Open chaotic Dirac billiards: Weak (anti ... - APS Link Manager

PHYSICAL REVIEW B 88, 245133 (2013)

Open chaotic Dirac billiards: Weak (anti)localization, conductance fluctuations, and decoherence M. S. M. Barros,1 A. J. Nascimento Júnior,1 A. F. Macedo-Junior,1 J. G. G. S. Ramos,2 and A. L. R. Barbosa1 1

Departamento de F´ısica, Universidade Federal Rural de Pernambuco, 52171-900 Recife, Pernambuco, Brazil 2 Departamento de Ciências Exatas, Universidade Federal da Para´ıba, 58297-000 Rio Tinto, Para´ıba, Brazil (Received 14 October 2013; revised manuscript received 8 December 2013; published 30 December 2013)

In this paper, we investigate the transport properties of open chaotic Dirac billiards and their intrinsic (chiral universal) symmetry classes. The prominent examples of these systems are some categories of topological insulators and graphene structures. We extend the diagrammatic method of integration over the unitary group and obtain analytical results for the semiclassical limit and for the high quantum limit in the universal regime. We show the emergence of quantum fingerprints characteristic of the chiral symmetries, which are amplified in the presence of a single open channel in each electronic terminals. We compare the chaotic Dirac billiards with the “Schrödinger billiards” in a myriad of regimes, exhibiting the differences between the chiral universal classes and the Wigner-Dyson classes. Two numerical methods were used to confirm our analytical findings, yielding also the distribution of conductances. We also investigate analytically the effect of dephasing using the characteristic time scales of the chaotic billiards and we show the appearance of peculiar numbers of chaos. DOI: 10.1103/PhysRevB.88.245133

PACS number(s): 73.23.−b, 73.21.La, 05.45.Mt

I. INTRODUCTION

The study of the electronic transport properties on disordered mesoscopic systems exhibits the survival of only time-reversal (TRS), spin-rotation (SRS), particle-hole (PHS), and chiral or sublattice (SLS) symmetries.1,2 They give rise to ten symmetries classes which are divided into WignerDyson,3 cChiral,4,5 and Altland-Zirnbauer6 random matrix ensembles.7 There are three Wigner-Dyson classes, namely, circular orthogonal ensemble (COE), characterized by the presence of TRS and SRS (β = 1), circular unitary ensemble (CUE), which has the TRS broken by external magnetic field (β = 2), and circular symplectic ensemble (CSE), which is characterized by both TRS and SRS broken by a spin-orbit interaction (β = 4). The chiral ensembles have also three classes with the same Wigner-Dyson symmetries: chiral circular orthogonal (chCOE), unitary (chCUE), and symmpletic (chCSE) ensembles. The difference between Wigner-Dyson and chiral ensembles is the validity of the SLS in chiral ensembles. Moreover, the absence of an external magnetic field keeps the particle-hole symmetry in both chCOE and chCSE. Lastly, there are four Altland-Zinbauer classes, which describe quantum electronic devices connected to superconductors.8 The recent control of novel materials allows the successful experimental study of all symmetries classes.9–11 Particularly, some studies have been focusing on chiral chaotic quantum billiards,12–16 also known as chaotic Dirac billiards.17 Through this device, the wave functions of the incident electrons are described by the massless Dirac equation of the corresponding relativistic quantum mechanics, instead of the Schrödinger equation. Furthermore, the mass term is contained in the Dirac equation if the SLS is broken.13,15 The massless Dirac equation is appropriate to describe the electronic states of two independent sublattice components, which generates additional constraints known as pseudospins.10 The prominent examples of these bipartite systems are square lattices, such as some topological insulators,18–21 and hexagonal lattices, whose main example are the graphene structures.10,17,18,22,23 The quantum coherent electronic transport and its corresponding effects of weak localization13,15,19,20,24 and universal 1098-0121/2013/88(24)/245133(10)

conductance fluctuations13,15,18,25–28 are the most common fundamental physical phenomena resulting from these symmetries. The role of the edges on the electronic wave functions gives rise to weak (anti)localization, which decreases (amplifies) the conductance when the TRS and SRS are unbroken (broken). On the other hand, the conductance fluctuations in the ballistic regime are a phenomenon of the multiple electronic scatterings on the edges or on the impurities of the chaotic Dirac billiard. Both effects were studied in the case of SLS broken by massive edges as in Refs. 13,15,18, and 22. The quantum interference effects are quite susceptible to quantum coherence break and are suppressed by a sufficiently large dephasing rate.11,19,29–34 Furthermore, decoherence allows the study of the transition from quantum to classical regimes of the electronic transport, being particularly special for a complete understanding of the relations between symmetries and quantum interferences.30,31,35 In this work, we present a complete analytical and numerical analysis of an open chaotic Dirac billiard, with unbroken SLS, connected to two terminals, in the presence of disorder. Using the random matrix theory,4,5 we obtain exact expressions for the average of conductance and its universal fluctuation correlations, in the noninteracting regime. We assume a dephasing time (τφ ) and an electronic dwell time (τD ) inside the billiard much larger than the Ehrenfest time, {τφ ,τD } τE , ensuring the holding of the universal regime. To calculate the averages over chiral ensembles with unbroken TRS, we introduce an extension of the diagrammatic method36 of integration over the unitary group. We study a large class of symmetries and, in particular, the analytical procedure is used to investigate the orthogonal group.37 We find that a chaotic Dirac billiard presents a conductance average and variance remarkably different from a “chaotic Schrödinger billiard”.35,38 These results are appreciable in the high quantum regime, defined in a setup of only one open channel in each terminal. However, we also show a very fast convergence of their behavior, as a function of the total number of open channels, to the “nonrelativistic” behavior. We show explicitly that the variance of conductance on the chiral ensemble converges to twice the variance of conductance on

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M. S. M. BARROS et al.


the Wigner-Dyson ensemble35,38 in the limit of a large number of open channels. We also investigate a myriad of electronic transport decoherence mechanisms using the model introduced in Refs. 39 and 40 for chaotic quantum billiards. As the experiments of Refs. 30–32 indicate, this model is highly successful. The model will consist of connecting a third terminal, with N¯ φ open channels, to the chaotic Dirac billiard and setting the average current through it to null.39 The electronic charge can leave through this terminal but it is reinjected inside the billiard with a randomized phase. As a consequence, the number of open channels in the third terminal can tune the dephasing rate (1/τφ ). More specifically, the dephasing rate is proportional to N¯ φ .29,30 In this regime, we obtain novel universal numbers of the relevant ratio between conductance variances, in different symmetries classes, for the chaotic Dirac billiard in the semiclassical limit. As remarkable examples, for N¯ φ 1, we obtain 2.5 and 0.44 for the ratios chCOE/chCUE and chCSE/chCUE, respectively. The same ratio evaluated to a chaotic quantum “Schrödinger” billiard (COE/CUE) is 3.0, and was first calculated in Ref. 30 with great accordance with experimental data. The work is divided as follows: In the Sec. II, we introduce the scattering matrix symmetries of the chiral ensembles and present a corresponding extension of the diagrammatic method of integration over the orthogonal group. We apply this method in the obtention of the exact results for the averages and variance of conductance of the three classes of chiral ensembles, for a chaotic Dirac billiard connected to two terminals. To confirm our results, we developed two independent numeric simulations raised on the corresponding Hamiltonian model41 and on the Cartan table.7 In Sec. III, we investigate the decoherence effects on of weak (anti)localization and universal conductance fluctuations. We show the universal numbers that strikingly distinguish between the chiral ensembles and the Wigner-Dyson ensembles. The conclusions are given in Sec. IV. II. OPEN CHAOTIC DIRAC BILLIARD

We follow Refs. 2 and 5 to represent the massless Dirac Hamiltonian with sublattice or chiral symmetry. The Hamiltonian satisfies the following anti-commutation relation: 0 1 . (1) H = −σz Hσz , σz = M 0 −1M The H matrix has dimension 2M × 2M and 1M is an M × M identity matrix. We can interpret the M of 1’s and −1’s in σz as the number of atoms in each sublattice,42 in a total of 2M atoms in the chaotic Dirac billiard. The Hamiltonian model for the scattering matrix, S, can be written as41 †

† −1

S() = 1 − 2π iW ( − H + iπ WW ) W.

(2)

The S matrix has dimension N¯ T × N¯ T , where N¯ T = N¯ 1 + N¯ 2 + · · · + N¯ m is the total number of open channels or atoms in the m terminals, which are connected to the chaotic Dirac billiard. The N¯ i = 2Ni is the number of open channels in the ith terminal. In this way, we define each sublattice with Ni open channels and, as the system is compounded by two sublattices, there are 2Ni open channels in the ith terminal. The

2M × N¯ T matrix W represents all combinations (interactions) of the chaotic Dirac billiard resonances coupled to the open channels of the m terminals. The scattering matrix is unitary S † ()S() = 1 due to the conservation of electronic charge. From Eqs. (1) and (2), the S matrix also satisfies the relation 1 0 S() = z S † (−)z , z = NT , (3) 0 −1NT where NT = N1 + N2 + N3 + · · · + Nm . We assume electronic transport through the chaotic Dirac billiard near zero energy or, equivalently, on the Dirac point,23,43,44 = 0. In this case, we can rewrite S = z S † z .

(4)

It is convenient to represent the S matrix as a function of transmission, t, and reflection, r, blocks as ⎤ ⎡ r11 t12 · · · t1m ⎢ t21 r22 · · · t2m ⎥ (5) S=⎢ .. .. ⎥ .. ⎦, ⎣ .. . . . . tm1 tm2 · · · rmm where tij and rij have dimension N¯ i × N¯ j , with i,j = 1, . . . ,m. The diagrammatic method of integration over the unitary group (for the systems with SLS symmetry) cannot be used directly to compute averages over chCUE, chCOE, and chCSE. The main reason is the introduction of new constraints, resulting from SLS, on the S matrix given by Eq. (4), which were not previously taken into account in Ref. 36. A first solution is founded on the decomposition of the S matrix as a function of the corresponding symmetry: an orthogonal matrix for chCOE, unitary matrix for chCUE, and symplectic matrix for chCSE, as described in Ref. 7 (see Table I). However, such a decomposition is insufficient for the diagrammatic method of integration over the orthogonal group specifically in chCOE and chCSE symmetries. While in the circular case, the COE does not contain orthogonal matrices (despite the name), but only matrices that are unitary and symmetric, in the chCOE we have actual orthogonal matrices. In the following, we extend the diagrammatic method to cover this problem. A. Integration over the orthogonal group and the average of conductance

In this section, we extend the diagrammatic method of integration over the unitary group.36 The extension will allow us to compute the average and the variance of conductance for the chaotic Dirac billiard connected to m terminals. The TABLE I. The S matrix is decomposed as a function of the U matrix, which is an orthogonal matrix for chCOE, unitary matrix for chCUE, and symplectic matrix for chCSE, as described in Ref. 7. The decomposition of the scattering matrices S satisfies Eq. (4). Symmetry

β

U

S

chCOE chCUE chCSE

1 2 4

O(2NT ) U(2NT ) Sp(4NT )

S = z U T z U S = z U † z U S = z U † z U

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OPEN CHAOTIC DIRAC BILLIARDS: WEAK . . .


integration procedure enables us to obtain the novel exact results over the orthogonal chiral group.37 We emphasize that this method is general and allows us to study all chiral classes. For pedagogical reasons, we start by analyzing the relevant configuration of two terminals (m = 2). We begin by considering the Landauer conductance transmission coefficient45,46 2e2 † 2e2 Tr tt = Tr(C1 SC2 S † ), (6) h h where t is the transmission block of the scattering matrix and the factor 2 is used to account for the spin degeneracy. The matrices C1,2 are projection matrices over their respective terminals, which are defined as 0 1¯ 0 0 C 1 = N1 . (7) , C2 = 0 0 0 1N¯2 G=

Here 1N¯i is a N¯ i × N¯ i identity matrix, and N¯ i = 2Ni is the number of open channels in the ith terminal. For the projection matrices, both relations C1 C2 = 0 and C1 + C2 = 1N¯ T hold. We replace in Eq. (6) the scattering matrix S = z U T z U characteristic of the chCOE, Eq. (4) (see also Table I), and obtain 2e2 G= Tr(C1 z U T z U C2 U T z U z ) h 2e2 Tr(C1 U T z U C2 U T z U ), (8) = h where the relation z C1 z = C1 was used. The U matrix is orthogonal, i.e., its entries are real numbers. We are now in a position to apply the diagrammatic method of integration over the orthogonal group to calculate the average of Eq. (8). The diagrammatic rules consist of accounting for all the possible contractions of indices of the scattering matrices. The basic elements are shown in Fig. 1(a). Black and white dots represent the indices of the U matrix, and their corresponding contractions δij are indicated as the thin lines connecting the dots. The first procedure is to represent G/(2e2 / h) diagrammatically as shown in Fig. 1(b) for which the scattering matrix is represented by thick dotted lines, • · · · ◦, and the projection matrices C1,2 are represented by directed thick solid lines, as can be seen in the Fig. 1(a). The second procedure is to perform the ensemble average directly on the diagrammatic representation by connecting the dots of the same color with thin lines in all topologically distinct ways. We find nine possible diagrams, as can be seen in Fig. 1(c), and, for more details, see Appendix A. From the diagrams of Fig. 1(c), we obtain

Gβ=1 = V11 Tr(C1 C2 )Tr(z )2 + Tr(C1 )Tr(C2 )Tr z2 2 2e / h

+ Tr(C1 C2 )Tr(z2 ) + V2 Tr(C1 C2 )Tr z2 + Tr(C1 )Tr(C2 )Tr(z ) + Tr(C1 C2 )Tr(z )

+ Tr(C1 C2 )Tr z2 + Tr(C1 C2 )Tr z2

(9) + Tr(C1 )Tr(C2 )Tr z2 2

2

where the diagrams indexed by 1, 2 and 3 in Fig. 1(c) have weights V11 = (2NT + 1)[2NT (2NT − 1)(2NT + 2)]−1

(a)

(b)

(c)

FIG. 1. (a) Basic objects of the diagrammatic rules. In decreasing order, a random element (Uij ) of the U matrix, a projector on the space of channels of the terminals (Ci ), and a Kronecker delta element, representing the contraction of indices (δij ). (b) Diagrammatic representation of Eq. (8). (c) The ensemble average of the conductance, Eq. (9). Notice that diagrams 1, 2, and 3 have weights V11 , and diagrams 4 to 9 have weights V2 . The weights were calculated in Ref. 37.

and the diagrams indexed by 4 to 9 have weights V2 = −[2NT (2NT − 1)(2NT + 2)]−1 . The weights were calculated in Ref. 37. The identity Tr(z ) = Tr(C1 C2 ) = 0 determines that some diagrams have null contribution. The only diagrams with nonnull contribution are diagrams 2 and 9 of Fig. 1(c). Using Eq. (9), we obtain the following average for the chCOE: Gβ=1 =

4N1 N2 NT 2e2 . h (2NT − 1) (NT + 1)

(10)

We can obtain the conductance average for the chCSE also using Eq. (9) and the following algebraic substitutions: Gβ=1 = 2e2 / h[· · · ] → Gβ=4 = 2e2 / h (−1/2) [· · · ]; NT → −2NT ; Tr → −2Tr. Performing this procedure, we obtain Gβ=4 =

16N1 N2 NT 2e2 . h (4NT + 1) (2NT − 1)

(11)

Finally, we analyze the chCUE ensemble for which the U matrix of Eq. (8) is unitary and the usual diagrammatic method36 can be used. For the chCUE ensemble, four diagrams of Fig. 1(c) contribute to the average,47 diagrams 1 and 2, with weight V11 = [(2NT − 1)(2NT + 1)]−1 , and diagrams 4 and 5, with weight V2 = −[2NT (2NT − 1)(2NT + 1)]−1 . We can represent the final results in a compact way through the following general equation for the conductance average of the

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three chiral ensembles: G =

4βN1 N2 NT 2e2 . h (βNT + 1) (2NT − 1)

(12)

Equation (12) is the first main result of this paper. It can be used to study some relevant limits. Firstly, taking the high quantum regime, N1 = N2 = 1, the conductance averages in 8 the scale 4e2 / h, G /(4e2 / h), are 49 for chCOE, 15 for the 16 chCUE, and 27 for the chCSE, as can be seen in Fig. 2(a). They are just distinct as compared with the known results for the “Schrödinger billiard” in all symmetries, COE, CUE, and CSE (see Appendix B). Secondly, the semiclassical limit can be obtained by expanding Eq. (12) in powers of NT with the following result: 2e2 N1 N2 N1 N2 2 G = 2 + 1− h N1 + N2 β (N1 + N2 )2 N1 N2 2 1 4 1− + 2 + + · · · . (13) 2 β β (N1 + N2 )3 Equation (13) can be alternatively rewritten as a function of N¯ i = 2Ni with the result ¯ ¯ 2e2 N1 N2 N¯ 1 N¯ 2 2 G = + 1− h N¯ 1 + N¯ 2 β (N¯ 1 + N¯ 2 )2 4 N¯ 1 N¯ 2 2 + 1− + 2 + · · · . (14) β β (N¯ 1 + N¯ 2 )3 The first term in Eq. (14) is Ohm’s law and the second is known as weak (anti)localization β = 1 (β = 4). Notice the second term of expansion is null for β = 2, but the third represents a non-null contribution to the interference corrections. In the symmetric configuration, N¯ = N¯ 1 = N¯ 2 , the two first terms

of Eq. (14) simplify to G = 2e2 / h N¯ /2 + (1 − 2/β)1/4 , which is in agreement with Refs. 13 and 15 for β = {1,2}. Another interesting point is to compare the result of Eq. (14) with the one resulting from Wigner-Dyson Ensembles:35 The first two terms of expansions (14) and (B2) are equivalent, as expected, but the results are completely different after the second term, essentially because of the SLS validity on the Dirac Billiards. B. Universal Conductance Fluctuations

We apply our previously developed diagrammatic method in the last subsection to obtain exact expressions for conductance variances on Chiral Ensembles. Firstly, the variance of conductance is defined as (15) var[G] = G2 − G2 , 2 where G was obtained in Eq. (12), and G is defined as G2 = [Tr(tt † )]2 = [Tr(C1 SC2 S † )]2 4e4 / h2 = [Tr(C1 z U T z U C2 U T z U z )]2

for chCOE (see Table I). We are in a position to use the developed diagrammatic method. After a cumbersome algebraic procedure of identifying all the possible contractions between black dots and white dots of the characteristic diagrams of Eq. (16), we obtained a total of 11025 diagrams. However, within this set of diagrams, we also identify the existence of only 540 non-null contributions to the variance, as can be seen in Appendix A and Fig. 7. After the sum over all contributions, we obtain the following exact expressions for variances

⎧ 16N1 N2 NT 3 + 2NT3 + 4N1 N2 NT2 − 4NT − 4N1 N2 − 5N12 − 5N22 ⎪ ⎪ ⎪ ⎪ ⎪ (2NT − 3)(2NT − 1)2 (NT + 3)(NT + 1)2 (2NT + 1) ⎪ ⎪

⎪ 4e4 ⎨ 8N1 N2 3 + 16N1 N2 NT2 − 6NT2 − 6N12 − 6N22 var[G] = 2 h ⎪ (2NT − 3)(2NT + 3)(2NT + 1)2 (2NT + 1)2 ⎪ ⎪ ⎪

⎪ ⎪ ⎪ 32N1 N2 NT 3 − 16N1 N2 + 8NT − 20N12 − 20N22 − 16NT3 + 64N1 N2 NT2 ⎪ ⎩ (4NT + 3)(4NT + 1)2 (2NT − 3)(2NT − 1)2 (4NT − 1)

Equation (17) is the second main result of this paper. Taking the high quantum regime, N1 = N2 = 1, the variances of 228 conductance (in units of 8e4 / h2 ), var[G]/(8e4 / h2 ), are 2025 124 416 for chCOE, 1575 for chCUE, and 8019 for chCSE, as can be seen also in Fig. 2(b). Once again, this second result is remarkably different from those obtained previously for COE, CUE, and CSE (see Appendix B). To study the semiclassical limit, we expand Eq. (17) in powers of NT and obtain var[G] 4 N12 N22 1 = +O 4e4 / h2 β (N1 + N2 )4 NT 4 N¯ 12 N¯ 22 1 . = + O β (N¯ 1 + N¯ 2 )4 N¯ T

(18)

(16)

(β = 1), (β = 2),

(17)

(β = 4).

Equation (18) is in agreement with Refs. 13 and 15, which solve the problem for β = {1,2}. Taking N¯ = N¯ 1 = N¯ 2 on Eq. (18), we obtain for the variance of conductance 14 1 for chCOE, 18 for chCUE, and 16 for chCSE, which are, 1 respectively, 4, 2, and 1 times 16 of the results previously obtained for the CUE. C. Numeric simulation

In order to confirm the analytical results, Eqs. (12) and (17), we develop two independent numerical simulations to generate the S-matrix chiral ensembles. Both ensembles are used to compute the average, the variance and, furthermore, the distribution of conductance of Eq. (6). The first numerical analysis is founded on the Hamiltonian model41 (HM). We use Eq. (2) to construct random scattering

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6

2

(a)

(c)

G (4e2 /h)

5 1.5

4 3

1 β=1 β=2 β=4

2 0.5

1

(b)

var[G] (8e4 /h2 )

0.12

0.09

0.06

0.06

0.03

0.03

0

0

(d)

0.12

0.09

2

4

6

8

0

10

QR HM DM

0

2

4

6

8

10

N2

N

FIG. 2. (Color online) The average and variance of conductance are plotted in units of 4e2 / h and 8e4 / h2 , respectively. (a),(b) Symmetric terminals, N = N1 = N2 , and (c),(d) asymmetric terminals with N1 = 2. The dotted lines are obtained from QR (Gram-Schmidt) factorization and from Hamiltonian model (HM) numeric simulations. Solid lines are the exact results of Eqs. (12) and (17).

matrices with SLS. The anti-commutation relation, Eq. (1), implies a kind of Hamiltonian that can be written as4,5 0 T . (19) H= T† 0 Here, the T -matrix block of the H matrix has dimension M × M. The random matrix theory establishes that the entries of the T matrix have Gaussian distribution,48 βM P (T ) ∝ exp − 2 T r(T † T ) , (20) 2λ where 2M is the number of resonances inside the chaotic Dirac billiard including both the two sublattice degrees of freedom. Also, λ = M/π is the variance related to the electronic single-particle level spacing, . The W = (W1 ,W2 , . . . ,Wm ) matrix is a 2M × 2NT deterministic matrix, describing the coupling of the resonances states of the chaotic Dirac billiard with the propagating modes in the m terminals. This deterministic matrix satisfies nondirect process, i.e., the orthogonality † condition Wp Wq = π1 δp,q holds. Accordingly, we consider the relation σz Wz = W, indicating the scattering matrix is symmetric (3). We consider the system on the Dirac point, = 0, and, to ensure the chaotic regime and consequently the universality of observables, the number of resonances inside the chaotic Dirac billiard is large (M NT ). The second model is founded on the Cartan table,7 Table I, used to rewrite the 2NT × 2NT scattering matrix as a function of the corresponding ensemble: unitary matrices for chCUE, orthogonal matrices for chCOE, and symplectic matrices for chCSE. To generate these ensembles, as described in Ref. 49, we use Gram-Schimidt factorization, also known as QR factorization.

The two numerical simulations produce Fig. 2, which shows both the average and variance of conductance obtained through 106 realizations compared with the analytical results, Eqs. (12) and (17), for each one of the three classes of chiral ensembles. We use the T matrices, with dimension 100 × 100 (M = 100), and the corresponding H matrices with dimension 200 × 200 (200 resonances). Figures 2(a) and 2(b) are plotted for symmetric terminals N = N1 = N2 , while Figs. 2(c) and 2(d) shows the behavior for asymmetric terminals, with N1 = 2. The average and variance of conductance are plotted in units of 4e2 / h and 8e4 / h2 , respectively. In Fig. 3 (left), we compare the variance var[G] (in units of 8e4 / h2 ) of the chaotic Dirac billiard for chiral ensembles, Eq. (17), with the variance var[G]wd (in units of 4e4 / h2 ) of the chaotic “Schrödinger billiard” (or Wigner-Dyson ensembles), Eq. (B3), for symmetric terminals N = N1 = N2 . These dimensionless variances present strong differences for N = 1 and converge to the same values for N 2. We can establish the following approximations between the two “kinds” of conductance variances in the semiclassical limit, var[G]β → 2 × var[G]wd,β .

(21)

To emphasize the peculiarities of the high quantum regime, N1 = N2 = 1, we compare numerically, using an ensemble of 106 realizations, the conductances distribution P (G) of the chaotic Dirac billiard (in units of 4e2 / h) with the distribution of the chaotic “Schrödinger billiard” (in units of 2e2 / h) of the Wigner-Dyson ensembles.38 The comparison is shown in Fig. 3 (right), which shows the formation of a quite different behavior between the two billiards. For example, Fig. 3 (right) exhibits the formation of a singularity for the unitary ensemble

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PHYSICAL REVIEW B 88, 245133 (2013) 10 QR HM

β=1

2.5

β=1

2 1.5

5

1

0.12

0.5 QR-CH HM-CH DM.-CH QR-WD HM-WD DM.-WD

0.09

var[G]

β=2

2

P (G)

β=1

2.5

β=2

1.5 1

1

0.5

0.06

0.5

β=4 2

β=4

2.5

β=4

2

1.5

0.03

1.5 1

1

0.5

0

β=2

2

1.5

0

2

4

6

8

0

10

0.5 0.2

N

0.4

0.6

G (2e2 /h)

0.8

1

0

0.2

0.4

0.6

0.8

1

G (4e2 /h)

FIG. 3. (Color online) (Left) The figure indicates the comparison between the variance var[G] (in units of 8e4 / h2 ) of a chaotic Dirac billiard of chiral ensembles (CH) with the variance var[G]wd (in unitos of 4e4 / h2 ) of Wigner-Dyson chaotic billiards (WD), both with symmetric terminals, N = N1 = N2 . The dotted lines are the plots of Eq. (17) (DM-CH) and the solid lines are the plots of Eq. (B3) (DM-WD). The symbols refer to QR (Gram-Schmidt) factorization (QR) and Hamiltonian model (HM) numeric simulations for chiral ensembles (CH) and Wigner-Dyson ensembles (WD). (Right) We compare the distribution of conductance P (G) in the high quantum regime (N1 = N2 = 1), in units of 4e2 / h, between a chaotic Dirac billiard of chiral ensembles (right column) and, in units of 2e2 / h, with the chaotic quantum billiard of Wigner-Dyson ensembles (Ref. 38) (left column) with 106 realizations of the random S matrix.

and a nonuniform conductance distribution in the orthogonal ensemble only on the Dirac billiard. III. DECOHERENCE IN THE CHAOTIC DIRAC BILLIARD

After the introduction of the diagrammatic method of integration over the three groups, which was used to obtain conductance averages and conductance variances of the chaotic Dirac billiards connected to two terminals, we study the electronic phase coherence breaking. We use the formulation proposed by Büttiker,39,40 which, as the experiments of Refs. 30 and 31 indicate, was successful in describing the chaotic quantum billiard. The method was originally used in the framework of random matrix theory, and more recently in Ref. 32. The formulation assumes a third fictitious terminal connected to the chaotic Dirac billiard which induces the phase coherence breaking or decoherence. The consistency condition is to maintain the average current in the third terminal null, I3 = 0, in a way that there is only effective current between the terminals labeled by 1 and 2. Another consistence condition is the electronic current conservation, I = I1 = −I2 . The basic mathematical formulation to incorporate both conditions is to change Eq. (6) by the addition of the third terminal such that it can be rewritten as30,31 2e2 g31 g23 , (22) g12 + G= h g31 + g32 †

where gij = Tr(tij tij ). The conductance given in Eq. (22) can be written as a function the of scattering matrix through the transmission coefficients gij = Tr(Ci SCj S † ), with i,j = 1,2,3. The S matrix has dimension N¯ T × N¯ T , where N¯ T =

N¯ 1 + N¯ 2 + N¯ φ and N¯ φ = 2Nφ is the number of open channels in the third terminal. The semiclassical regime is reached by expanding Eq. (22) in powers of N¯ T and taking its average, g31 g23 G 1 g . (23) = + + O 12 g31 + g32 2e2 / h NT2 The averaged transmission coefficients gij can be obtained from Eq. (9) as discussed in Appendix A. We replace Eq. (A1) in Eq. (23) and obtain 2e2 N1 N2 G = 2 h N1 + N2 N1 N2 1 2 . (24) + 1− β N1 + N2 N1 + N2 + Nφ We are in the position to introduce the relation N¯ φ /(N¯ 1 + ¯ N2 ) = τD /τ φ, where τD = 2π mA/ h(N¯ 1 + N¯ 2 ) is the dwell time through the chaotic Dirac billiard and τφ = 2π mA/ hN¯ φ is the dephasing time of the electronic transport. Furthermore, m and A are the electronic mass and the lithographic area of the Dirac billiard, respectively. We can rewrite Eq. (24) as a function of the dephasing time and open channels numbers in the ith terminal, N¯ i = 2Ni , as N¯ 1 N¯ 2 2e2 N¯ 1 N¯ 2 1 2 G = . + 1− h N¯ 1 + N¯ 2 β (N¯ 1 + N¯ 2 )2 1 + ττDφ (25) We can obtain two limits using Eq. (25) to test its consistence. The first one is for τφ τD for which we recover Eq. (14). The second one is for τD τφ for which the weak (anti)localization corrections are suppressed and only

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OPEN CHAOTIC DIRAC BILLIARDS: WEAK . . . 0

PHYSICAL REVIEW B 88, 245133 (2013) 0.12

var[G] (8e4 /h2 )

(a) β = 1

-0.014 -0.028 -0.042

QR HM

-0.07

(b) β = 4

0.098

var[G]β /var[G]β =2

δG (4e2 /h)

-0.056

0.084 0.07 0.056 0.042 0.028 0.014 0

0

10

5

Nφ

20

15

FIG. 4. (Color online) (a) Week localization and (b) antilocalization corrections as a function of Nφ for the high quantum regime, N1 = N2 = 1. The symbols denote QR (Gram-Schmidt) factorization (QR) and Hamiltonian model (HM) numeric simulations for chiral ensembles with 106 realizations. Solid lines are interpolation formulas.

the Ohm’s law term survives. In Figs. 4(a) and 4(b), we plot the weak (anti)localization corrections, δG (in units of 4e2 / h), as a function of Nφ in the high quantum regime, N1 = N2 = 1. For β = 2, as expected δG = 0. We observe a monotonically decreasing weak localization correction as a function of Nφ for β = 4, and a peculiar transition decreasing and monotonically increasing for β = 1, in the region of high quantum regime. The numerical simulations of Figs. 4(a) and 4(b) are valid also for the semiclassical regime for which Eq. (25) is valid. The universal conductance fluctuations from Eq. (22) can been written as32 g32 4 var[g31 ] + g31 4 var[g32 ] var[G] = var[g ] + 12 4e4 / h2 (g32 + g31 )4 g32 g31 co var[g31 ,g32 ] +2 (g32 + g31 )4 2

+2

2

g31 2 co var[g21 ,g23 ] (g32 + g31 )2

g32 2 co var[g21 ,g31 ] +2 +O (g32 + g31 )2

QR HM

0.08 0.06 0.04 0.02 0 2.5

(b) β=1

2 1.5 1

β=4

0.5 0

25

(a)

0.1

0

5

10

Nφ

20

15

FIG. 5. (Color online) (a) Variance of the conductance as a function of Nφ in the high quantum regime, N1 = N2 = 1. Top to bottom curves specify β = 1,2,4 behavior, respectively. (b) The ratios between variance of conductances for Nφ 1. They tend to 2.5 due to chCOE/chCUE, and 0.44 due to chCSE/chCUE, as obtained in Eq. (30). The symbols designate the numeric simulations of QR (Gram-Schmidt) factorization (QR) and Hamiltonian model (HM) chiral ensembles, both with 106 realizations. Solid lines are interpolation formulas.

In the symmetric terminals case, N = N1 = N2 , Eq. (27) simplifies to 2 1 τφ var[G] 1 τφ 2 1 2 −1 = + 4 2 4e / h 4β τD β β 16N τD 2 1N 1 2 N = + . (28) −1 β Nφ2 4β β Nφ2 We obtain the variance of conductance in regime (N = 1) using Eq. (28). We obtain ⎧ τ 2 φ 5 ⎪ = 54 N12 ⎪ ⎪ 16 τD φ 4 ⎨ 4e 1 τφ 2 1 1 = var[G] = 2 8 τD 2 Nφ2 h ⎪ ⎪

⎪ 2 τ φ 7 7 1 ⎩ = 32 128 τD N2 φ

1 NT2

.

(26)

25

the high quantum (β = 1), (β = 2),

(29)

(β = 4).

Taking the ratios between variance of conductances from Eq. (29), we obtain the following novel universal numbers: 5 (β = 1), var[G]β (30) = 27 var[G]β=2 (β = 4). 16

Using the same typical diagrams for the conductance variances used in Sec. II B and also Eq. (A1), we obtain in the limit Nφ 1(τφ /τD 1) the result 2 τφ var[G] 4 N12 N22 = 4 2 4 4e / h β (N1 + N2 ) τD

2 N1 N2 N12 − N1 N2 + N22 τφ 2 2 −1 + . 5 β β (N1 + N2 ) τD (27)

Previously obtained results for chaotic Schrödinger billiards (Wigner-Dyson ensembles) exhibit a ratio of variances of conductance of 3.0 for COE/CUE,30,31 while we obtain 2.5 for chCOE/chCUE as shown in Eq. (30). In Fig. 5, we plot the variance of conductance for the high quantum regime, N1 = N2 = 1, as a function of Nφ , and the ratios between variance of conductances. In the limit Nφ 1, the ratios tend to 2.5 for chCOE/chCUE and 0.44 for chCSE/chCUE, confirming nicely the results of Eq. (30). In addition, Fig. 6 shows the convergence of the conductance distribution of the chaotic Dirac billiard to a Gaussian

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6

6

β=1

4

β=1

Nφ=0 Nφ=1

4

Nφ=2 Nφ=3 Nφ=4

2 0

2

Nφ=5

0

0.2

0.4

0.6

0.8

1

P (G)

8 QR

β=2

0.2

0.4

0.6

0.8

1

0.4

0.6

0.8

1

0.4

0.6

0.8

1

β=2

6

QR HM HM

4

4

2

2 0

0.2

0.4

0.6

0.8

1

12

0

0

0.2

12

10

10

β=4

8

6

4

4

2

2 0

0.2

β=4

8

6

0

0

8

6

0

0

0.4

0.6

0.8

1

0

G (2e2 /h)

0

0.2

G (4e2 /h)

FIG. 6. (Color online) Conductance distributions for the chaotic Dirac billiards (right column) and for the chaotic quantum billiard (left column). The distribution P (G) tends to the Gaussian as Nφ is increased. The symbols are for numerical simulations of QR (Gram-Schmidt) factorization (QR) and Hamiltonian model (HM) for chiral ensembles and Wigner-Dyson ensembles (Ref. 31), both with 106 realizations.

behavior as Nφ increases, in the high quantum regime (N1 = N2 = 1). In this case, P (G) is described by its mean and variance, Eqs. (24) and (29). In the limit Nφ = 0, P (G) is highly non-Gaussian as we have shown in Fig. 3 (right). On another side, in the limit Nφ = 1, P (G) has an essentially Gaussian behavior for both chCUE and chCSE. The same behavior occurs for chCOE in the different limit Nφ = 2. The distributions of conductance of the chaotic quantum (Schrödinger) billiard31 was plotted in the same Fig. 6 of the chaotic Dirac billiard for a direct comparison. Finally, we consider the semiclassical limit N¯ 1 + N¯ 2 1 in the number of channels of Eq. 26 and we obtain 1 var[G] 4 N¯ 12 N¯ 22 =

. 4e4 / h2 β (N¯ 1 + N¯ 2 )4 1 + τD 2

(31)

τφ

In the limit τφ τD , we recover the Eq. (18), while in the limit τD τφ the variance is suppressed. IV. CONCLUSIONS

In this work, we present an analytical and numerical study of an open chaotic Dirac billiard in the presence of disorder. Using the random matrix theory and a conception of a diagrammatic method extension, we obtain exact expressions for the conductance averages and the corresponding variances (or universal conductance fluctuation amplitudes). The results are obtained for all chiral ensembles after the integration over the unitary group through the diagrammatic method. We study and compare a large class of symmetries and, in particular, the orthogonal group. We found that a chaotic

Dirac billiard presents a conductance average and variance remarkably different from a “chaotic Schrödinger billiard”. This results is appreciable in the high quantum regime, defined in the case of a single open channel in each terminal. However, we also show a very fast convergence of their behavior as a function of the total number of open channels. We also show a variance of conductance in chiral ensemble with convergence (in the limit of a large number of open channels) to twice the variance of conductance in the Wigner-Dyson ensemble.35,38 We also investigated the electronic transport through the chaotic Dirac billiard in the presence of decoherence mechanisms. We apply the diagrammatic method extension concomitantly with the model of three terminals introduced in Refs. 39 and 40. These calculations will permit us, in future investigations, to study through the third terminal the effects of both the interacting particles regime and the Coulomb blockade.50 Future investigations could include the effect of dephasing mechanisms on the noise following Ref. 51, which can generate additional fingerprints of the chaotic Dirac billiard. We also obtain novel universal numbers of the relevant ratio between conductance variances, in different symmetry classes, for the chaotic Dirac billiard in the semiclassical limit. Another interesting investigation includes the magnetic and spin-orbit crossovers52 between the universal chiral classes of symmetry. In this last case, the novel numbers of chaos encoded in the conductance variances ratios will introduce the competition of crossover mechanisms and dephasing characteristic numbers. We believe our results will be useful to control experimentally novel degrees of freedom in mesoscopic quantum dots such as graphene and topological insulators.

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ACKNOWLEDGMENTS

This work was partially supported by CNPq, CAPES and FACEPE (Brazilian Agencies). APPENDIX A: DIAGRAMMATIC METHOD FOR INTEGRATION OVER THE ORTHOGONAL GROUP

The diagrammatic rules consist of accounting for all possible contractions of indices of the U matrices, as we can see in Fig. 1(a). Black and white dots represent the indices, and its contractions are indicated as lines connecting the dots. A summary of the basic rules is: (1) Draw the basic elements according to Fig. 1(a). (2) Connect all the possible pairings of black dots of U matrices and do the same for the white dots; see Fig. 1(c). (3) Denote the U-cycle for any closed circuit in the diagram with an alternating sequence of thick dotted lines and thin lines. U-cycles have weight given by VP , where P characterizes the permutation of indices. (4) Denote a T-cycle any closed circuit with an alternating sequence of directed thick solid line and thin lines. T-cycles correspond to the trace of the matrices in the circuit. From the basic rules above, we concluded that diagrams 1, 2, and 3 of Fig. 1(c) have two U-cycles and consequently weights V11 , while diagrams 4 to 9 have one U-cycle and consequently weights V2 . The weights for the orthogonal group were calculated in Ref. 37. Referring to the T-cycle, diagrams 1, 2, 6, and 9 all have three T-cycles, diagrams 3, 4, 5, 7, and 8 have two T-cycles, and diagram 5 has four T-cycles. Using this information, we obtain Eq. (9) with a little algebra. From Eq. (9), we can calculate usually the transmission † coefficient averages gij = Tr(tij tij ) = Tr(Ci SCj S † ), with i = j , performing the substitutions C1 to Ci and C2 to Cj . Notice that Ci Cj = 0 and m C = 1N¯ T . Using this prescription, i=1 i we obtain 4βNi Nj NT (βNT + 1)(2NT − 1) β N¯ i N¯ j N¯ T = , i = j, (β N¯ T + 2)(N¯ T − 1)

FIG. 7. The figures show examples of the diagrams resulting from Eq. (16), each one having weight V1111 , V1,1,2 , V2,2 , V1,3 , and V4 , for diagrams 1, 2, 3, 4, and 5, respectively. The weights were calculated in Ref. 37.

Wigner-Dyson universal ensembles to compare with results obtained in this work. The averages of conductance for the three classes of Wigner-Dyson are Gwd =

Gwd (A1)

APPENDIX B: QUANTUM CHAOTIC BILLIARD FROM WIGNER-DYSON ENSEMBLES

(B1)

where NT = N1 + N2 and β = {1,2,4}. In the limit of high quantum regime, N1 = N2 = 1, the averages of conductance, in units of 2e2 / h, G /(2e2 / h), are 13 for COE, 12 for CUE, and 23 for CSE. Equation (B1) can be expanded in power of NT as

gij =

where N¯ T = N¯ 1 + · · · + N¯ m and N¯ T = 2NT . To calculate the variance of conductance, first it is necessary to compute the average of Eq. (16). In Fig. 7, we show five examples of the diagrams obtained from Eq. (16), each one with the weights V1111 , V1,1,2 , V2,2 , V1,3 , and V4 for diagrams 1, 2, 3, 4, and 5, respectively. The weights were calculated in Ref. 37 on page 794. We obtain a total of 540 non-null diagrams, 9 of them with weight V1111 , 78 with weight V1,1,2 , 63 with weight V2,2 , 168 with weight V1,3 , and 222 with weight V4 . From these diagrams, we can obtain Eq. (17) for three classes of chiral ensembles.

N1 N2 2e2 . h NT − 1 + 2/β

N1 N2 N1 N2 2e2 2 = + 1− h N1 + N2 β (N1 + N2 )2 2 N1 N2 2 + 1− + ··· (B2) β (N1 + N2 )3

For variance of conductance, the result is var[G]wd 4e4 / h2 =

2βN1 N2 (βN1 + 2 − β)(βN2 + 2 − β) (βNT + 2 − 2β)(βNT + 2 − β)2 (βNT + 4 − β) (B3)

In the high quantum regime, N1 = N2 = 1, the variances of 4 1 conductance, var[G]/(4e2 / h), are 45 for COE, 12 for chCUE, 1 and 18 for chCSE. Taking the semiclassical limit, NT 1, Eq. (B3) simplifies to

The chaotic quantum billiard is well understood in the literature, as can be seen in Refs. 35 and 38. In this section, we list the results for average and variance of conductance of the 245133-9

var[G]wd 2 N12 N22 = +O 4 2 4e / h β (N1 + N2 )4

1 NT

.

(B4)

M. S. M. BARROS et al. 1


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