Hurwitz zeta 4  2 

function and   3 / 2  is the Riemman function. Note that the term proportional to the E field appearing in (5.54) when expressed per electron,

e2 E

2B  2 1  hNE Lx    2     , (5.55) nA h  2  8 BLy

gives the following magnetization (which is also proportional to E):

e 2 E

2 1  hNE  Lx   2  2     2 (5.56) nA h  2  8 B Ly

When all π L.Ls are fully occupied, then

B  nA0 / 2  2   1 , and this term becomes: Lx  e2  4   1  E  2n A  h

Lx   E (5.57) 2n A 2

which is in accordance to the non - relativistic case (see corresponding term of (5.11)).

Although

this

    4  1 e2 / h differs

from

eq.

(5.44)

where

    2  1 2e2 / h this point needing further clarification. Let us then check some limits: When the electric field is absent, E=0, the total energy per electron becomes (in units of Fermi energy in the absence of B  F  uF  nA ): 3    2   2 TOT B   F 4   2 n   2   3  N   nA 0   n 1 

 B      1   2     nA 0 

1

2



   1   

(5.58)

186

5. High electric field influence on QHE systems

This is the result we would have gotten had we solved the problem from the beginning without the E-field.

Fig. 5.10: Energy in units of  F and magnetization in units of Bohr magneton  B as functions of magnetic field in the absence of an electric field. When B is at a critical value, B  nA0 / 2  2   1 , (all π L.Ls fully occupied) then: TOT

1  2 N uF  2   1

 2 N uF

nA 2

 2   1

 2

3

 n 1

nA 2    2 n  2   1   n 1 n  2 N uF



   1  

nA 2    1

 2   1

(5.59)

3

If π=1 then TOT  0 , as one would expected, because all electrons are placed at the lowest L.L. having zero energy.   If π=2 TOT  2 eBuF  4  and so on. In all cases, we can clearly see the  0  difference between the case of 2DEG in conventional semiconductors (note, for example, that at the critical values of B the total energy is no longer equal to the Fermi energy but keeps rising up – see fig. 5.10 and its differences with fig. 5.2).

5.3.2 Stronger E-field The inequality 2e Bu F  iF  1  iF   eELx , 1  2 4 1   

is the criterion where no overlap occurs, as long as   1  iF . Ννw, if we have „more‟ electrons and have to place them in available states, we will finally reach the

187

5. High electric field influence on QHE systems

condition   iF  1. In this case, states indexed by n  iF  2.....  1 , overlap, and the energy gap closes at Fermi energy. This means that graphene gains a metallic Efield induced character and Hall conductivity is modified (analogous to the nonrelativistic case) by a term containing both electric and magnetic field. Recall from previous work [13] that the following relation also holds:

 1

E  1  E  uF B , uF B

(so that no collapse of L.Ls occurs), which is always true as long as we treat particle number as our variable, and keep E and B-fields constant. Or, equivalently, we may keep N and B fixed and treat E as our main variable, starting from zero until reaching maximum value EMAX  uF B . We remind here the reader of the limitations of this problem regarding also the magnetic field, which has to be such that the magnetic length is much greater than lattice constant (i.e. we will not study any Hofstadter effects in this work). Either way, our results and predictions can be obtained by satisfying all possible limitations. For example, we may prepare an experiment where E is strong enough for inter L.L. overlap to occur, and at the same time B is also strong enough to satisfy the above inequality.

Fig. 5.11: Schematic Representation of energy levels in graphene. Figure shows

 n ,k

y

vs

X0. The inter L.L. energy gap gets smaller by increasing Landau Level index n. States n=0,1 and 2 do not overlap with each other, but states n=3,4,5… and so on, do. If the Fermi energy is located at n=0,1 or 2, we expect that Hall conductivity will still be quantized in units of

    2  1 2e2 / h , with π=1,2, and 3. We examine now the case appearing in Fig. 5.11. Landau levels n=0,1 and 2 can be occupied independently, while further occupations above n=2 L.L. leads to the unavoidable mixing of states makes the problem interesting, in the sense that peculiarities may arise both in thermodynamic and transport properties. We fix Fermi energy  F at L.L n=π-1:

188

5. High electric field influence on QHE systems

2    1 eBuF

F 

1    2

1

 eEX 0 (5.60)

4

with X 0  the guiding center position of the last electron in the Fermi level n=π-1. We consider now the simplest case: From Fig. (5.11), we will examine the case where all states are occupied up to the Fermi energy (eq. 5.60). The Fermi energy is located at n=4 with X 0    Lx / 2 :

F 

8 eBu F

1    2

1



4

eELx (5.61) 2

To determine the number of states in L.L. n=3, we examine its intersection with Fermi energy (see Fig. 5.11): 6 eBu F

1    2

1

4

 eEX 0 F 

8 eBu F

1    2

1



4

eELx L eBu F  X 0F   x  2 2 eE 1   2 1 4  



8 6



(5.62) Now, we determine the starting ( l0 ) and final ( l F ) point of l for this case: l0  

Ly  6 (5.63)  2 0 2 lB 1   2 1 4  

X 0  lB2

2 lF 6lB   1 Ly 1   2  4

Ly B eBuF   lF    2 0 h E 1   2 14  





8 6 

(5.64)

Ly E

e 6

huF eB 1   2 

1

4

The number of states in L.L. n=3 is given by: Number of states: =

lF  l0 

Ly B eBu F h E 1   2 14









8  6 , (5.65)

and the total number of states under Fermi energy is:

g

  Ly B eBuF 2  0  0 h E 1   2 14  





8  6 (5.66)

n 0

189

5. High electric field influence on QHE systems

If all states are filled with electrons, then the electron number is (considering that we have four electrons in each case, except for n=0, in which we put 2 electrons in each state): N  10

Ly B eBu F  4 0 h E 1   2 14  





8  6 (5.67)

This relation modifies Hall conductivity as:

 

nAe eBu F e e2 e N  10  4 B SB h hLx E 1   2 1 4  





8  6 (5.68)

Now, by directly using eq. (5.41) and substituting in (5.68): 2e Bu F

1    2

1

4

z  eELx  E 

2e Bu F

eLx 1  

2



1

z (5.69) 4

we conclude to:

 

nAe e e2 e2 N  10  4 B SB h zh





4  3 (5.70)

demonstrating the possibility of irrational quantization of the Hall conductance.

5.4 Conclusions We have shown that by finding the optimal energy at zero temperature in the case of conventional semiconductors, fractional Hall values appear at the points of jumps of the Fermi energy. This does not necessarily mean that plateaux appear. Inclusion of impurity potential may lead to quantized plateau structure even in the case of strong E field, due to further broadening of L.Ls, isolating extended from localized states. We should point out that it is not the gap closing that causes plateau disappearance, but rather the fact that we have ignored the impurity potential in our calculations. We have also calculated analytically, using ground state energy considerations the total internal energy, magnetization and polarization as functions of the electromagnetic field. The associated de Haas-van Alphen oscillation periods are also influenced by the presence of the electric field in a specific quantitative manner. A corresponding exact calculation in a pseudo-relativistic system, such as Graphene, is more involved but has been carried out in detail. An immediate result of our toy model is the possibility of irrational Hall values, although further investigation is required (i.e. inclusion of disorder) in order to see if these effects survive under realistic conditions.

190

5. High electric field influence on QHE systems

Chapter References [1] See i.e. D. Yoshioka, “The Quantum Hall Effect”, Springer (2002), pp. 28-29 [2] K. Moulopoulos, “Gauge nonlocality in planar quantum-coherent systems”, arXiv:1308.6277, see in particular Appendix H on Axions [3] R. E. Prange & S. M. Girvin, “The Quantum Hall Effect”, Springer (1990) [4] R. B. Laughlin, “Quantized Hall conductivity on two dimensions”, Phys. Rev. B 23, 5632 (1981) [5] D. J. Thouless, “Quantization of particle transport”, Phys. Rev. B 27, 6083 (1983) [6] G. De Nittis and H. Schulz-Baldes, “Spectral Flows Associated to Flux Tubes”, Ann. Henri Poincare 17 1–35 (2016) [7] M. Z. Hasan and C. L. Kane, Colloquium: “Topological insulators”, Rev. Mod. Phys. 82, 3045 (2010); X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors”, Rev. Mod. Phys. 83, 1057 (2011) [8] D. Shoenberg, “Magnetic Oscillations in Metals”, Cambridge University Press (1984) [9] G. Konstantinou and K. Moulopoulos, “Thickness-induced violation of de Haasvan Alphen effect through exact analytical solutions at a one-electron and a onecomposite fermion level”, Eur. Phys. Journ. B 86, 326 (2013) [10] B. I. Halperin, “Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential”, Phys. Rev. B 25, 2185 (1982) [11] M. I. Katsnelson, “Graphene: Carbon in Two Dimensions”, Cambridge University Press (2012) [12] N. M. R. Peres and E. V. Castro, “Algebraic solution of a graphene layer in a transverse electric and perpendicular magnetic fields”, Journ. Phys. Condens. Matt. 19, 406231 (2007) 191

5. High electric field influence on QHE systems

[13] V. Lukose, R. Shankar, and G. Baskaran, “Novel Electric Field Effects on Landau Levels in Graphene”, Phys. Rev. Lett. 98, 116802 (2007)

192

Thesis Conclusions

Thesis Conclusions This work has investigated potential consequences of very strong electromagnetic fields imposed on electronic systems that move in certain geometric (and in some cases topologically nontrivial) configurations. Topology emerges as a twisted and linked mathematical structure that comes into play through Berry connections and Berry curvatures, which in turn give robust topological invariants, such as the socalled first Chern number (in the Gauss-Bonnet-Chern theory). The main focus, however, of the present work – in terms of concrete analytical calculations that are (or were, at the time of publication) original, and that lead to concrete predictions – has been to carry out in detail mainly thermodynamic calculations of the above interesting electronic systems, both for applications and from a fundamental Physics point of view; such systems are semiconducting heterostructures, or interfaces, or films, with a finite thickness (and Zeeman splitting), placed in a perpendicular magnetic field, or, electrons on a cylindrical surface that encloses a magnetic flux. It has been shown here that by varying the thickness of a semiconducting heterostrucure, when placed in a strong perpendicular magnetic field, the system experiences both magnetic and topological transitions. Magnetic, because the magnetic susceptibility oscillates as the thickness is varied between positive and negative values in a discontinuous way. Topological, because the corresponding Chern number also oscillates with the thickness. These phenomena arise due to the nontrivial optimization of the total energy, where one has to carefully fill the available states at zero temperature, up to the Fermi energy. The complexity of the various crossover-scenarios that naturally showed up in the solution of this problem brings to light important hidden consequences: deviations from standard dHvA periodicities (a well-known semiclassical result) which states that thermodynamic quantities must be periodic in the interval of values of the inverse magnetic field; the deviations found analytically were shown to have quite intricate patterns, that were here analyzed in detail. Similar results were obtained by taking into account the spin of electrons through proper Zeeman terms. Once again we have observed discontinuities in global Magnetization for certain values of thickness of the interface, and violations of dHvA periodicities. Our next step was to solve the 3D version of this problem, namely, the thermodynamics of a noninteracting electron gas in full 3D space, in a homogeneous magnetic field. In that case, the system‟s behavior was rapidly converging to the dHvA periodicities, and this was occurring in a broad range of (not so strong) magnetic field values, while in the exceedingly strong field-case large deviations occurred. This demonstrated the purely quantum character of our exact solution, which is more pronounced for strong magnetic fields (and where the semiclassical approximation is indeed less accurate). We note that all thermodynamic properties depended only on the Imaginary Part of Hurwitz Zeta functions, the generalized version of normal Riemann functions, and we gave through this formulation an 193

Thesis Conclusions

estimate of the magnetic field value so that the system may be considered as „semiclassical‟. Related results were extracted by taking into account the Zeeman term. We also estimated the electronic density of a hypothetical solid, in which the predicted strong violations of the dHvA effect would show up in the lab. Our line of reasoning is quite general when it comes to one-electron Physics, and we devoted for completeness a section (describing a corresponding analytical calculation) to 3D exotic materials, the Topological Insulators (or, better, the Topological Band Insulators, that have been defined and studied during the last decade within the framework of one-electron Physics, i.e. the well-known Band Theory). There, by considering a slab of topological material of finite small thickness, we investigated the critical thickness where the first transition (a one-electron crossover) occurs in full detail. To complete our analysis, Aharonov-Bohm electronic configurations were also investigated, in which the electric current was found to have the standard saw-tooth behavior (for 2D systems in a magnetic field), namely periodic with respect to a flux quantum. In other parts of this work, again in the framework of one-electron Physics, we proposed a method for more appropriate handling of the gauge transformations connecting two different gauges. This method actually “cures” an old issue raised in the literature on quantum gauge transformations (as a warning against a naive handling of gauge transformations that has led to erroneous results). In the framework of Landau Level Physics we thoroughly analysed the issue and we gave a complete resolution by taking advantage of two rather “forgotten” quantities, the pseudomomentum and a pseudo-angular momentum. Our method was expanded to cases where spatially inhomogeneous magnetic fields are present, with the previous pseudomomentum quantities transformed into line integral-operators and therefore generalized (appearing in this form, to our knowledge, for the first time). At some step of our research activity, my advisor had noticed a boundary nonHermitian influence that was spontaneously showing up in two well-known theorems of Quantum Mechanics, the Hellmann-Feynman and Ehrenfest theorems, resolving earlier (but also new) paradoxes. Indeed, a modification of Hellmann-Feynman and Ehrenfest theorems that goes beyond the status quo of ordinary Quantum Mechanics was apparently showing up naturally, with an interesting structure based on generalized boundary currents (defined by a feedback operator) that do not always satisfy a continuity equation (this depending on the properties of this feedback operator). My contribution to this issue consisted in the working out of certain applications of these non-Hermitian boundary terms, such as the problem of quantum motion in a crystal (the standard Bloch theory), where it corrected a textbook error in the literature, concerning the average value of momentum with respect to Bloch wavefunctions. As already mentioned, these boundary terms “cure” several earlier Quantum Mechanical paradoxes (some of them appearing in Quantum Chemistry in the form of a violation of the so-called Hypervirial theorem), with some of the simplest of these paradoxes being discussed in this work in order to make this odd 194

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issue as clear as possible. An even more generalized version of the HellmannFeynman theorem was also presented, originally noted and derived by my colleague Kyriakos Kyriakou, in which the parameters are allowed to have time-dependence (and not a necessarily adiabatic one). The results involve two quantities: a Berry curvature and a Berry electric field, that both act on a generalized parameter spacetime. Ιn this, I identified the Berry electric field and I observed that together with the Berry “magnetic field” (which is a dynamic generalization of the Berry curvature) they obey the exact analog of Maxwell equations, but now in the parameter space. As a consequence, a fictitious particle in the parameter space with charge equal to Planck‟s constant is born, and it is responsible for the quantization of “magnetic charge” (actually related to the well-known Chern number quantisation in the framework of the Gauss-Bonnet theorem, that is actually discussed as background information in the beginning of this Dissertation). Last but not least, I have always wondered about the thermodynamics of the 2D Quantum Hall Effect system in an extremely strong electric field E. We do know that when the E-field is weak enough, in such a way that an energy gap is well defined, the Hall conductivity at low temperatures is quantized in universal values. But when the E-field becomes strong enough energy gaps are destroyed, and the Quantum Hall plateaus are expected to simultaneously disappear. Motivated by this not well-studied problem, I analysed the thermodynamics – first for a conventional system, with parabolic spectrum – and showed that the total energy depends on several nonsymmetric terms, some of which did not appear in the low E-field limit. For example, a term that depends on the product EB emerges that is only present when there are fully occupied L.Ls. Some (E-dependent) deviations from the standard dHvA periodicities were observed in our results, and explained. At the end, a quantisation of Hall conductivity was also derived, although, in order for one to comment on the observability of the resulting plateaus one needs to add or incorporate disorder, or even finite edges (and an associated confinement potential) in this system, so that the solutions are capable of describing realistic conditions. This needs to be done in future works, and the manner in which one can proceed is commented on further below. Apart from the conventional system, a corresponding calculation for a pseudorelativistic system (with a Dirac-type of spectrum) such as Graphene was also carried out in full detail, demonstrating its own particularities and with very different analytical patterns and predictions. As already mentioned emphatically a number of times, a fuller treatment (possibly with different analytical and numerical tools) would be a natural extension of the above work, in order to provide more secure predictions for a realistic system that may contain defects, impurities, edges with confining potentials and of course spin, parameters that deserve to be considered and worked out in the future

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