Spectrum of Graphene Quantum Dots

Pontifícia Universidad Católica de Chile Facultad de Física

Spectrum of Graphene Quantum Dots Hanne Van Den Bosch

Tesis presentada a la Facultad de Física de la Ponticia Universidad Católica de Chile para optar al grado de doctor en Física.

Profesor Guía Comisión Informante

Rafael D. Benguria Gustavo Düring Søren Fournais Edgardo Stockmeyer

Agosto 2017 Santiago, Chile

©

2017, Hanne Van Den Bosch.

Se autoriza la reproducción total o parcial, con nes académicos, por cualquier medio o procedimiento, incluyendo la cita bibliográca del documento. The content of this thesis may be reproduced partially or entirely for academical purposes when a reference to the original is included.

iii

Voor Vake.

Acknowledgements First of all I would like to thank my advisor, Rafael Benguria, for the warm welcome in Chile and for his kind guidance throughout the last four years. It has been a real privilege to learn from his profound knowledge on almost every topic in mathematics, physics and history of science, and even more so since he was always eager to teach something. In addition to these gems of knowledge, I hope to take some of his optimism, enthusiasm and open-mindedness into the rest of my life. I am also very grateful to Edgardo Stockmeyer for introducing me to the subject of graphene, for the agreeable work together and his wise advice, and to Søren Fournais for many interesting discussions, for inviting me to Aarhus and for teaching me about semi-classical limits. I would like to thank Gustavo Düring for accepting to read this thesis and to provide a connection to the physics-side of the spectrum. The research work leading to this thesis has beneted from many discussions, courses and seminars.

In particular, I would like to thank Martín

Chuaqui for his comments on conformal maps. It has been specially enriching to visit IST Austria, the University of Stuttgart and the University of Aarhus, and I am grateful to Robert Seiringer and Phan Thành Nam, Marcel Griesemer and Timo Weidl, and Søren Fournais, for welcoming me in their research groups. I am also very grateful to thank Phan Thành Nam and Rupert Frank for sharing their insight on the ionization problem. I would like to say thank you to the administrative sta in Chile and abroad for their ecient help in providing the necessary conditions for this work. It has been a great experience to share an oce with physicists from dierent areas and I am thankful to each of them for keeping me up to date with string theory, susy, solid state experiments and Champion's League football, among other topics. Many friends from inside and outside the university encouraged me during the PhD. Thanks to all of them! I am grateful to my family in Belgium for their care and support during the last twenty-seven years, to my parents for encouraging me in dicult moments and very specially to my grandfather Vake, an eternal teacher who passed on his love for logic through many of his unforgettable stories and riddles. Warm thanks to my Chilean family for adopting me and particularly to Noelia and Gustavo for babysitting while I am writing these lines. And of course, to Pedro for his invaluable and steady support and to little Inés for her heart-warming smile. Finally, nancial support from the Vicerectoría de Investigación, CONICYT through scholarship 63140048 and FONDECYT project #1120836 and the Iniciativa Cientíca Milennio through the Milennium nucleus Física Matemática, RC120002 is acknowledged.

v

Contents

Acknowledgements

v

Contents

vii

Abstract

ix

1 Introduction

1

1.1

Electronic excitations in graphene

. . . . . . . . . . . . . . . .

3

1.2

Boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

Denitions and notations

6

. . . . . . . . . . . . . . . . . . . . .

2 Domain of the Dirac operator with boundary conditions

7

2.1

Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . .

10

2.2

Construction of a singular Weyl sequence

. . . . . . . . . . . .

16

2.3

4-component description . . . . . . . . . . . . . . . . . . . . . .

17

3 Spectral gap of graphene quantum dots

21

3.1

Proof of Theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . .

3.2

Spectral gap for multiply connected domains

. . . . . . . . . .

27

3.3

Spectral gap for 2 valleys. . . . . . . . . . . . . . . . . . . . . .

30

4 Weyl asymptotics

24

31

4.1

General tools

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

Local asymptotics

4.3

Localizing, straightening the boundary and conclusion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 35 42

Bibliography

47

Appendices

53

A Computations in the tight-binding model

55

A.1

From band structure to Dirac operator . . . . . . . . . . . . . .

55

A.2

Armchair and zigzag boundary conditions . . . . . . . . . . . .

57

B Comparison of eigenvalues

63 vii

Contents

viii

C The invertible double of Ω

65

C.1

Adding a curved extension . . . . . . . . . . . . . . . . . . . . .

65

C.2

Constructing the Dirac operator

. . . . . . . . . . . . . . . . .

66

C.3

Construction of the invertible double . . . . . . . . . . . . . . .

68

C.4

Doubling spinors with innite mass boundary conditions . . . .

69

D Eigenvalues for an annulus or disc

71

E Technical lemmas from Chapter 4

77

E.1

General tools

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

E.2

Local asymptotics

E.3

Localizing, straightening the boundary and conclusion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 79 81

Abstract

The goal of this thesis is to study Dirac operators describing low-energy electronic operators in graphene, in the case where these excitations are conned to a bounded (mesoscopic) region: a quantum dot. A rst result is a classication of local boundary conditions and their inuence on the domain of denition of the operators. We obtain a domain included in the rst Sobolev space, which implies an elliptic regularity result, for an explicit family of boundary conditions, including the physically important

armchair

and

innite mass boundary

conditions. Such a regularity result fails for the other type of boundary conditions from the physics literature, the

zigzag

boundary conditions.

For boundary conditions satisfying the regularity result, the spectrum is discrete and we can study some quantitative properties of the eigenvalues

λk .

Under some conditions, we prove the existence of a gap in the spectrum around zero. When the boundary conditions are in addition uniform and the domain is simply connected, an explicit bound relating the size of the spectral gap to the size of the quantum dot is obtained.

We discuss a weaker result for

one-connected domains and illustrate the phenomenon by some explicit computations for the eigenvalues of a disc or annulus. Finally, we study the average asymptotic behavior of the large eigenvalues (squared) by computing a two-term asymptotic expansion of the Riesz mean classical parameter

h

tends to zero.

ix

2 2 k [h λk

P

− 1]−

as the semi-

Chapter 1 Introduction

The goal of this thesis is to present a mathematical study of the operators describing (in a certain limit) excitations in graphene quantum dots. The results described have been obtained in collaboration with Rafael Benguria, Søren Fournais and Edgardo Stockmeyer (Chapters 2 and 3) and with Søren Fournais (Chapter 4).

Most of the mathematical results have been published in

[BFSVDB17b, BFSVDB17a], but the thesis contains a more detailed description, some alternative proofs and a few secondary results that have not been published. Graphene is a eycomb lattice.

2-dimensional

material consisting of carbon atoms on a hon-

+

Its experimental realization in 2003 [NGM 04] motivated a

vast amount of research in experimental and theoretical solid stated physics and was awarded with the 2010 Nobel prize in physics.

This is because the

two-dimensional nature and the special symmetries of the honeycomb lattice

+

give graphene extraordinary properties (see e.g., [Per10], [CNGP 09] for a review), such as huge mechanical resistance, transparency, excellent conductivity of heat and electricity. For this thesis, only this last point will be of importance. It has been predicted on the basis of band structure computations [Wal47]

+

and conrmed experimentally [NGM 05] that the electrons conducting electricity in graphene are described in a low-energy, non-interacting limit as massless Dirac fermions. In other words, their kinetic energy is given by the Dirac operator

D = −i~vf σ·∇ , where

(1.1)

σ = (σ1 , σ2 ) is the vector of Pauli matrices (see below) and the operaC2 -valued wavefunctions (spinors). The constant vf refers to the 6 −1 velocity in graphene, approximately 10 m s . Since vf is two orders

tor acts on Fermi

of magnitude smaller than the velocity of light, pseudo-relativistic eects show up in graphene at low energies, and graphene provides a playground to observe some characteristic phenomena from quantum electrodynamics, see [KN07] for an overview. Here, the objective is to study a graphene quantum dot: a piece of graphene large enough to use the description by a continuum Dirac operator but small 1

Chapter 1. Introduction

2

enough to see eects related to the connement of electrons, as observed ex-

+

perimentally in [PSK 08]. In this case the boundary conditions play a crucial role. In the physics literature, several boundary conditions are used, mostly the so-called

zigzag, armchair and innite mass boundary conditions.

A rst math-

ematical result in this thesis is a description of all possible boundary conditions and the domain of the corresponding self-adjoint operators, see Chapter 2. If one requires the operator to be self-adjoint on a domain in the rst Sobolev space

H 1,

which is the space of functions with square-integrable rst deriva-

1

tives, the boundary conditions have to satisfy some restrictions . In particular,

zigzag

boundary conditions are not suitable.

Under suitable boundary conditions, the Dirac operator will have discrete spectrum. The goal of Chapters 3 and 4 is to study this spectrum quantitatively.

A rst result is an explicit bound on the size of the spectral gap or

the distance of the spectrum to zero.

A gap is of crucial importance to use

graphene quantum dots in electronic components such as a single-electron transistors.

For

armchair

obtain in Theorem

and

innite mass

boundary conditions, the bound we

λ reads r 2π . |λ| ≥ ~vf area

3.3 on eigenvalues

(1.2)

Introducing numerical values,

√ |λ| area ≥ 2.5 × (6.5 × 10−16 eV s) × (106 m s−1 ) ≈ 1.5 eV nm. 0.1 eV,

This means that, in order to obtain a gap of of graphene of about

30 nm

of diameter.

one would need a piece

This is large compared to the dis-

tance between the carbon atoms which is of the order of

0.1 nm

, and thus the

continuum description is justied. The spectral gap is a way of studying the eigenvalues closest to zero. Complementary to this, we can study the behavior of large eigenvalues in the socalled semi-classical limit. If we enumerate the eigenvalues by

λj ,

with

j ∈ Z,

a natural object to study is the Riesz mean

X j

[h2 λ2j − 1]− ,

in the limit where the semi-classical parameter

h tends to zero.

Using the stan-

dard tools to study these so-called Weyl asymptotics, we obtain an expansion of the form

X 4π [h2 λ2j − 1]− = h−2 + h−1 × boundary area j

term

+ o(h−1 ),

where the boundary term depends on the length of the boundary and the boundary conditions, see Theorem 4.1 in Chapter 4 for the exact expression.

1 see

Theorem 2.3 for the precise statement.

1.1. Electronic excitations in graphene

a1

3

k1 k2

a2

K0

K

Figure 1.1: The honeycomb lattice (left) consists of two triangular sub-lattices generated by lattice vectors

a1 , a2 .

A unit cell is shaded in gray.

The reciprocal lattice (right) is an equilateral triangular lattice generated by basis vectors is shaded in gray. denoted

K

and

k1 , k2 .

The hexagonal rst Brillouin zone

The corners of the hexagon are Dirac points,

K 0.

The rest of this introduction consists of a short discussion of electronic excitations in graphene, the boundary conditions used in the physics literature, and an introduction of the notations that will be used throughout the thesis. A more detailed (formal) derivation of (1.1) from the tight-binding model for graphene can be found in Appendix A.

1.1 Electronic excitations in graphene The honeycomb lattice (see Figure 1.1) is not a Brillouin lattice. It consists of a superposition of two triangular lattices or, equivalently, of translations by lattice vectors

a1 , a2

of a unit cell containing two carbon atoms.

When

computing the band structure (originally computed by Wallace in [Wal47]) in the tight-binding model including only nearest-neighbor interactions (see Appendix A), the energy levels are

E± (k) = ±t 1 + eik·a1 + eik·a2 . This means the two bands have conical intersections at zero energy at the corners of the Brillouin zone, the so-called Dirac cones. Around these cones,

2

E± (K + q) = ±vf |q| + O(|q| ). Here,

vf ≈ 106 m s−1

is the Fermi velocity in graphene. Electronic excitations

in graphene thus obey a ultra-relativistic dispersion relation, but with the role of the velocity of light replaced by

vf .

More generally, Feerman and Weinstein

[FW12] proved that conical intersections occur for Schrödinger operators with


4

Figure 1.2: A terminated honeycomb lattice. The upper boundary has a zigzag termination, the right side has an armchair termination. dashed sites are the

missing

The

sites where the wavefunction should

vanish.

generic potentials that have the symmetries of the honeycomb lattice. (See also [FLTW16] for a rigorous derivation of the tight-binding band structure.) There are two inequivalent Dirac points

K

and

K 0,

also called

valleys.

If

one takes as an ansatz for the electronic wavefunction an oscillating part with momentum

K

or

K0

and a slowly varying envelope, the envelope function

(a two-component function corresponding to each of the sub-lattices) satises in the continuum limit a Dirac equation.

The details of this computation

can be found again in Appendix A. There are two envelopes corresponding to expansions with momenta close two each of we need a pair of spinors or a

C

4

K

and

K 0 , so for a full description,

-valued function.

1.2 Boundary conditions. When describing a bounded piece of graphene, boundary conditions have to be imposed.

In Section 2.3, this point will be studied from a mathematical

viewpoint. There exist two families of boundary conditions depending on two and three continuous parameters respectively. In the physics literature however, authors commonly study conditions.

zigzag, armchair

and

innite mass

boundary

1.2. Boundary conditions.

5

Zigzag boundary conditions arise naturally from the lattice termination when the border is perpendicular to one of the three nearest-neighbor bonds, as shown in Figure 1.2. This boundary condition requires the component on one of the sub-lattices to vanish at the boundary.

If the lattice termination

is parallel to nearest-neighbor bonds, armchair boundary conditions show up. These conditions require a vanishing of components on both sub-lattices and to achieve this one needs a peculiar relation between the contributions from both valleys to hold at the boundary. Innite mass boundary conditions were rst introduced in [BM87] for theoretical reasons. These boundary conditions do not arise from the lattice termination, but can be generated when a mass term

M σ3 ,

where

M

is supported outside the domain under consideration, is added

to the Dirac operator. As

M

tends to

±∞,

the resulting operator converges in

a suitable sense (see [SV16]) to a Dirac operator in the domain with boundary conditions that relate both pseudo-spin components. The mathematical results we obtain are the following: on a bounded domain, the operators with innite mass and armchair boundary conditions have

±∞. As noted in [Sch95], the 0 in its essential spectrum. For conditions, 0 is never an eigenvalue. If

discrete eigenvalues that accumulate only at

operator with zigzag boundary conditions has armchair and innite mass boundary

in addition, the domain is simply connected, the bound (1.2) holds. For the detailed statement, see Theorem 3.3 and Proposition 3.6. Zigzag boundary conditions are considered to be

generic

when the car-

bon lattice terminates in rational directions not exactly parallel to the bonds, [AB08, FLTW16]. These boundary conditions give rise to states with almost zero energy localized close to the edges, which translates mathematically in an operator that is not elliptic. Also numerical simulations [LSB09] suggest these edge states survive even in the presence of disorder. Armchair boundary conditions for graphene akes and nanoribbons have been observed to open a gap in the spectrum around zero, [BF06, ORZ13,

+

ZWL 07]. The eigenstates corresponding to these low energies are rather delocalized. Innite mass boundary conditions can be seen as an idealization of the operator with a nite mass term localized outside the domain under consideration.

Physically, such a mass term comes from a potential with opposite

signs on both sub-lattices.

Various strategies to obtain such a mass by in-

+

teraction between graphene and its substrate have been proposed [ZGF 07,

+

+

GKB 07, RPM 10].

Experimentally, eective masses of about

0.2 eV

have

been obtained. In [GRS15], the inuence of such a mass term localized outside a circular quantum dot is investigated. Interestingly, for this small, nite mass term, the authors obtain gaps decreasing as disc, and

C

C/r,

where

r

2

is the radius of the

is of the same order of magnitude as in (1.2).

2 Also for the experimental realization of quantum dots [PSK+ 08] etched out of a graphene slice, a typical spacing between energy levels (multiplied by the dot's diameter) of this order of magnitude is observed. However, from the experimental data presented it is not possible to determine if this spacing occurs exactly between the rst negative and the rst positive energy level.


6

1.3 Denitions and notations We will mainly study the

2-dimensional

Dirac operator dened by the dier-

ential expression

T = −iσ·∇ = −i

0 ∂1 + i∂2

∂1 − i∂2 , 0

~ = vf = 1. The Pauli matrices are 0 1 0 −i 1 σ1 = , σ2 = , σ3 = 1 0 i 0 0

in units where

0 . −1

They satisfy the (anti)-commutation relations

{σj , σk } = 2δjk , where

δjk

[σj , σk ] = 2ijkl σl , jkl

is the Kronecker delta and

totally anti-symmetric and normalized by

j, k, l ∈ {1, 2, 3},

is the Levi-Civita symbol, which is

123 = 1.

In general, a dot will denote the scalar product in

X

a·b=

R2

or

R3 ,

or

aj bj .

j Even if

a

has complex entries, there is no complex conjugation. An asterisk

refers to the complex conjugate of a complex number or the adjoint of an operator. For

C

2

v ≡ (v1 , v2 )> ∈ C2 ,

we will occasionally write

(·, ·)C2 .

-inner product will be denoted by

v ∗ ≡ (v1∗ , v2∗ ).

The

During most of the work, we will

Ω and when this causes no confusion, k·k and (·, ·) will L2 (Ω, C) or L2 (Ω, C2 ). A particular domain will be denoted by D, and the unit circle, its boundary,

consider a xed domain

be the norm and scalar product in is the unit disc, which by

S1 . When discussing the application of our results to the physics of graphene,

we will also consider

V ≡

acting on

L2 (Ω, C4 ),

T 0

0 T

,

which takes into account contributions of both valleys

(after a suitable unitary transformation, see Appendix A). We will frequently use the abbreviations z.z., a.c.,

∞-m.

to refer to zigzag, armchair, and innite

mass boundary conditions respectively.

Chapter 2 Domain of the Dirac operator with boundary conditions

As we have seen before, our goal is to study operators with dierential expression

T ≡ −iσ·∇

in

2

R

acting on

L2 (Ω, R2 ),

where

Ω is a suciently regular domain C ∞ (Ω, C2 ). In this case, we can

. We start by dening this expression on

integrate by parts and obtain

Z Ω

(u, T v)C2 = −i

XZ Ω

j

(σj u, ∂j v)C2

Z = −i

∇ · (σu, v)C2 + i

Ω

XZ Ω

j

Z

(σj ∂j u, v)C2

Z

= −i ∂Ω

(n·σ u, v)C2 +

Ω

(T u, v)C2 .

(2.1)

If we look for local boundary conditions, we can write them in each point of the boundary as projections on the eigenspace associated to two-by-two matrix

A.

So if we assume that, on

(u, n·σ v)C2 =

∂Ω, Au = u

1 (u, {n·σ , A}v)C2 2

on

1

of a self-adjoint

and

∂Ω.

In order to obtain a symmetric operator, the boundary operator commute with

n·σ .

Av = v ,

A should anti-

If we want to construct a self-adjoint operator, in addition

we will need a matrix with exactly one eigenvalue equal to

1.

generality we can assume that the other eigenvalue is equal to

Without loss of

−1

and thus we

are looking for a linear combination of the Pauli matrices. There is only one free parameter left and we dene

Aη = sin(η)σ3 + cos(η)t·σ .

(2.2)

If we are interested in the two-valley description, we can look, in a similar way, for a four-by-four, unitary and traceless matrix

n·σ 0

A

anti-commuting with

0 . n·σ 7

(2.3)

Chapter 2. Domain of the Dirac operator with boundary conditions

8

This case will be studied in Section 2.3. Restricting for the moment to the one-valley description, we have constructed symmetric operators

Dη

acting on

D(Dη ) ≡ {u ∈ H 1 (Ω, C2 )|Aη γu = γu}, where

γ

(2.4)

Ω. Not all operators dened in this η = π/2, we obtain the zigzag boundary

is the trace at the boundary of

way are self-adjoint. For instance, if

conditions that have been studied in [Sch95]. In this case, all functions of the

u = (f (x + iy), 0)T , with f holomorphic, T u = 0. This shows that zero is in the essential form

are in the domain and satisfy spectrum of the operator with

zigzag boundary conditions. If the operator would be self-adjoint, its spectrum should consist of eigenvalues of nite multiplicity that only accumulate at

±∞,

see Proposition 3.1. In this case, in order to obtain a self-adjoint operator, the

H 1 (Ω).

domain should be extended to include functions that are not in

This

domain is described implicitly in the following proposition.

Proposition 2.1. Let Ω ⊂ R2 be a C 2 -domain and η : ∂Ω 7→ R a C 1 -function. Denote by γ the trace on ∂Ω. Then Dη dened in

(2.4)

is symmetric and

D(Dη∗ ) = {u ∈ L2 (Ω, C2 )|T u ∈ L2 (Ω, C2 ), Aη γE u = γE u ∈ H −1/2 (∂Ω, C2 )},

where γE coincides with γ on H 1 (Ω) and is dened in Proof.

That

Dη

.

(2.5)

is symmetric follows from the construction of

Aη .

First we

make precise in which sense the trace operator can be extended to more general

E : H 1/2 (∂Ω) 7→ H 1 (Ω) a bounded extension operator 1/2 7.35]) and dene for φ ∈ H (Ω) and v ∈ L2 (Ω) such

functions. We denote by (see [Ada75][Theorem that

T v ∈ L2 (Ω), γE v[φ] = i ((v, T Eσ·n φ) − (T v, Eσ·n φ)) .

By (2.1), this coincides with the usual trace

γ

for all

(2.5)

v ∈ H 1 (Ω).

This de-

nition is independent of the choice of the extension operator, since if

e − E)σ·n φ can be other extension operator, (E ∞ 2 ∞ 2 functions in C0 (Ω, C ). For u ∈ C0 (Ω, C ),

approximated in

H

1

e E

is an

-sense by

(T v, u) − (v, T u) = 0 by the denition of the distributional derivative. We rst prove the inclusion

D(D∗ ) ⊂ {u ∈ L2 (Ω, C2 )|T u ∈ L2 (Ω, C2 ), AγE u = γE u ∈ H −1/2 (∂Ω, C2 )}. Fix

v ∈ D(D∗ ).

For all

u ∈ C0∞ (Ω, C2 ) ⊂ D(D), (D∗ v, u) = (v, Du) = (v, T u) = T v[u]

9

so we may identify the distribution

γE v

Tv

with

D∗ v ∈ L2 (Ω).

Thus we can bound

dened in (2.5),

|γE v[φ]| ≤ kvkL2 (Ω) kEσ·n φkH 1 (Ω) + kD∗ vkL2 (Ω) kEσ·n φkL2 (Ω) ≤ CE (kvkL2 (Ω) + kD∗ vkL2 (Ω) )kφkH 1/2 (∂Ω) , and

γE v ∈ H −1/2 (∂Ω)

as required.

γE v (1 − A)γE v = 0.

Finally, we check that

boundary conditions, which can be written as

satises the We start by

computing

(1 − A)γEσ·n (1 − A)φ = (1 − A)σ·n (1 − A)φ = (1 − A)(1 + A)σ·n φ = 0, φ ∈ H 1/2 (∂Ω), the extension Eσ·n (1 − A)φ ∈ D(D). (1 − A)γE v [φ] = γE v[(1 − A)φ]

so for all

By (2.5),

= −i ((v, DEσ·n (1 − A)φ) − (D∗ v, Eσ·n (1 − A)φ)) = 0. We still have to prove the opposite inclusion. We x

v ∈ {u ∈ L2 (Ω, C2 )|T u ∈ L2 (Ω, C2 ), AγE u = γE u ∈ H −1/2 (∂Ω, C2 )}, and dene

D∗ v ≡ T v.

u ∈ D(D),

Now we have to check that, for all

(T v, u) = (v, Du) . Since the extension operator is arbitrary, we can assume using the properties of

Eγu = u.

Then by

A,

−iγE v[σ·n γu] = −i/2 (AγE v[σ·n γu] + γE v[σ·n Aγu]) = −i/2γE v[{A, σ·n }γu] = 0 Therefore, for all

u ∈ D(D),

we have

γE v[σ·n γu] = 0

and

(T v, u) − (v, Du) = iγE v[σ·n γu] =0 Now we come to the main result of this chapter. If operator is self-adjoint with domain in

cos η 6= 0

on

∂Ω,

our

H 1 (Ω).

Theorem 2.2. Let

Ω ⊂ R2 be a C 2 -domain and η : ∂Ω 7→ R a C 1 -function such that cos η 6= 0. Then Dη as dened in (2.4) is self-adjoint.

Remark 1.

If the boundary and the boundary function

η

are

C ∞,

then this

theorem can be proven by studying the interplay between the principal symbol of the Calderon projection [BBW93, Chapter 12] and the projection dening


10

the boundary conditions. (See also [Pro13, BBLZ09]). The Calderon projector is a projection in

L2 (∂Ω)

that projects on the space of boundary values of

harmonic spinors i.e., solutions u of T u = 0 in Ω.

If this projection is, in a cer-

tain sense, complementary to the projection dening the boundary conditions,

T

then the operator

together with the boundary conditions

Aη

form an elliptic

boundary value problem. This implies, among other things, that the operator

Dη

is self-adjoint. Here, we give a proof that is long but self-contained and does

not require advanced tools from the analysis of pseudo-dierential operators.

Remark 2.

Our result is an elliptic regularity result. Implicitly, we establish

the following type of estimate: for

C2 ≥ 0

depending on

Ω

and

η

v ∈ D(Dη ),

there exist constants

k∇vk ≤ C1 kDη vk + C2 kvk

Remark 3.

The hypothesis

C1 > 0,

such that (2.6)

cos η 6= 0 can not be relaxed much.

If

cos η

tends to

zero at least quadratically at some point of the boundary, we show in Section 2.2

0 ∈ σess (Dη ). H (Ω).

that in

As explained above, this means that

1

Remark 4.

D(Dη∗ ) is not contained

cos η = 0 at a part of the boundary, we can still conclude D(Dη∗ ) are H 1 away from this part of the boundary. This can ∗ be seen by applying the nal argument in the proof (see page 14) to v ∈ D(Dη ) Finally, if

that functions in

multiplied by a localization function that cancels on the part of the boundary where

cos η = 0.

The remainder of this chapter contains the proof of Theorem 2.2 and the construction of a singular Weyl sequence as claimed in Remark 3. For the full description with 4 components and boundary conditions that are not blockdiagonal, the following generalization of Theorem 2.2 will be proven in Section 2.3.

Theorem 2.3. Fix a 2

2

cos (θ) 6= sin (ζ)

and

C 2 -domain Ω and C1 -functions θ, ζ, φ : ∂Ω 7→ R. cos(θ) 6= 0, the operator Vm is self-adjoint on

If

D(Vm ) ≡ {ψ ∈ H 1 (Ω, C4 )|Am γψ = γψ},

where Am (θ, φ, ζ) =

sin θ(cos ζ σ3 + sin ζ t·σ ) e−iφ cos θ(sin ζ σ3 − cos ζ t·σ ) . eiφ cos θ(sin ζ σ3 − cos ζ t·σ ) sin θ(cos ζ σ3 + sin ζ t·σ )

2.1 Proof of Theorem 2.2 D(Dη∗ ) ⊂ D(Dη ). In view of Proposition 2.1, we only ⊂ H 1 (Ω). The next Lemma says that for this it is sucient ∗ 1/2 to show that traces of spinors in D(Dη ) are in H (∂Ω). In the rest of this section we drop the subscript η .

We have to show that

∗ have to show D(Dη )

Lemma 2.4. If v ∈ D(D∗ ) and γE v ∈ H 1/2 (∂Ω, C2 ), then v ∈ H 1 (Ω, C2 ).

2.1. Proof of Theorem 2.2

Proof.

v ∈ D(D∗ )

11

γE v ∈ H 1/2 (∂Ω). As before, we denote by E 1 a bounded extension operator. Since EγE v ∈ H (Ω), we may restrict our 2 2 attention to u ≡ v − EγE v . We dene w ≡ T u and denote by ue , we ∈ L (R ) the respective extensions by zero. First, we check that T ue = we . For all test ∞ 2 functions φ ∈ Cc (R ), Fix

with

T ue [φ] = (ue , T φ)L2 (R2 ) = (u, T φ)L2 (Ω) = (T u, φ)L2 (Ω) − iγE u[σ·n γφ] = (w, φ)L2 (Ω) = (we , φ)L2 (R2 ) , where we have used that

γE u = 0

by denition. In

R2 ,

we can apply a Fourier

transform to compute

kT ue k2L2 (R2 ) =

Z

2

2

|k| |c ue (k)| dk = k∇ue k2L2 (R2 ) ≥ k∇ue k2L2 (Ω)

Having established this, we may restrict our attention to the local behavior of functions in

D(D∗ ) close to the boundary.

We will rst solve the problem on

the unit disc and then take advantage of the special form of the Dirac operator and the Riemann mapping theorem to translate this results to our problem at hand. This strategy is very specic for the two-dimensional case. A more general but also longer approach would be to restrict to local balls, straighten the boundary and use Fourier transforms as is done in [BBW93].

Regularity for traces on the unit circle In this subsection, we identify

R2

with

C

in order to take advantage of the

structure of the Dirac operator. In this notation, we have

0 T u(z) = −2i ∂z∗

∂z ∂z u2 (z) u(z) = −2i , 0 ∂z∗ u1 (z)

where the Cauchy-Riemann derivatives are

1/2(∂x + i∂y ).

∂z = 1/2(∂x − i∂y )

and

∂z∗ =

We also introduce the Cauchy kernel and its complex conjugate,

that map functions on

(Kf )(z) =

1 2πi

S1 Z

to (anti)-holomorphic functions in

S1

f (ζ) dζ, ζ −z

(Kf )(z) =

−1 2πi

Z S1

D

respectively,

f (z) dζ ∗ . ζ ∗ − z∗

With these two extension operators, we dene

S≡ T Sv = 0 in D. 2 1 basis for L (S , C),

such that mal

K 0

0 , K

In the standard coordinates on

en (θ) = (2π)−1/2 einθ ∈ L2 (S1 ).

S1 , we x an orthonor-


12

The action of

K

and

K

on these basis functions is

( (2π)−1/2 z n if n ≥ 0 , (Ken )(z) = 0 else. ( (2π)−1/2 (z ∗ )n if n ≤ 0 (Ken )(z) = . 0 else.

(2.7)

The following lemma groups the properties of the Cauchy kernel that we will need.

Lemma 2.5. If K and K are dened as above, then for all s ∈ [−1/2, 1/2] i) K and K extend to bounded operators from H −1/2 (S1 ) to L2 (D). ii) For all f ∈ H s (S1 ) we have ∂z∗ Kf = 0 and ∂z Kf = 0 with derivatives taken in the sense of distributions. iii) γK and γK extend to bounded operators on H s (S1 ) and they are selfadjoint projections onto span{en |n ≥ 0} and span{en |n ≤ 0}, respectively. iv) γK + γK = 1 + (e0 , ·) e0 when acting on H s (S1 ). v) For β ∈ C 1 (S1 ) and s = −1/2 or s = 0 the commutators [β, γK] and [β, γK] are bounded from H s (S1 ) to H s+1/2 (S1 ). Proof. for

K

Point iv) follows directly from iii). We will prove the remaining items only since the proof for

K

is identical. We can also restrict our attention

to continuous functions, since the statement for general functions in follows by density. For

(en , f )

2

1

f ∈ L (S )

we dene the Fourier coecients

H s (S1 ) fb(n) =

and we take as a denition

kf k2H s =

X

2 (1 + |n|)2s fb(n) .

n∈Z Now we compute

1 X b ∗b f (n) f (m) (z n , z m ) 2π n,m≥0 2 X = fb(n) (1 + 2n)−1 ≤ kf kH −1/2 ,

kKf k2L2 =

n≥0 which establishes i). Point ii) is clear for continuous functions. Point iii) follows

s = −1/2 or s = 0, x f ∈ C 1 (∂Ω) and [β, γK]f = βγKf − γK(βf ),

from (2.7). To see v), we take the Fourier coecients of

√

2π [β, γK]f

∧

(P (n) =

k≥0

P

k≥0

b − k)fb(k) − P b b β(n k∈Z β(n − k)f (k), b − k)fb(k), β(n

compute

n ≥ 0, n < 0.

2.1. Proof of Theorem 2.2

13

By Cauchy-Schwarz,

∧ 2 2π [β, γK]f (n)  2 P b P b 2   β(n − k) (|k| + 1)−2s f (k) (|k| + 1)2s ,  k 0,

2 |k| β −νkβy 1 e b b2 + 1

so that

De ek = λk eek , We will dene

eek ≡ 0

for

νk < 0,

λk =

2bk . b2 + 1

in order not to have to distinguish the cases.

With these denitions we have

0

δ(x − x )IC2

Z Z 1 X 1 ∗ 0 = eσ,k,l (x)eσ,k,l (x ) dk dl + eek (x)e e∗k (x0 ) dk 8π 2 σ=± R2+ 2π R

As before, these generalized eigenfunctions allow to compute the trace up to an error term independent of

h.

Lemma 4.7. If φ ∈ Cc∞ (R2 , R+ ) , then Z Z 2 2 φ (x) TrC2 [H+,b ]− (x, x) ≤ C |∇φ| , Tr[φH+,b φ]− − 2 R+

where TrC2 [H+,b ]− (x, x) =

Proof.

Z Z 1 X 2 [h2 (k 2 + l2 ) − 1]− |eσ,k,l | (x) dl dk 8π 2 σ=± R R+ Z 1 2 [h2 λ2k − 1]− |e ek | (x) dk + 2π R

This closely follows the proof of Lemma 4.5

So, everything reduces to a computation of the integral of

TrC2 [H+,b ]− (x, x)

and an estimation of the dominant terms. This is done in the following lemma.

4.2. Local asymptotics

39

Lemma 4.8. For all φ ∈ C ∞ (R2 , R+ ) satisfying supp(φ) ⊂ B(0, 1), kφk∞ + k∇φk∞ ≤ 1, we have Z Z Z 1 φ2 (x) TrC2 [H+,b ]− (x, x) − h−2 φ2 (x) dx − h−1 Θ(b) φ2 (x, 0) dx R2+ 4π R2 R

≤ Cβ −1 (1 + [ln(βh−1 )]+ ).

Proof. X

First of all, we compute

2

|eσ,k,l | (x, y) = 8 − 4(1 − b2 ) Re e2ily (η 2 − b2 )−1 + (1 − (η ∗ b)2 )−1

,

σ=±

h,

After scaling the variables to extract the dependence on

Z

we obtain

1

1 − h−2 R(l) cos(2lyh−1 ) + I(l) sin(2lyh−1 ) dl 4π 0 e + h−2 R(y/h), (4.4)

TrC2 [H+,b ]− (x, x) = h−2

where we have dened

Z (1 − b2 ) [k 2 + l2 − 1]− Re (η 2 − b2 )−1 + (1 − (η ∗ b)2 )−1 dk, 2 2π R Z (1 − b2 ) I(l) = − [k 2 + l2 − 1]− Im (η 2 − b2 )−1 + (1 − (η ∗ b)2 )−1 dk, 2 2π R Z 1 e = R(t) [λ2 − 1]− 2kβe−2kβt dk. 2π R+ k R(l) =

We have

(η 2 − b2 )−1 + (1 − (η ∗ b)2 )−1 = (k 2 + l2 ) ((1 − b2 )(k 2 + l2 ) + 2ilk)−1 + ((1 − b2 )(k 2 + l2 ) + 2ilkb)−1 , and since the imaginary part of both terms is odd in

k , I(l) = 0.

We now extract the boundary term and estimate the errors for each part separately.

Contribution of Re.

We will show that

1 b2 + 1 e φ (x, y)R(y/h) dx dy − h−1 3π 2 |b| R2+

Z −2 h

2

≤ C(β

−1

−1

+ 1 + [ln(βh

Z R

φ2 (x, 0) dx

)]+ ).

(4.5)

We start by changing variables in order to extract the boundary contribution,

−2

Z

h

R2+

e φ (x, y)R(y/h) dx dy = h−1 2

Z Z R

−1

Z

2

=h

φ (x, 0)

Z

R

+ h−1

Z Z R

0

h−1

Z 0

h−1

e dy dx φ2 (x, hy)R(y)

0

+∞

0 hy

Z e dy − R(y)

+∞

h−1

e dy dx R(y)

e dy dx. ∂2 φ2 (x, s) dsR(y)

Chapter 4. Weyl asymptotics

40

The rst term is easily calculated and gives

+∞

Z

e dy = 1 R(y) 2π

0

Z R+

[λ2k − 1]− dk =

1 b2 + 1 . 3π 2 |b|

This is the boundary contribution in (4.5). For the rst error term, we use, similarly,

+∞

Z −1 1 e R(y) dy = [λ2k − 1]− e−2kβh dk 2π −1 + h ZR −1 1 ≤ e−2kβh dk = C(βh−1 )−1 2π R+ R 2 and by the bounds on φ, φ (x, 0) ≤ C . For the last error term R 2 using ∂2 φ (x, s) ≤ 1 and supp(φ) ⊂ B(0, 1), Z

h−1

h−1

Z Z 0

R

hy

Z

e dy dx ≤ Ch−1 h ∂2 φ2 (x, s) dsR(y)

h−1

Z

0

we have, by

e dy y R(y) 0

Now

Z

h−1

1 e y R(y) = 2π

0 We set

t = 2kβh−1 Z

Z R+

[λ2k

Z − 1]−

h−1

2kβye−2kβy dy dk.

0

and calculate

h−1

2kβye−2kβy dy = h−1 t−1 (1 − e−t ) − e−t ,

0 so

h−1

Z

e y R(y) = Cβ −1

0

Z

t+

[1 − (t/t+ )2 ]t−1 1 − e−t − te−t dt,

0

t+ = t/λk = 1 − b2 / |b| h.

where we have dened

Z

t+

Finally, we bound

t−1 1 − e−t − te−t dt ≤ C(1 + [ln(t+ )]+ )

0 to obtain

h−1

Z

e y R(y) ≤ C(1 + [ln(βh−1 )]+ ).

0 Putting everything together, this gives the estimate (4.5).

Contribution of R. Z −2 h

We now turn to the contribution of

φ2 (x, y)

Z

R2+

1

R(l) cos(2lyh−1 ) dl dx dy − h−1

0

≤ Cβ

−1

−1

(1 + ln([βh

)]+ ).

R,

which is given by

1 3π

Z R

φ2 (x, 0) dx (4.6)

4.2. Local asymptotics

41

The rst step is the same as before. We set

R1 0

b R(l) cos(2ly) dl = R(y)

and

write

h−2

Z R2+

b φ2 (x, y)R(y/h) dx dy = h−1

Z Z

Z

φ2 (x, 0)

Z

R

+ h−1

R Now we note that

h−1

Z Z

Z

b dy dx φ2 (x, hy)R(y)

0

R

= h−1

h−1

+∞

+∞

Z b dy − R(y)

h−1

0 hy

b dy dx R(y)

b dy dx ∂2 φ2 (x, s) dsR(y)

0

0

b R(y) can be seen as the Fourier transform of the function R as an even function to the negative real axis. Since

obtained by extending

this function is continuous, the Fourier transform can be inverted (taking care of some constants) and therefore

Z

b dy = 1 R(y) 2 R+

Z

b dy = π R(0) = 1 . R(y) 4 3π R

We have the bound

Z −2 h

φ2 (x, y)

R2+

Z

1

Z 1 φ2 (x, 0) dx 3π R Z h−1 Z +∞ b b −1 ≤h y R(y) dy. R(y) dy +

R(l) cos(2lyh−1 ) dl dx dy − h−1

0

h−1

0

In order to estimate the error terms, we need good bounds on

b. R

If

β

is

bounded away from zero, these bounds are not dicult to obtain by interpreting

1 b R(y) as the Fourier transform of a compactly supported C and piece-wise 2 C -function. In order to extract the dependence of the error terms on β , unfortunately the best strategy seems to compute the integral explicitly. This gives

2 l3 4 2 3/2 (1 − l ) − θ+ tan2 θ+ − tan θ+ + θ+ , 2 2 3 3π π β √ ≡ arctan(β 1 − l2 /l). For y = 6 0, we may integrate by parts

R(l) = where

θ+

and

compute

Z

1

R(l) cos(ly) dl = −y

b R(y) =

−1

0

Z

1

R0 (l) sin(yl) dl

0

=y

−2

0

0

−2

Z

(R (1) cos(y) − R (0)) − t

1

R00 (l) cos(ly) dl.

0 From the explicit expression of

R,

we see that

R0 (1) = 0,

while

Expanding the arctangent, we obtain

( Cβ −2 |R00 (l)| ≤ Cβ 2 l−4 (1 − l2 )5/2

if if

l≤β l ≥ β.

R0 (0) = Cβ −1 .


42

Therefore, (since

β≤1 Z

by denition)

1

|R00 (l)| dl ≤ Cβ −2 β + Cβ 2 β −3 ≤ Cβ −1 .

0 In summary, we obtain the bound

b R(y) ≤ Cβ −1 y −2 .

(4.7)

This is sucient to treat the rst error term :

Z

+∞

h−1

b dy ≤ Chβ −1 . R(y)

For the second integral, we also use

( |R(l)| ≤ b R(y) ≤ Cβ

to obtain

Z

l≤β l≥β

min(β −1 ,h−1 )

b dy ≤ y R(y) 0

if if

and combining with (4.7),

h−1

Z

C Cβ 2 l−2

"Z

h−1

Cβy dy +

# β

−1 −1

y

dy

β −1

0

+

≤ Cβ −1 (1 + [ln(βh−1 )]+ ). This gives the required estimate (4.6). Inserting (4.6) and (4.5) in (4.4) nally gives the Lemma

4.3 Localizing, straightening the boundary and conclusion In order to apply the results in the previous section to our problem at hand, we have to pick localization functions. Following [FG11], we dene a positiondependent length scale for the localization functions. For

p d(x)2 + l02 1 , l(x) = p 2 d(x)2 + l02 + 1

we set

d(x) ≡ dist(x, R2 \ Ω),

h in the end. In this way, l(x) l(x) ≈ l0 /2, since l0 is small, for x ∈ ∂Ω. NowRwe pick a smooth, nonnegative function g with support in the unit ball and g 2 = 1 and set for ξ ∈ R2 x−ξ p φξ (x) = g J(x, ξ). l(ξ) where

l0

x ∈ R2 ,

tends to

will be chosen as a small power of

1/2

far away from the boundary and

4.3. Localizing, straightening the boundary and conclusion Here,

J(x, ξ)

is the Jacobian of the transformation

ξ 7→

x−ξ l(ξ) .

43

As shown in

Appendix E,

1 3 l(ξ)−2 ≤ J(x, ξ) ≤ l(ξ)−2 2 2

for

x ∈ B(ξ, l(ξ)),

kφξ kL∞ ≤ Cl(ξ)−1 , and kφξ kL2 ≤ C . In Appendix E, we show that the family of functions {φξ : ξ ∈ Ωe } satises the hypothesis of Lemma 4.4, and therefore, where

Ωe ≡ {ξ ∈ R2 : B(ξ, l(ξ)) ∩ Ω 6= ∅}, and

Z

Z

2

|∇φξ | (x) dξ ≤ C Ωe

φ2ξ (x)l−2 (ξ) dξ

Ωe

Now we have to compute traces of

φξ Hφξ .

If

supp φξ

boundary, this can be done by applying Lemma 4.5. If

does not intersect the

supp φξ

does intersect

the boundary, we need to apply a coordinate transformation to transform the boundary into a straight line. There are many ways to do this, but since we are working with the Dirac operator in two dimensions, it is particularly convenient to use conformal mappings.

Lemma 4.9. Take

and l > 0 such that B(ξ, l) intersects ∂Ω. Then for all v ∈ D(D) with support in B(ξ, l) , we can construct vt ∈ D(D+,b ) with support in B(0, (1 + Cl)l) such that ξ ∈ R2

kvk2 − kvt k2 ≤ Clkvk2 kT vk2 − kT vt k2 ≤ ClkT vk2 + Ckvk2 .

Here b = B(s), where s is the point on ∂Ω closest to ξ . The proof is given in Appendix E. This Lemma allows to compute the trace contribution from the boundary balls by a comparison argument as the one used in the proof of Lemma 4.6.

Finally, we have all the necessary tools to

conclude.

Proof of Theorem 4.1 .

Z Tr[H]− −

Ωe

When

τ (ξ) = Cl(ξ)−2 , Z Tr[φξ Hφξ ]− dξ ≤ C l−2 (ξ) dξ.

By Lemma 4.4 with

Ωe

ξ is such that B(ξ, l(ξ)) ⊂ Ω, we can apply Lemma 4.5 to write Z Z 2 2 Tr[φξ Hφξ ]− − h−2 1 φ (x) dx ≤ C |∇φξ | (x) dx 4π R2 ξ R2 ≤ l(ξ)−2 .

(4.8)


44

We will write

Ωi = {x ∈ Ω|B(x, l(x)) ⊂ Ω}

and

Ωb = Ωe \ Ωi .

ξ ∈ Ωb ,

For

we

apply a coordinate transformation as in Lemma 4.9. By a comparison argument as in the proof of Lemma 4.6, see Appendix E for details, we obtain

Tr[φξ Hφξ ]+ − Tr[φξ,T H+,b(ξ) φξ,T ]+ ≤ Ch−2 l(ξ)

Z

φ2ξ (x) dx

Ω

≤ Ch−2 l(ξ), where

b(ξ)

b(ξ)

is the value of

is close to

±1,

B

(4.9)

at the point of the boundary closest to

ξ.

Now if

we apply the improved version of Lemma 4.6, Lemma E.2

from Appendix E to obtain

Z 2 Tr[φξ,t H+,b(ξ) φξ,t ]+ − h−2 1 φξ,t 4π R+ Z 1/2 2 ≤ C |∇φξ,t | + Ch−1 l(ξ)−1 1 − b2 (ξ) and therefore, by using the inverse coordinate transformation and the bounds

φξ , Z Z 2 −1 Tr[φξ Hφξ ]+ − h−2 1 φ + h ξ 4π

on

Ω

∂Ω

Θ(B)φ2ξ

1/2 ≤ Cl(ξ)−2 + Ch−2 l(ξ) + Ch−1 1 − b2 (ξ) l(ξ)−1 + Ch−1

Z

Θ(B)φ2ξ

∂Ω

≤ Cl(ξ)−2 + Ch−2 l(ξ) + Ch−1 β 1/2 l(ξ)−1 + Ch−1 l(ξ)−1 β On the other hand, if

b(ξ)

is not close to

±1

(4.10)

we use Lemma 4.8 and note

that

|Θ(B(s)) − Θ(b(ξ))| ≤ Cl(ξ)

for

s ∈ B(ξ, l(ξ)) ∩ ∂Ω.

Thus we obtain in a similar way

Z Z 2 −1 Tr[φξ Hφξ ]+ − h−2 1 φξ − h 4π

Θ(B)φ2ξ

Ω ∂Ω ≤ Cl(ξ)h−2 + Cl(ξ)−2 1 + β −1 (ξ)(1 + [ln(β(ξ)l(ξ)h−1 )]+ ) + Ch−1 . (4.11)

β ≤ (h/l(ξ))2/3 and the bound the opposite case, we obtain for all ξ ∈ Ωb Z Z 2 −1 2 Tr[φξ Hφξ ]+ − h−2 1 φξ − h Θ(B(s)))φξ (s) ds 4π Ω ∂Ω By picking the bound (4.10) if

in (4.11) in

≤ Cl(ξ)h−2 + Cl(ξ)−2 + Cl(ξ)−4/3 h−2/3 (1 + |ln(h)|) + Ch−1 . By bounding

l(ξ)−2 ≤ C(max(l0 , d(ξ))−2

(4.12)

4.3. Localizing, straightening the boundary and conclusion we obtain

Z

l−2 (ξ) dξ ≤ Cl0−1 .

Ωe Since

p l(ξ) ≤ 1/2 d(ξ)2 + l02 ,

45

for

ξ ∈ Ωb

Z

we can bound

d(ξ) ≤

√

3l0 ,

so

l(ξ)k ≤ Cl0k+1 .

Ωb using this and putting together (4.8) and (4.12), we nally obtain

Z

Tr[φξ Hφξ ]+ dξ − h−2

Ωe 2

≤ C(l0 /h) + ≤ C(l0 /h)2 + where we have used l0 tional to

−6/7

h

|ln(h)|.

≤ h.

1 − h−1 4π

Z

−1/3 −2/3 Cl0−1 + Cl0 h −1/3 −2/3 Cl0 h |ln(h)| Choosing l0

= h4/7

∂Ω

Θ(B(s))) ds

|ln(h)| + C(l0 /h)

nally gives an error propor-

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Appendices

53

Appendix A Computations in the tight-binding model

A.1 From band structure to Dirac operator The honeycomb lattice

Γ

consists of a superposition of two triangular sublat-

tices. We choose as basis vectors (see Figure A.1) for the triangular Bravais lattice

a1 = a

where

−3/2 p , − 3/2

a2 = a

3/2 p , − 3/2

a ≈ 1.42 Å is the distance between neighboring carbon atoms. The recipk1 , k2 satisfying

rocal lattice is again a triangular lattice generated by vectors

(am , kn ) = 2πδmn , and explicitly

k1 =

2π a

−1/3 √ , −1/ 3

k2 =

2π a

1/3 √ . −1/ 3

The Hamiltonian describing the energy of an electron in the lattice is then

H = −∆ + V, where

V

V (x + aj ) = V (x),

is the eective potential created by the carbon nuclei and the inner

electrons, which has the periodicity of the lattice. In the tight-binding model, one makes the ansatz for the electron state

ψ(x) =

X

cr φat (x − r),

r∈Γ where

φ

is a state close to the groundstate for a single atoms and the discrete

coecients

cr

have to be determined for each lattice site. Equivalently,

ψ(x) =

X

an,m φat (x − xn,m ) + bn,m φat (x − ae1 − xn,m ),

(n,m)∈Z2 55

Appendix A. Computations in the tight-binding model

56

so

an,m

and

bn,m

represent the magnitude of the electron wavefunction on each

xn,m = √ na1 + ma2 . If one ignores the |y| ≥ 3a, and assume the potential rotations by 2π/3 around the origin, as is the

of the sublattices and we have dened cross product between and

φat

φ

and

are invariant under

φ(· − y)

for

honeycomb lattice, we obtain the tight-binding Hamiltonian including only rst order

hopping,

(ψ, Hψ) ≈

X

2

2

|an,m | + |bn,m |

(φat , Hφat ) +

n,m

+

X

an,m b∗n,m + an,m b∗n−1,m + an,m b∗n,m−1 (φat , Hφat (· − e1 )) + c.

c.

n,m

≡ −t

X

an,m b∗n,m + an,m b∗n−1,m + an,m b∗n,m−1 + c.

c.+

n,m

+U

X

2 2 |an,m | + |bn,m | .

n,m Experimentally, the hopping parameter

t ≈ 2.8 eV

and the parameter for next-

nearest-neighbor interaction is about one order of magnitude smaller. We shift the energy to set

U = 0 and then we can compute the band structure. Formally, (ψk )n,m ≡ e−ik·xn,m (ak , bk )> . For a math-

this is just computing the energy of

ematical treatment, see [RS78][Chapter 16]. Since the Hamiltonian is invariant under translations by lattice vectors, only values of

B

k

in the rst Brillouin zone

need to be considered. We obtain the tight-binding Hamiltonian

(Ht.b. ψk )n,m = te

−ik·xn,m

0 1 + e−ik·a1 + e−ik·a2

1 + eik·a1 + eik·a2 0

ak . bk

Diagonalizing the matrix gives the two energy bands

E± (k) = ±t 1 + eik·a1 + eik·a2 .

(A.1)

The important feature is that these bands intersects at zero energy at the corners of the rst Brillouin zone, the so-called Dirac points

−2π K= a

4π 1/3 √ = 2 a1 , 1/3 3 9a

−2π K = a 0

4π −1/3 √ = 2 a2 . 1/3 3 9a

Expanding the energy bands to rst order around these points gives

E± (K + δ) ≈ ± e−i2π/3 δ·a1 + ei2π/3 δ·a2 = ±3/2ta |δ| ≡ ~vf |δ| , so the dispersion relation can be approximated by a circular cone, as for massless relativistic particles. This expression also gives the relation between

vf ,

vf = 3/2~−1 ta ≈ 106 m s−1 .

t

and

A.2. Armchair and zigzag boundary conditions

57

If we assume to be at low energy, we can take a further ansatz for

bn,m

in the form of an oscillating part with momentum

K

and

K0

an,m

and

and a slowly

varying envelope function,

an,m = e−iK ·( xn,m )α(xn,m ) + e−iK bn,m = e−iK ·( xn,m )β(xn,m ) + e−iK where we identify

α, α0 , β

and

β0

0

0

·xn,m

·xn,m

α0 (xn,m )

β 0 (xn,m ),

with functions on all of

R2 .

Then, we have

Ht.b. ψ n,m −iK ·x n,m e β(xn,m ) + e−iK ·xn−1,m β(xn−1,m ) + e−iK ·xn,m−1 β(xn,m−1 ) = −t −iK ·xn,m e α(xn,m ) + e−iK ·xn+1,m α(xn+1,m ) + e−iK ·xn,m−1 α(xn,m−1 ) −iK 0 ·x 0 0 n,m e β 0 (xn,m ) + e−iK ·xn−1,m β 0 (xn−1,m ) + e−iK ·xn,m−1 β 0 (xn,m−1 ) − t −iK 0 ·xn,m 0 0 0 e α (xn,m ) + e−iK ·xn+1,m α0 (xn+1,m ) + e−iK ·xn,m−1 α0 (xn,m−1 ) a, we have f (x ± al ) = f (x) ± al ·∇f (x) . Ht.b. ψ = Eψ as     −e−i2π/3 a1 − e+i2π/3 a2 ·∇β α +i2π/3  β  a1 + e−i2π/3 a2 ·∇α   = −t  +e  E 0 +i2π/3  −e α  a1 − e−i2π/3 a2 ·∇β 0  β0 +e−i2π/3 a1 + e+i2π/3 a2 ·∇α0   (+i∂1 + ∂2 )β(xn,m ) 3  (−i∂1 + ∂2 )α(xn,m )  . = −i ta  2 (+i∂1 − ∂2 )β 0 (xn,m ) (−i∂1 − ∂2 )α0 (xn,m )

To rst order in the lattice scale This means we can rewrite

Dening nally

 −iπ/4 e  0 φ(x) =   0 0

0 eiπ/4 0 0

0 0 0 e

−iπ/4

  0 α β  0   , eiπ/4  α0  β0 0

(A.2)

we obtain the usual Dirac equation

Eφ = vf

−iσ·∇ 0 φ. 0 −iσ·∇

A.2 Armchair and zigzag boundary conditions If the lattice has a horizontal zigzag edge passing through the origin, we require the amplitude to vanish at the which happen to lie on the

missing

sites, the dashed sites in Figure A.3 ,

B -sublattice.

The boundary condition is

β(xn,−n ) = β 0 (xn,−n ) = 0,

n ∈ Z.


58

After the transformation (A.2), we obtain

Az.z. φ(0, x2 ) = φ(0, x2 ),

x2 ∈ R,

Az.z.

σ3 ≡ 0

0 . −σ3

For a vertical armchair edge passing through the origin, we require

bn,n = 0.

an,n =

This means

α(xn,n ) + α0 (xn,n ) = 0,

β(xn,n ) + β 0 (xn,n ) = 0,

n ∈ Z.

In matrix form, this can be rewritten as

   α(x1 , 0) α(x1 , 0)   β(x1 , 0)  0 −I  1 , 0)  .  β(x = 0  −I 0 α (x1 , 0) α0 (x1 , 0) β 0 (x1 , 0) β 0 (x1 , 0) 

Applying again the unitary transformation (A.2), we obtain

Aac. φ(x1 , 0) = φ(x1 , 0),

x1 ∈ R,

Aac. ≡

0 −σ2

−σ2 0

=

0 σ·t , σ·t 0,

since we have chosen the orientation where the tangent is vertical.


a1

59

a2

k1 k2

K

K0

Figure A.1: Choice of basis vectors for the computations in this appendix in direct space (upper image) and momentum space (lower image). The lled circles in the honeycomb lattice represent atoms on sublattice

B

and the white circles are on sublattice

A.


60

z x

y

Figure A.2: Dispersion surface in the tight-binding model with only nearestneighbor hopping, as in (A.1). The light gray polygon shows the boundary of the rst Brillouin zone


61

Figure A.3: A terminated honeycomb lattice. The upper boundary has a zigzag termination, the right side has an armchair termination. dashed sites are the vanish.

missing

The

sites where the wavefunction should

Appendix B Comparison of eigenvalues

As a by-product of the proof of Theorem 3.3, we obtain the following comparison of eigenvalues for the operator with innite mass boundary conditions.

Proposition B.1. Fix a C 2 -domain Ω, denote by (µk )k∈N the non-decreasing

sequence of eigenvalues of D02 , counted with multiplicity. Furthermore, denote by µDk and µNk the sequences of eigenvalues of the Laplacian on L2 (Ω, C) with Dirichlet and Neumann boundary conditions respectively. Then µ2k = µ2k+1 ≤ µD k.

and, if Ω is convex, Proof.

The fact that

µ2k = µ2k+1 ≥ µN k. µ2k = µ2k+1

is because of the spectrum

D0

is symmet-

ric around zero, as noted in [BM87] or in the proof of Proposition 3.6. inequalities follow from a comparison of quadratic forms. We denote by curvature of

∂Ω

the

and dene

Z q1 (u, v) = (∇u, ∇v)L2 (Ω,C2 ) +

∂Ω

κ (u, v)C2 ,

u, v ∈ C ∞ (Ω, C2 ) ∩ D(D0 ),

κ (u, v)C2 ,

u, v ∈ C0∞ (Ω, C2 ),

Z q2 (u, v) = (∇u, ∇v)L2 (Ω,C2 ) +

∂Ω

u, v ∈ C ∞ (Ω, C2 ).

q3 (u, v) = (∇u, ∇v)L2 (Ω,C2 ) The forms

The

κ

q1 , q2

and

q3

are positive, densely dened and closable.

By

Friedrich's extension Theorem [RS75][Theorem X.23], they correspond to pos-

A1 , A2 , A3 respectively. By equation (3.3), q1 ⊂ (D0 ·, D0 ·) and therefore A1 ⊂ D02 . Since both operators are self-adjoint, A1 = D02 . Its sequence of eigenvalues is µk On the other hand, q2 (u, v) = (∇u, ∇v) and thus A2 consists of two copies itive self-adjoint operators

of the Laplacian with Dirichlet boundary conditions. Its eigenvalues (counted with multiplicity) are

D D D µD 0 , µ0 , µ1 , µ1 , · · ·

Since

q2 ⊂ q1 ,

ciple,

µ2k = µ2k+1 ≤ µD k. 63

by the min-max prin-

Appendix B. Comparison of eigenvalues

64

It is well-known that the operator corresponding to of the Neumann Laplacian. If

Ω

is convex,

κ≥0

eigenvalues, this translates into

µ2k = µ2k+1 ≥ µD k.

q3

consists of two copies

and we have

q3 ≤ q1 .

For

Appendix C The invertible double of

Ω

In this appendix, we show an alternative way to deduce Theorem 3.3 from Bär's result [Bär92] for closed surfaces. This is by explicitly constructing an (approximate) invertible double of a at domain

Ω

and showing how the in-

nite mass boundary conditions allow to extend spinors on

Ω

to spinors on the

invertible double. An accessible introduction on spinors and Dirac operators on surfaces can be found in the introduction of Erdös and Solovej's paper [ES01]. The construction here is more involved than the proof of Theorem 3.3 given before, but it is instructive to learn about spinors on manifolds. Since the goal is mainly to illustrate the method, some parts of this appendix will be rather sketchy. We will use Einstein's convention for summing over repeated indices.

C.1 Adding a curved extension R3 . Take a parametrization of ∂Ω by arclength γ(τ ), τ ∈ [0, L]. Here, n(τ ) and t(τ ) will be the outward normal and tangent to ∂Ω. In a neighborhood of the boundary of Ω, its interior can be parametrized by x(ν, τ ) = γ(τ ) + νn(τ ), ν ∈ [−a, 0) and τ ∈ [0, L], for some suciently small a. Now consider Ω as a We will extend

Ω

into a curved surface imbedded in

Figure C.1: The plane domain

Ω

embedded in

65

R3 ,

and a curved extension.

Appendix C. The invertible double of Ω

66

z = h in R3 and dene a surface by adding a fringe γ(τ ) + h sin(ν/h)n(τ ) x(ν, τ ) = for ν ∈ [0, hπ/2). h/2 + h cos(ν/h)

subset of the plane

The resulting surface is a

C 1 -manifold ΩE ,

see Figure C.1, which suces to

dene a Dirac operator on it. As a basis for the tangent space we can take

xν =

cos(ν/h)n(τ ) − sin(ν/h)

κ is the curvature ρ = 1 + κh sin(u/h) Then, where

,

xτ = (1 + κh sin(u/h))

of the boundary.

t(τ ) 0

,

It will be convenient to dene

the metric reads

g(ν, τ ) =

1 0

0 ρ2

,

from which we can calculate the nonzero Christoel symbols entering the covariant derivative :

Γντ,τ = −ρκ cos(u/h), Γττ,ν = Γτν,τ = κ cos(u/h)/ρ Γττ,τ = κ0 h sin(u/h)/ρ. By denition, the covariant derivative (Levi-Civita connection) of a tangent vector eld

Y = Y ν xν + Y τ xτ

reads

j j k ∇LC xl Y = (∂l Y + Γlk Y )xj .

C.2 Constructing the Dirac operator In order to dene Cliord multiplication on spinors, it is convenient to introduce three antihermitian matrices

1

− sin( hν ) n∗ cos( hν ) , Σu = −i n cos( hν ) sin( hν ) 0 −in∗ Σt = −iρ , in 0 cos( hν ) n∗ sin( hν ) ΣN = −i . n sin( hν ) − cos( hν )

Here,

n = n(τ )

is the tangent vector (in the plane) to

∂Ω

considered as a

complex number. Each pair of these matrices anticommute and

Σν Στ = ρΣN .

Now we can set

Σ(Y )φ = Y j Σj φ. 1 In the dierential geometry literature, it is conventional to dene Cliord multiplication with antihermitian operators, instead of the hermitian Pauli matrices we have been using so far.

C.2. Constructing the Dirac operator

67

This is simply the restriction of Cliord multiplication in three dimensions to the tangent space.

spinors, and compatible with the metric in the sense that

−(Y , Y )g φ.

C2 -product of Σ(Y )(Σ(Y )φ) =

It is skew-symmetric with respect to the

We now need to nd a covariant derivative

∇Cl (Cliord

connec-

tion) for spinor elds in order to nd out what the Dirac operator looks like. This connection needs to be i) compatible with the

C2 -product

:

Cl d hφ, χi (Y ) = ∇Cl Y φ, χ + φ, ∇Y χ

ii) compatible with Cliord multiplication :

Σ(X)(∇Cl Y φ).

LC ∇Cl Y (Σ(X)φ) = Σ(∇Y X)φ +

These conditions need to hold for all tangent elds elds

φ, χ,

xν , xτ .

X, Y

and for all spinor

but by linearity, it is sucient to check them for the basis vectors

The rst condition implies that

Cl ∇Cl xi φ = ∂i φ + Γi φ, where

ΓCl i , i = ν, τ

are antihermitian matrices. The second condition can then

be written out explicitly :

LC Cl ∇Cl xi xj φ = Σ(∇xi xj )φ + Σ(xj ∇xi φ k Cl ⇔ (∂i Σj )φ + Σj ∂i φ + ΓCl i Σj φ = Γij Σ(xk )φ + Σj ∂i φ + Σj Γi φ

= Γkij Σk φ + Σj ∂i φ + Σj ΓCl i φ, so the four equations determining

∇Cl

are

∂i Σj − Γkij Σk = [Σj , ΓCl i ]. We obtain

Cl [Σν , ∇Cl τ ] = [Στ , ∇ν ] = 0, 1 ΣN , [Σν , ∇Cl ν ]=− 2h ρ(ρ − 1) [Στ , ∇Cl ΣN . τ ]=− 2h This nally allows to write the Dirac operator using the dual basis

eν = xν ,

eτ = ρ−1 xτ , −2 DΩC = Σ(ei )∇Cl Στ ∂τ − ei = Σν ∂ν + ρ

1 (2 − ρ−1 )ΣN . 2h

In the same coordinates, the Dirac operator on the original at domain

Ω reads

DΩ = Σν ∂ν + ρ−2 Στ ∂τ . Note that the coecients of the derivatives are continuous across

∂Ω,

while

there is a jump in the zeroth-order part of the operator. A calculation shows


68

that multiplication by the unitary matrix

iΣN

DΩE .

anticommutes with

allows to split the spinor bundle in the eigenspaces of

iΣN

This

and decompose the

Dirac operator accordingly in two mutually adjoint operators. On the at part of

ΩE , iΣN = σ3 DΩ = −i

and this corresponds to the obvious decomposition

0 n∗ (∂ν − iρ−1 ∂τ ) −1 n(∂ν + iρ ∂τ ) 0

On the curved part, the eigenspinors of

≡

A− Ω 0

0 A+ Ω

.

iΣN are

sin θ , φ+ (hθ, τ ) = p 2(1 − cos θ) n(1 − cos θ) ∗ 1 −n (1 − cos θ) φ− (hθ, τ ) = p . sin θ 2(1 − cos θ) 1

This choice of eigenvectors continuously extends the choice

0 1

1 , 0

φ+ =

used for the at piece. The corresponding decomposition of the

DΩE

φ− =

reads

DΩE (f φ+ + gφ− ) = (A+ f )φ− + (A− g)φ+ , ΩE ΩE where

κ(1 − cos(ν/h)) i f = −in ∂ν + ∂τ + ρ 2ρ

(C.1)

i κ(1 − cos(ν/h)) ∗ g(ν, τ ) = in ∂ − ∂ + A− g. ν τ ΩE ρ 2ρ

(C.2)

A+ f (ν, τ ) ΩE and

Here, it is important to note that all coecients are bounded if

h & 0.

C.3 Construction of the invertible double In order to construct the invertible double, we take a copy of across the plane

z = y,

ΩE

and reect it

then glue the two copies to a cylindrical overlap region

h to obtain ΩI , consisting of an upper part ΩuI ≈ ΩE ∪ C and a l lower part ΩI ≈ ΩE,z7→−z ∪ C , see Figure C.2. Our previous choice of coordinates (ν, τ ) extends to the cylindrical region C by setting γ(τ ) + hn(τ ) x(ν, τ ) = for ν ∈ [hπ/2, hπ/2 + h). h/2 − (ν − hπ/2)

C

of height

The Dirac operator constructed previously extends to

C

by dening

± A± C (ν, τ ) ≡ AΩE (hπ/2, τ ). The goal of this is to obtain an overlap region where all the objects involved have the structure of a direct product. Specifying a spinor (a section of the spinor bundle) on

ΩI

consists of specifying a spinor on

ΩuI

and

ΩlI

satisfying a

C.4. Doubling spinors with innite mass boundary conditions

Figure C.2: Construction of the invertible double consisting of a

69

ΩE

and its

reection joined by a cylindrical overlap region.

compatibility condition in is a spinor on

ΩI

C.

if

vu = Then, we dene

v ≡ (vu , vl ) ∈ C 1 (ΩuI , C2 ) × C 1 (ΩlI , C2 )

We say

DΩI

0 in

in∗ vl , 0

on

C.

(C.3)

by

DΩI v = (DΩuI v u , DΩlI v l ). The point of this construction is that

DΩI u

will still satisfy the compatibility

condition, as can be checked by an explicit computation.

C.4 Doubling spinors with innite mass boundary conditions We start from a spinor

w

satisfying innite mass boundary conditions, or

w2 = inw1 First,

w

can be extended to

ΩE

∂Ω.

on

by writing

w(ν, τ ) = w1 (0, τ ) (φ+ (ν, τ ) + inφ− (ν, τ )) . On

C,

we extend

w(ν, τ ) = w(hπ/2, τ )

and note that

w1 (τ ) w(hπ/2, τ ) = √ 2

1−i . n(1 + i)


70

Finally, we dene

wD ≡ (w, w) on ΩI .

It can be checked easily that

wD

satises

the compatibility condition (C.3). We can therefore use Bär's Theorem (strictly speaking, we should further approximate by a smooth manifold) to write

kDwI k2L2 (ΩI ) ≥ But for small

4π kwI k2L2 (ΩI ) . Vol ΩI

h, kwI k2 ≥ 2kwk2 − Chkγwk2 . On the other D in (C.1) and (C.2), we can also bound

(C.4)

hand, by using the

representation of

kDwI k2L2 (ΩI ) ≤ 2kDwk2L2 (Ω) + Chkγwk2H 1 (∂Ω) . Thus, by taking the limit all

h→0

on both sides of (C.4), we nally obtain for

w ∈ D(D) kDwk2L2 (Ω) ≥

which is the case

B=1

4π kwk2L2 (Ω) , 2kΩk

of Theorem 3.3.

Appendix D Eigenvalues for an annulus or disc

Here, we explicitly compute eigenfunctions and eigenvalues of

D

in the case of

spherical symmetry, to gain some intuition about the behavior of our operator. We will consider

Ω=D

and

Ω = D \ B[0, ρ] for some ρ ∈ (0, 1) and introduce Ω is invariant by rotations about the origin. If

polar coordinates. In this case, in addition,

η

is constant on each boundary component, the rotation operator

(Rα φ)(r, θ) = leaves

D(Dη )

invariant and

e+iα/2 0

0 e−iα/2

φ(r, θ + α)

Rα (Dη φ) = Dη (Rα φ).

The generator of these

spin-rotations is the total angular momentum operator which commutes with

Dη .

( u2 (1, θ) = ieiθ B1 u1 (1, θ) u2 (ρ, θ) = −ieiθ Bρ u1 (ρ, θ) Here, we allow

n ∈ Z,

B1

or

Bρ

J = −i∂θ IC2 + σ3 /2,

We write the boundary conditions in the form

to be zero.

Ω = D \ B[0, ρ].

if

The eigenvalues of

J in D(D) inθ e a(r) u(r, θ) = i(n+1)θ , e b(r)

and the corresponding eigenfunctions of

J

are

n + 1/2,

have the form

with boundary conditions

  b(1) = iB1 a(1), b(ρ) = −iBρ a(ρ)   |a(r)| ≤ Cr|n| , |b(r)| ≤ Cr|n+1|

if if

Ω = D \ B(0, ρ), . Ω = D.

In polar coordinates,

T = −i

0 e−iθ (∂r − ir−1 ∂θ ) eiθ (∂r + ir−1 ∂θ ) 0 71

for

Appendix D. Eigenvalues for an annulus or disc

72

The eigenvalue equation for

(

Dη ,

in each eigenspace of

J,

then becomes

−i(b0 (r) + (n + 1)r−1 b(r)) = λa(r) −i(a0 (r) − nr−1 a(r)) = λb(r).

Zero eigenvalues. If

λ = 0,

the solutions are

a = C1 rn ,

b = C2 r−n−1 .

These solutions satisfy the boundary conditions only in special cases. When

Ω = D, this is the case for B1 = 0, and we obtain solutions for all n ≥ 0, with C2 = 0. As noted before, we obtain all monomials in z as eigenfunctions. When Ω = D \ B[0, ρ], the boundary conditions read C2 = iB1 C1 ,

C2 ρ−n−1 = −iBρ C1 ρn .

n ∈ Z if B1 = Bρ = 0 and an exceptional solution B1 /Bρ = −ρ2n+1 . In accordance with Proposition 3.2, zero if never an eigenvalue if B 6= 0 and B has constant sign.

There are solutions for all if

Other eigenvalues. If

λ 6= 0,

we can rewrite the system to obtain

(

a00 (r) + r−1 a0 (r) − n2 r−2 a(r) = λ2 a(r) b(r) = −iλ−1 (a0 (r) − nr−1 a(r)).

The general solutions are

a(r) = C3 Jn (|λ| r) + C4 Yn (|λ| r), b(r) = i sign(λ) (C3 Jn+1 (|λ| r) + C4 Yn+1 (|λ| r)) , Where

Jn

and

Yn

are the Bessel functions of order

n

of rst and second kind,

respectively.

Solution for the disc.

On D, the regularity conditions at C4 = 0 and we obtain eigenfunctions inθ e Jn (λn,k,+ r) φn,k,+ = , iei(n+1)θ Jn (λn,k,+ r) einθ Jn (λn,k,− r) φn,k,− = −iei(n+1)θ Jn (λn,k,− r)

automatically satised if

for all

n ∈ Z,

and where

λn,k,±

is the

k -th

positive solution of

Jn+1 (λn,k,± ) = ±B1 Jn (λn,k,± ). With this denition,

Dφn,k,± = ±λn,k,± φn,k,± .

the origin are

73

Case

B = 1.

If in addition

studied in [BFS16].

B1 = 1,

the properties of

λn,k,±

have been

In this case, we might also use a comparison argument

as in Appendix B with the Dirichlet and Neumann Laplacian restricted to

J

eigenspaces of

to obtain, by using induction on

k,

0 0 0 jn,k ≤ λn,k,+ ≤ jn+1,k ≤ jn,k ≤ λn,k,− ≤ jn+1,k ≤ jn,k+1 ,

n ≥ 0,

using the notations for zeroes of Bessel functions from [AS65]. For negative a similar inequality holds, since in this case

General B . Assuming that

where

jn,k

For the case of general

B1 > 0

is the

k -th

B , we can at least give a lower bound.

for deniteness, we have

λn,k,− ≥ jn,1

when

n ≥ 0,

λn,k,+ ≥ j|n|−1,1

when

n ≤ −1,

zero of

Jn .

The rst inequality holds since

have equal signs close to the origin, and analogous using

n,

λn,k,± = λ|n|−1,k,∓ .

J−n = (−1)n Jn .

jn,1 < jn+1,1 .

Jn

and

Jn+1

The second one is

To study the other cases we can use the

recurrence relations for Bessel functions ([AS65][eq.9.1.27])

n+1 0 Jn+1 (λn,k,± ) = Jn (λn,k,± ) − Jn+1 (λn,k,± ) λn,k,± (n + 1)B1 = 1∓ Jn (λn,k,± ) λn,k,± n Jn (λn,k,± ) Jn0 (λn,k,± ) = −Jn+1 (λn,k,± ) + λn,k,± n = ∓B1 + Jn (λn,k,± ). λn,k,± Thus if

n ≥ 0, 0 Jn+1 (λn,k,+ ) λn,k,+ − (n + 1)B1 = . Jn0 (λn,k,+ ) −λn,k,+ B1 + n J0

(λ

)

n,k,+ λn,k,+ ≥ min((n+1)B1 , nB1−1 ) or Jn+1 ≤ 0, which in 0 n (λn,k,+ ) 0 turn implies λn,k,+ ≥ jn,1 . An analogous reasoning holds for λn,k,− for n ≤ −1.

This implies either

We obtain the lower bounds

0 λn,k,+ ≥ min (n + 1)B1 , nB1−1 , jn,1

λn,k,− ≥ jn,1 λn,k,+ ≥ j|n|−1,1

0 λn,k,− ≥ min |n| B1−1 , (|n| − 1)B1 , j|n|−1,1

when

n≥0

when

n≥0

when

when

n ≤ −1 n ≤ −1.


74

Solutions for the annulus. many eigenvalues

±λn,k,±

where

n

In this case, for each

λn,k,±

is the

k -th

there are countably

positive zero of

Jn+1 (λn,k,± ) ∓ B1 Jn (λn,k,± ) Yn+1 (λn,k,± ) ∓ B1 Yn (λn,k,± ) det . Jn+1 (ρλn,k,± ) ± Bρ Jn (ρλn,k,± ) Yn+1 (ρλn,k,± ) ± Bρ Yn (ρλn,k,± ) It seems to be complicated to extract useful information from this complicated expression, but the roots can be computed numerically. We will just study the cases

B1 = Bρ = 1

and

B1 = −Bρ = 1.

In the rst case, if

ρ

is small the

presence of a hole increases the lowest eigenvalue, while in the second case the

ρ ∈ (0, 1).

presence of a hole decreases the lowest eigenvalue for any

Constant boundary conditions. for negative for

We will study the case

n it suces to change λn,k± ↔ λ|n|−1,k∓ .

z ≡ λn,k,+

n ≥ 0,

since

In this case, the equation

reads

Jn+1 (z) − Jn (z) Yn+1 (ρz) + Yn (ρz) = Yn+1 (z) − Yn (z) Jn+1 (ρz) + Jn (ρz) (D.1)

µ± n for the rst positive zero of Jn+1 ∓ Jn Yn+1 ∓ Yn . Since µ± n are the eigenvalues

We will write positive zero of

and

νn±

for the rst

for the problem on

the disc, by the argument in the previous subsection, we have

0 0 − jn,1 ≤ µ+ n ≤ jn+1,1 ≤ jn,1 ≤ µn ≤ jn+1,1 . On the other hand,

yn,1 ≤ νn− ≤ yn+1,1 yn,k and 0 zero of Yn and Yn respectively. We can also show

by continuity, where we use the notations

0 yn,k

for the

k -th

positive

yn+1,1 ≤ νn+ ≤ yn,2 . The upper bound holds by continuity. The lower bound is obvious from a plot of Bessel functions of the second kind.

1 As a consequence

0 νn,+ ≥ yn+1,1 ≥ jn+1,1 ≥ µn,+ With these preliminaries, we return to (D.1). Close to the origin, its left hand side is positive and its right hand side negative. solution, one of the sides should change sign. We take

ρ

In order to obtain a large enough to have

+ 1 It can be shown as follows. Since y n+1,1 > yn,1 , either µn ≥ yn+1,1 , or Yn+1 (z)−Yn (z) has two zeroes in the interval (0, yn,1 ). We will see the latter is impossible. By substracting the recurrence relations, 0 Yn+1 (z) − Yz0 = Yn (z) − Yn+1 (z) −

n+1 n Yn+1 (z) − Yn (z). z z

0 Therefore, if νe is a zero of Yn+1 (z)−Yn (z) in the interval (0, yn,1 ), then Yn+1 (e ν )−Yn0 (e ν ) > 0. Since the derivative of an analytic function changes sign between successive zeroes, it is impossible to have more than one zero in this interval.

75

νn− /ρ > νn+

+ µ− n /ρ > νn . Then, the factors in brackets on both sides don't + + (0, νn ). The rst solution λn,k,+ occurs in the interval (µ+ n , νn )

and

change sign in

and in particular

λn,k,+ > µ+ n. We now turn our attention to

z ≡ λn,k,− .

This eigenvalue satises the

equation

Jn+1 (z)+Jn (z) Yn+1 (ρz)−Yn (ρz) = Yn+1 (z)+Yn (z) Jn+1 (ρz)−Jn (ρz) . (D.2) Now close to the origin the left hand side is negative and the right hand side positive. Again, we assume

ρ

is large enough so that

− − min(νn+ /ρ, µ+ n /ρ) > max(µn , νn ). − µ− n and νn , but unfortunately we don't − + > µn , we conclude that λn,k,− > µ− n > µn

Again, the rst zero occurs between

2 − know which of these is larger. If νn

and we are done. In the opposite case, we rewrite (D.2) in the form

Jn+1 (z) + Jn (z) = Yn+1 (z) + Yn (z) For all

z ∈ (νn− , µ− n ),

Jn+1 (ρz) − Jn (ρz) . Yn+1 (ρz) − Yn (ρz)

by choosing a small value of

ρ,

the fraction on the left

hand side can be made arbitrarily close to zero. The only possibility for this to occur while keeping particular, if

ρ

z ∈ (ν− , ν+ ),

µ− n.

In

small enough (depending on

n),

is by having

z

arbitrarily close to

is suciently small,

λn,k,− ≥ µ+ n. In summary, we have shown that for

ρ>0

λn,k,± ≥ µ+ n. Since the smallest eigenvalue occurs for small

n,

this shows that a small hole

tends to increase the lowest eigenvalue.

Alternating boundary conditions. λn,k,+

In this case, the equation for

z≡

reads

Jn+1 (z)−Jn (z) Yn+1 (ρz)−Yn (ρz) = Yn+1 (z)−Yn (z) Jn+1 (ρz)−Jn (ρz) . z ∼ 0, the left hand side diverges as Cn z n (ρz)−n−1 and the right hand side n −n−1 as Cn (ρz) z , with some positive coecient Cn . Thus, as z & 0, Jn+1 (z)−Jn (z) Yn+1 (ρz)−Yn (ρz) > Yn+1 (z)−Yn (z) Jn+1 (ρz)−Jn (ρz) . For

2 Numerically,

λn,k,− ≥ µ+ n.

at least for 0 ≤ n ≤ 4 we have νn− < µ+ n , which directly gives the bound


76

On the other hand, at

+ z = µ+ n < νn ,

+ + + 0 = Jn+1 (µ+ n ) − Jn (µn ) Yn+1 (ρµn ) − Yn (ρµn ) + + + < Yn+1 (µ+ n ) − Yn (µn ) Jn+1 (ρµn ) − Jn (ρµn ) . By continuity, there is at least one intersection

λn,0,+

in

(0, µ+ n ).

Thus, when

the boundary conditions change sign, a hole of any size decreases the lowest eigenvalue.

Appendix E Technical lemmas from Chapter 4

E.1 General tools This section contains the proof of the variant of the variational principle for traces and the IMS-type formula from section 4.1.

Lemma E.1 (simple case of Lemma 4.2). Let (M, µ) be a measure space, L a

separable Hilbert space and consider a measurable family of functions vα ∈ L for all α ∈ M , such that Z

2

|hvα , ui| dµ(α) ≤ kuk2 ,

M

for all u ∈ L. Consider a self-adjoint operator A on H with quadratic form a, bounded from below and with compact resolvent. Then Z − Tr[A]− ≤

a(vα ) dµ(α),

if the latter integral exist. Proof in simple case. not accumulate, by

Pn

A

is bounded from below and its eigenvalues do

is the nite sum of negative eigenvalues. We denote

n λj .

the projection on the rst

associated to the

γn

Since

Tr[A]− j -th

eigenvalue

eigenvalues and by

wj

the eigenfunction

We dene a sequence of density matrices

by

Z vα hvα , Pn ui dµ(α).

γ n u = Pn M Then

0 ≤ γn ≤ 1

and the range of

γn

is in

D(A).

variational principle (4.1) and write

− Tr[A]− ≤ Tr(Aγn ). 77

Therefore, we can apply the

Appendix E. Technical lemmas from Chapter 4

78

Writing out the projections explicitly, we nd

Tr(Aγn ) =

n Z X

hwj , Awk i hwk , vα i hvα , wj i dµ(α)

M

j;k=1

Z =

a(Pn vα ) dµ(α) M 2

a(Pn vα ) = a(Pn−1 vα ) + λn |hwn , vα i| , the integrand is decreasing in n soon as n is larger than the number of negative eigenvalues. We apply the

Since as

monotone convergence theorem to write

Z − Tr[A]− ≤

lim a(Pn vα ) dµ(α)

M n→∞

Z ≤−

a(vα ) dµ(α) M

The last inequality holds since quadratic forms corresponding to self-adjoint operators are closed, and

Pn v α → v α

Proof of Lemma 4.4. Upper bound Z γ≡ N

in

L

and since

vα ∈ D(a).

We dene a density matrix

φβ [φβ f Hf φβ ]0− φβ dν(β)

and apply the variational principle (4.1).

0≤γ≤1

with

[φβ f Hf φβ ]0−

By the normalization of the

φβ ,

is the projection on the (nite dimensional)

subspace of the negative eigenvalues of the operator inside the brackets, the range of

γ

is in

D(f Hf ).

Thus,

Z − Tr[f Hf ]− ≤

Lower bound wj

N

− Tr [φβ f Hf φβ ]− dν(β)

Fix a density matrix

γ

D(f Hf ) γj ∈ [0, 1].

with range in

its eigenfunctions, corresponding to eigenvalues

and denote by Then, we can

write

Tr(f Hf γ) =

X

γj (h2 kT f wj k2 − kf wj k2 )

j

=

Z X N

Using

γj (h2 Re T f wj , T φ2β f wj − kφβ f wj k2 ) dν(β)

j

φT v = i(σ· ∇φ)v + T (φv)

and

2

(σ·a )2 = |a| IC2

gives

1 T f wj , T φ2β f wj C2 + T φ2β f wj , T f wj C2 2 2 2 2 = |T φβ f wj |C2 − |∇φβ |R2 |f wj |C2 .

E.2. Local asymptotics

79

Now we use the hypothesis (4.2) and bound

h2

Z Z N

R2

Z

2

2

|∇φβ |R2 |f wj |C2 dx dν(β) ≤

h2 τ (β)

Z

2

R2

N

φ2β |f wj |C2 dx dν(β)

Putting everything together,

Tr(f Hf γ) ≥

Z X N

γj h2 kDφβ f wj k2 − (1 + h2 τ (β))kφβ f wj k2 ) dν(β).

j (E.1)

φβ f Hf φβ , we write

In order to transform this back into traces of the operators

X

γj h2 kDφβ f wj k2 − (1 + h2 τ (β))kφβ f wj k2 )

j

= (1 − h2 τ (β))

X

γj h2 kDφβ f wj k2 − kφβ f wj k2

j 2

+ 2h τ (β)

X

γj h2 /2kDφβ f wj k2 − kφβ f wj k2 .

j In view of the a-priori bound of Lemma 4.3, it is sucient to consider small

(1 − h2 τ (β)) > 0. Since wj ∈ D(Df ) ⊂ Q(φβ f Hf φβ ) 1/2 4.2 with M ≡ N, vj ≡ λj wj . Thus,

and we can assume can apply Lemma

X

γj h2 kDφβ f wj k2 − (1 + h2 τ (β))kφβ f wj k2 )

h

we

j

≥ −(1 − h2 τ (β)) Tr[φβ f Hf φβ ]− − 2h2 τ (β) Tr[φβ f Hh7→h/√2 f φβ ]− Z ≥ − Tr[φβ f Hf φβ ]− − h2 τ (β)Ch−2 f 2 φ2β , R2

where the last line follows from Lemma 4.3.

Inserting this bound back into

(E.1), we nally nd

Z Tr[f Hf γ] ≥ −

Tr[φβ f Hf φβ ]− dν(β) Z Z −C τ (β) f 2 (x)φ2β (x) dx dν(β) N R2 Z Z Z 2 =− Tr[φβ f Hf φβ ]− − C f (x) τ (β)φβ (x)2 dν(β) dx N

N

R2

Taking the inmum over admissible density matrices

N

γ

gives the desired bound.

E.2 Local asymptotics This section contains an improved version of the estimate on a halfplane with boundary constant close to one. This is necessary in order to obtain the correct


80

B is either equal B , we do not need

error order at the end. Note that all of this is not necessary if to

±1 or bounded away from ±1.

In particular, for constant

the next lemma and the estimate on the order of the remainder improves to

O(h−2/3 ).

Lemma E.2. For all satisfying φ ≤ Cl , −2

2

φ ∈ Cc∞ (R2+ , R+ ), with support for all h ≤ 1 and b ∈ R \ {0},

in a ball of size l and

Z −2 1 2 φ (x) dx Tr[φH+,b φ]− − h 4π R2+ Z 1/2 2 ≤C |∇φ| + C min h−2 1 − b2 , l−1 h−1 1 − b2 . R2+

Proof.

The rst bound is the statement of Lemma 4.6. In order to obtain the

second bound, we introduce an additional partition of unity in the vertical coordinate with scale we let

r

r.

In order to obtain the correct order for the error term,

depend again on the distance to the border of the domain. For

t ∈ R,

we set

q

[t]2+ + r02 1 r(t) = q , 2 [t]2 + r2 + 1 +

0

g

As before, we pick a smooth, nonnegative function ball and

R

g2 = 1

and set for

with support in the unit

t∈R

χt (x) = g

x2 − t r(t)

p

j(x2 , t).

Here,

j(x2 , t) = r

−1

x − t 0 (t) 1 + r (t) , r(t) t 7→

is the Jacobian of one-dimensional transformation formula with

τ (t) = cr

−2

(t),

x2 −t r(t) .

we have

Z +∞ Tr[φH+,b φ]− − Tr[χ φH φχ ] dt t +,b t − −r0 Z Z +∞ ≤C φ2 (x) r−2 (t)χt (x)2 dt dx R2+

≤ Cl ≤

−2

Z

−r0 l

−l −1 −1 Cr0 l .

Z

+∞

dx1

r −r0

−2

Z (t) dt 0

l

χt (x)2 dx2

By the IMS-

E.3. Localizing, straightening the boundary and conclusion We apply Lemma 4.5 when

|t| > r(t)

and Lemma 4.6 when

|t| ≤ r(t)

81

to obtain

Z Z +∞ 1 −2 Tr[χt φH+,b φχt ]− dt − h φ2 (x) dx −r0 4π R2+ Z +∞ Z 2 ≤C |∇(χt (x)φ(x))| dx dt R2+

−r0

+ C 1 − b2 h−2

3r0

Z R+ 2

χ2t (x)φ2 (x) dx dt

2

≤C R2+

r0

√

−r0

Z

Optimizing over

Z

|∇φ(x)| dx + Cr0−1 l−1 + C 1 − b2 h−2 r0 l−1 . −1/2 r0 = h 1 − b2

with

gives the correct bound.

E.3 Localizing, straightening the boundary and conclusion We rst provide some additional details on the construction of the localization function. This section also contains the proof of Lemma 4.9 and some more details on how this statement about transplanted functions allows to obtain a comparison between our original operator and the operator on a halfplane. Recall the denitions

p d(x)2 + l02 1 l(x) = p , 2 d(x)2 + l02 + 1

d(x) ≡ dist(x, R2 \ Ω),

and

φξ (x) = g for some smooth normalized mation

ξ 7→

and with

p

J(x, ξ).

J(x, ξ)

the Jacobian of the transfor-

x−ξ l(ξ) . We have

∇l(x) = where

g,

x−ξ l(ξ)

n(x) = ∇d(x)

1 d(x)n(x) p p , 2 2 d (x) + l02 ( d(x)2 + l02 + 1)2

is (almost everywhere in

Ω)

a unit vector pointing away

from the boundary. Thus

J(x, ξ) = det −l

−1

(ξ)δi,j − l

= l(ξ)−2

−2

1 d(ξ)n(ξ)j p (ξ)(x − ξ)i p 2 d(ξ)2 + l02 ( d(ξ)2 + l02 + 1)2

! i,j

1 (x − ξ)·n(ξ) d(ξ) p p 1+ . 2 2 2 2 2 2 l(ξ) d(ξ) + l0 ( d(ξ) + l0 + 1)


82

The last fraction is less or equal than one, while within the support of have

|x − ξ ·n(ξ) | ≤ 1,

φξ ,

we

so

1 3 l(ξ)−2 ≤ J(x, ξ) ≤ l(ξ)−2 . 2 2 {φξ |ξ ∈ Ωe }

Now we check that the family

satises hypothesis (4.2) of

Lemma 4.4. We have

|∇φξ (x)| ≤ l(ξ)−1

p J(x, ξ) |g 0 | ( x−ξ ) + g( x−ξ )l(ξ)−1 l(ξ) l(ξ)

≤ Cl(ξ)−2 IB(ξ,l(ξ)) (x). Then,

Z

Z

2

|∇φξ | (x) dξ ≤ C Ωe

Ωe

Z ≤C

l(ξ)−4 IB(ξ,l(ξ)) (x) dξ Z l(ξ)−4 φ2ζ (x)IB(ξ,l(ξ)) (x) dζ dξ.

Ωe

|∇l| ≤ 1/2,

By using

Ωe

φ2ζ (x)IB(ξ,l(ξ)) (x) 6= 0

we obtain if

|ζ − ξ| ≤ |ζ − x| + |x − ξ| ≤ l(ζ) + l(ξ) ≤ 2l(ζ) + 1/2 |ζ − ξ| . So we rewrite

Z

Z

2

|∇φξ | dξ ≤ C Ωe and for

φ2ζ (x)

Z

Ωe

l(ξ)−4 dξ dζ,

B(ζ,4l(ζ))

ξ ∈ B(ζ, 4l(ζ)) l(ξ)−4 ≤ l(ζ)−4 + 2l(ζ)−5 |ξ − ζ| ≤ 9l(ζ)−4 .

so nally

Z

Z

2

|∇φξ | dξ ≤ C Ωe

Ωe

Proof of Lemma 4.9. mapping that sends of

Ω

φ2ζ (x)l(ζ)−2 dζ.

2

We identify R with C. ∂Ω ∩ B(ξ, l(ξ)) to a part

to the upper half plane.

We need to dene a conformal of the real axis and the interior

To show that the constants involved do not

ξ , we rst choose for each connected component ∂Ωj of ∂Ω a conformal map Fj that maps ∂Ωj to the unit circle and Ω to (part of ) 1 the interior of the unit circle. Such a Fj exist and are C up to the boundary, [Pom91, Theorem 3.5, p. 48]. For each s ∈ ∂Ω, we dene depend on the point

Gs (z) =

2i Fk (s) − Fk (z) , |Fk0 (s)| Fk (z) + Fk (s)

s ∈ ∂Ωk .

Fk with a Möbius transformation Fk (s) to 0 and the unit disc to the upper halfplane. The factor |Fk0 (s)|

This is the composition of the corresponding that maps

with

E.3. Localizing, straightening the boundary and conclusion is chosen to obtain all

x ∈ Ω ∩ B(ξ, l),

|G0s (s)| = 1.

We also set

we have

83

Ck = kFk0 k∞,Ω + kFk00 (x)k∞,Ω

For

|Fk (x) − Fk (s)| ≤ Ck |x − s| ≤ Ck l, such that, when

l

is suciently small

|F (x) + F (s)| ≥ 2 |F (x)| − |F (x) − F (s)| ≥ 2 − Ck l. Therefore, when

l

is suciently small

|G0s (x) − G0s (s)| ≤ 2Ck |x − s| ,

|Gs (x)| ≤ 2Ck |x − s| ,

∈ Ω ∩ B(ξ, l).

H 0 (z) 6= 0 in R2+ , H 0−1/2 can be dened as a holo2 morphic function on R+ . 1 We also dene a C -extension of the boundary function B to the interior We set

of

Ω.

Hs = G−1 s .

for allx

Since

v ∈ D(D), we dene −1 0∗ −1/2 (z) 0 Bb (Hs ) v(Hs (z)). vt (z) = 0 (Hs0 )−1/2 (z)

Now for

x 7→ Hs (x) gives a parametrization of the boundary, for all s0 ∈ ∂Ω, t(s ) = Hs0 (Gs (s0 ))/ |Hs0 | (Gs (s0 )). Thus vt satises the correct boundary con0 −1/2 − 1 ≤ Cl, ditions. Since |B/b − 1| ≤ Cl and (Hs ) kvk2 − kvt k2 ≤ Clkvk2 .

Since

0

Also,

T vt =

(Hs0 )1/2 0

∂z∗ B (Hs0 )−1/2 (z)v1 0 (T v) ◦ Hs − 2i , 0 B/b(Hs0∗ )1/2 b

and thus

kT vk2 − kT vt k2 ≤ Clkvk2 .

Finally, we provide some details about the comparison argument used to straighten the boundary.

Proof of Equation 4.9. H+,b and H in a small b ≡ b(ξ) and l ≡ l(ξ).

We use the same approach as before in order to compare ball. In order to simplify notations, we write

φ ≡ φξ ,

P γ = j γj vj vj∗ , with vj in the domain of φHφ. As in Lemma 4.9, we construct vj,t in the domain of D+,b , which coincides with the form domain of φξ,t H+,b φξ,t . By Lemma 4.9, there exist constants C1 , C2 such that kφt vj,t k2 − kφvj k2 ≤ C1 lkφt vj,t k2 kD+,b φt vj,t k2 − kDφvj k2 ≤ C1 lkD+,b φt vj,t k2 + C2 kφt vj,t k2 . For the lower bound, we x a density matrix


84

We write

Tr(φHφγ) =

X

≥

X

γj h2 kDφvj k2 − kφvj k2

j

γj h2 (1 − C1 l)kD+,b φt vj,t k2 − (1 + h2 C2 + C1 l)kφt vj,t k2

j

2 ≥ (1 + C1 l)(1 + h2 C2 + C1 l) Tr (φt e h2 D+,b φt − φ2t )γt , where we have dened

2 e 1l h2 = 1+h1−C 2 C +C l h 2 1

γt = (1 + C1 l)−1

and a density matrix

N X

∗ γj vj,t vj,t ≤ 1.

j=1 By the variational principle in Lemma 4.2 and the bounds in Lemmas 4.6 and 4.8,

2 − Tr[φHφ]− ≥ −(1 + C1 l)(1 + h2 C2 + C1 l) Tr[φt e h2 D+,b φt − φ2t ]− Z ≥ Tr[φt H+,b φt ]− − C(C1 l + C2 h2 )h−2 φ2t .

e h2 = 2 We x a density matrix now with range in the domain of φt D+,b φt . slight abuse of notation we denote by uj,t its eigenfunctions and by γj

For the upper bound, we use the same strategy but now, we take 1+C1 l h2 . 1−h2 C2 −C1 l

With a

the correspondig engenvalues. The rest of the argument goes through as before. We obtain the bound

2 Tr (φt e h2 D+,b φt − φ2t )γ ≤ −(1 − C1 l)

1 − (C1 l)2 Tr[φHφ]− , 1 − h2 C2 − C1 l

so nally

2 − Tr[φHφ]− ≤ −(1 + C(C1 l + C2 h2 )) Tr[φt e h2 D+,b φt − φ2t ]− Z ≤ − Tr[φt H+,b φt ]− + C(C1 l + C2 h2 )h−2 φ2t .

Spectrum of Graphene Quantum Dots

Spectrum of Graphene Quantum Dots

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