PHYSICAL REVIEW E 88, 052912 (2013)

Statistics of quantum transport in weakly nonideal chaotic cavities

Sergio Rodrıguez-Perez,1 Ricardo Marino,2 Marcel Novaes,1 and Pierpaolo Vivo2

1Departamento de Fısica, Universidade Federal de Sao Carlos, Sao Carlos, SP, 13565-905, Brazil2Laboratoire de Physique Theorique et Modeles Statistiques, UMR CNRS 8626, Universite Paris-Sud, 91405 Orsay, France

(Received 13 August 2013; published 19 November 2013)

We consider statistics of electronic transport in chaotic cavities where time-reversal symmetry is broken andone of the leads is weakly nonideal; that is, it contains tunnel barriers characterized by tunneling probabilities �i .Using symmetric function expansions and a generalized Selberg integral, we develop a systematic perturbationtheory in 1 − �i valid for an arbitrary number of channels and obtain explicit formulas up to second order for theaverage and variance of the conductance and for the average shot noise. Higher moments of the conductance areconsidered to leading order.

DOI: 10.1103/PhysRevE.88.052912 PACS number(s): 73.23.−b, 05.45.Mt, 73.63.Kv

I. INTRODUCTION

At low temperatures and applied voltage, provided thatthe average electron dwell time is well in excess of theEhrenfest time, statistical properties of electronic transportin mesoscopic cavities exhibiting chaotic classical dynamicsare universal. Random matrix theory (RMT) has proven verysuccessful in describing this universality [1]. In this approach,the scattering S matrix of the cavity is modeled by a randomunitary matrix [2,3] (see [4–6] for the most recent analyticalresults on distribution of S).

We consider a chaotic cavity attached to two leads, havingN1 and N2 (with N1 � N2) open channels in each lead, anddenote by N = N1 + N2 the total number of channels. TheS matrix can be written in the usual block form S = ( r t ′

t r ′ ),in terms of reflection and transmission matrices. Landauer-Buttiker theory [7–11] expresses physical observables in termsof the eigenvalues {T1, . . . ,TN1} of the Hermitian matrix t t†. Afigure of merit is the conductance G(T ) = G0

∑N1i=1 Ti , where

G0 = 2e2/h is the conductance quantum. The assumptionthat S is a random matrix implies that the Tj ’s are correlatedrandom variables characterized by a certain joint probabilitydensity (jpd), and as a consequence, every observable becomesa random variable whose statistics is of paramount interest.

When the leads attached to the cavity are ideal, S isuniformly distributed in one of Dyson’s circular ensemblesof random matrices, which are labeled by a parameter β: it isunitary and symmetric for β = 1 (corresponding to systemsthat are invariant under time reversal), just unitary for β = 2(broken time-reversal invariance), and unitary self-dual forβ = 4 (antiunitary time-reversal invariance). In this ideal casethe jpd of reflection eigenvalues Ri = 1 − Ti is given [2,12,13]by the Jacobi ensemble of RMT, namely,

P(0)β (R) ∝ |�(R)|β

N1∏i=1

(1 − Ri)β

2 (N2−N1+1)−1, (1)

where �(R) = ∏j<k(Rk − Rj ) is the Vandermonde determi-

nant. The average and variance of the conductance were stud-ied, using perturbation theory in 1/N , long ago [2,3,14,15].In particular, as N → ∞, this variance becomes a constantdepending only on the symmetry class, a phenomenon thathas been dubbed universal conductance fluctuations. Morerecently, a fruitful approach based on the theory of the Selberg

integral was developed [16] and afterwards extended [17–20]to compute transport statistics nonperturbatively. The fulldistribution of G, known to be strongly non-Gaussian for asmall number of channels [17,20–22], was studied in [23–26].The statistics of other observables was studied in [19,27–30].The integrable theory of quantum transport in the ideal case,pioneered in [23,24] for β = 2, has been recently completedincluding the other symmetry classes [31].

A more generic situation occurs when the leads are notideal but contain tunnel barriers. A simple barrier, which doesnot mix transversal modes, is represented by a set of tunnelingprobabilities {�i}, one for each open channel. In this casethe distribution of S is given by the so-called Poisson kernel[21,32,33]

Pβ(S) = [det(1 − SS†) det(1 − SS†)]β/2−1−βN/2, (2)

which depends only on β and the average scattering matrixS (whose singular values are determined by the tunnelingprobabilities of each lead). In the limit �i → 1 we have S = 0,and one recovers the ideal case. Even though controllablebarriers in the leads are by now an established experimentalprotocol [34–37], few explicit theoretical predictions areavailable due to the complicated nature of the Poisson kernel.For instance, the average and variance of conductance areknown perturbatively in the limit �iN � 1 for all i [15,38,39].In this same limit the shot noise has also been obtained [40,41];systems with very small leads were considered in [42,43].Semiclassical studies of transport in the large-N limit andnonideal setting have recently appeared [44–46].

A more systematic RMT theoretical investigation wasinitiated when Vidal and Kanzieper [47] obtained the jpdof reflection eigenvalues for β = 2 and only one nonideallead. In this work we characterize those N1 nonideal channelsby a diagonal matrix γ = diag(γ1, . . . ,γN1 ), with γi = 1 − �i

(these are not the same γi which appear in [47]; the definitionsdiffer by a square root). The other lead is kept ideal. Our goalis to use symmetric function expansions and a generalizedSelberg integral to develop a systematic perturbation theoryin γ of this problem. In this framework we present explicitformulas for the most useful transport statistics. In contrast tomost previous results, ours are valid for an arbitrary numberof channels in the two leads; that is, they are not restricted tothe large-N limit.

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RODRIGUEZ-PEREZ, MARINO, NOVAES, AND VIVO PHYSICAL REVIEW E 88, 052912 (2013)

II. PERTURBATIVE γ EXPANSION

Let (a)n = a(a + 1) · · · (a + n − 1) be the rising factorialand let

2F1(a,b; c; x) =∑n�0

(a)n(b)n(c)nn!

xn (3)

be the hypergeometric function. Let F be the N1 × N1 matrixwhose elements are

Fij = 2F1(N2 + 1,N2 + 1; 1; γiRj ). (4)

When the nonideal lead supports N1 channels, the jpd ofreflection eigenvalues is given by [47]

P(γ )2 (R) = Z detN (1 − γ )

�(R)

�(γ )det(F)

N1∏i=1

(1 − Ri)N2−N1 ,

(5)

where Z is a normalization constant,

Z = N !

N1!N2!

N1∏i=1

(N2!)2

(N2 + i)!(N2 − i)!. (6)

Expression (5) is hardly operational. We therefore start bywriting it in a perturbative way, i.e., as an infinite series in γ .

Let a nonincreasing sequence of positive integers λ1,λ2, . . .

be called a partition of n if∑

i λi = n, and let this be denotedby λ � n. The number of parts in λ is �(λ), and we set λm = 0if m > �(λ). Partitions can be used to label a very importantset of symmetric polynomials known as Schur polynomials,which are denoted by sλ. Assuming N1 variables, they aredefined by

sλ(x) = 1

�(x)det

(x

λi−i+N1j

). (7)

For example, the first few such polynomials are given by

s0(x) = 1, s1(x) =N1∑i=1

xi, (8)

s11(x) =N1∑i<j

xixj , s2(x) = s11(x) +N1∑i=1

x2i . (9)

If we define

αλ =N1∏i=1

(N + λi − i

N2

)2

, (10)

the following expansion can be established:

det(F) = �(γ )�(R)∑

λ

αλsλ(γ )sλ(R), (11)

where the infinite sum is over all possible partitions. Thisfollows from the nice structure of Fij , which depends on theindices ij only through the combination γiRj . An account ofthis and similar identities can be found, for example, in thebook by Hua [48].

In order to use (11) to express the jpd of reflectioneigenvalues, it is useful to factor out the α0 term and noticethat

αλ

α0= [N ]2

λ

[N1]2λ

, (12)

where

[N ]λ =�(λ)∏i=1

(N + λi − i)!

(N − i)!(13)

is a generalization of the rising factorial. The normalizationconstant then simplifies as

Z′ = Z α0 =N1∏i=1

(N − i)!

(N1 − i)!(N2 − i)!i!. (14)

This is precisely the normalization constant missing from (1).Finally, combining (1), (5), (11), and (12), we get the finalresult,

P(γ )2 (R)

P(0)2 (R)

= Z′detN (1 − γ )∑

λ

[N ]2λ

[N1]2λ

sλ(γ )sλ(R). (15)

III. COMPUTING OBSERVABLES

Since any observable is a symmetric function of thereflection eigenvalues, it must be expressible as a linearcombination of Schur polynomials; hence it suffices to obtainthe average value of sμ(R) for an arbitrary partition μ. In thisway we are led to consider the multiple integral

∫ 1

0�2(R)sλ(R)sμ(R)

N1∏i=1

(1 − Ri)N2−N1dR (16)

(where dR ≡ ∏j dRj ), which is a generalization of Selberg’s

integral [49]. However, this is difficult to evaluate directly.One way to proceed is to express the product of twoSchur polynomials again as a linear combination of Schurpolynomials,

sλ(R)sμ(R) =∑

ν

Cνλμsν(R), (17)

where the constants Cνλμ are known as Littlewood-Richardson

coefficients [50]. There is no explicit formula for them, butthey can be computed using some recursive algorithms, andthere are tables for the first ones. For instance, the coefficientswith ν up to 4 are given by

s0sλ = sλ, s1s1 = s2 + s11,

s2s1 = s3 + s21, s11s1 = s111 + s21,

s3s1 = s4 + s31, s21s1 = s31 + s22 + s211,

s2s2 = s4 + s31 + s22, s2s11 = s31 + s211,

s111s1 = s1111 + s211, s11s11 = s1111 + s211 + s22.

By means of the Littlewood-Richardson coefficients, weonly need to consider the simpler integral

Iν =∫ 1

0�2(R)sν(R)

N1∏i=1

(1 − Ri)N2−N1dR, (18)

which is known to be given by [51,52]

Iν = sν(1N1 )N1∏i=1

i!(N1 + νi − i)!(N2 − i)!

(N + νi − i)!, (19)

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STATISTICS OF QUANTUM TRANSPORT IN WEAKLY . . . PHYSICAL REVIEW E 88, 052912 (2013)

where sν(1N1 ) is the value of a Schur polynomial when all itsarguments are equal to unity.

If we combine the above result with Z′, we get a substantialsimplification, which is manifestly a rational function of N1

and N2; that is, the variable N1 no longer appears as the limitto products. The result is

Z′Iν = [N1]2νχν(1)

[N ]ν |ν|! , (20)

where ν � |ν| and χ is the character function in the permuta-tion group, so χν(1) is the dimension of the irreducible repre-sentation of that group associated with partition ν [to arrive atthis result we have used that sν(1N1 ) = χν(1)[N1]ν/n!].

The final result is that the average value of sμ(R), withrespect to the jpd (15), is given by

〈sμ(R)〉γ = detN (1 − γ )∑

λ

Dμλsλ(γ ), (21)

with

Dμλ = [N ]2λ

[N1]2λ

∑ν

Cνλ,μ

[N1]2νχν(1)

[N ]ν |ν|! . (22)

IV. THE LEADING ORDER

The jpd (15) equals the jpd of the ideal case (1) times acorrection which can be systematically expanded in powersof γ . In this way any observable in the finite-γ regime canbe expressed in terms of observables computed in the idealregime. For example, to leading order we have

P(γ )2 (R)

P(0)2 (R)

∝[

1 + N

N1

(N

N1s1(R) − N1

)Trγ

]. (23)

As a first application, let 〈Gn〉γ be the average value of thenth moment of the conductance in the nonideal case. Using(23) and the fact that s1(R) = N1 − G, it is easy to see thatthe difference between the weakly nonideal case and the idealcase is given to leading order by

〈Gn〉γ − 〈Gn〉0 ≈ N

N1Trγ

[N2〈Gn〉0 − N

N1〈Gn+1〉0

]. (24)

A similar estimate holds for other transport statistics.

V. STATISTICS OF CONDUCTANCEUP TO SECOND ORDER

Using the approach presented here, the average value ofany observable can, in principle, be found to any order in γ .Consider, for instance, the average conductance. In the idealcase, it is given by 〈G〉0 = N1N2/N . Up to second order in γ ,the calculations we just outlined provide

〈G〉γ ≈ N1N2

N− N2

2 Trγ

N2 − 1+ N2

2 [2Trγ 2 − N (Trγ )2]

(N2 − 1)(N2 − 4). (25)

We stress that this result is exact for any number of channelsN1 and N2. It is interesting to check that it is compatible withthe available results in the literature [38], which are, on thecontrary, exact in γ but perturbative in N1,N2. Taking intoaccount that traces are of order N1, we see that each term in(25) scales linearly with N . For example, when N1 = N2, wehave

limN1=N2→∞

〈G〉γN1

≈ 1

2− trγ

4− (trγ )2

8, (26)

where tr is the normalized trace,

trX = limN1→∞

1

N1TrX. (27)

The limiting law (26) perfectly matches the result by Brouwerand Beenakker (Eq. 6.17 in [38]), which reads

〈G〉(BB)γ ≈ g1g

′1

g1 + g′1

+ O(N−1), (28)

where in our notation g1 = N1 − Trγ and g′1 = N2. Comput-

ing the same limit as in (26), we get

limN1=N2→∞

〈G〉(BB)γ

N1≈ 1 − trγ

2 − trγ, (29)

whose expansion in γ up to the second order precisely re-produces (26). Notice that the average conductance decreaseswith γ , as should be expected.

The variance of conductance, on the other hand, is given inthe ideal case by

var0G = N21 N2

2

N2(N2 − 1). (30)

In the nonideal case, up to second order in γ , it becomes

varγ (G) ≈ N21 N2

2

N2(N2 − 1)+ 2 N2

2 (N1 − N2)2Trγ

N (N2 − 1)(N2 − 4)

+ N22 [A1N (Trγ )2 + 2B1(N2 − 1)Trγ 2]

N (N2 − 1)2(N2 − 4)(N2 − 9), (31)

where

A1 = (N1 − 2N2)(3N1 − 8N2)N2 + 20N1N2 − 37N22 − 3,

(32)

and

B1 = (N1 − 2N2)(N2N2 + N2 − 5N1). (33)

Again, the result is exact for any N1,N2. Notice that each orderin γ attains a finite value in the large N limit. For instance,when N1 = N2, the first order vanishes identically, and wehave

limN1=N2→∞

varγ (G) ≈ 1

16+ 5(trγ )2 − 4trγ 2

64, (34)

which perfectly matches the first terms of the expansion in γ

of formula 6.24 in [38], where in our notation g1 = N1 − Trγ ,g2 = N1 − 2Trγ + Trγ 2, g3 = N1 − 3Trγ + 3Trγ 2 − Trγ 3,and g′

p = N2 for all p.As a further check, we consider the case N1 = 1, for which

the full density of conductance is known [47] in terms of a

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RODRIGUEZ-PEREZ, MARINO, NOVAES, AND VIVO PHYSICAL REVIEW E 88, 052912 (2013)

FIG. 1. Average and variance of conductance as functions of γ forN1 = 1 and N2 = 5. Solid and dashed lines are, respectively, exactresults and our approximations. For the average, the difference isminimal. For the variance, the approximation predicts a nonphysicalnegative result at high γ but is excellent up to moderate values of γ .

single scalar opacity parameter γ ,

fγ (G) = N2GN2−1φγ (G), (35)

where

φγ (G) = (1 − γ )N2+12F1(N2 + 1,N2 + 1; 1; γ (1 − G)).

(36)

The average conductance (and similarly for the variance)is given by the integral 〈G〉γ = ∫ 1

0 dG Gfγ (G). Expandingφγ (G) up to second order in γ and computing the integralorder by order, we obtain

〈G〉γ ≈ N2

1 + N2− N2

2 + N2γ − N2

(2 + N2)(3 + N2)γ 2, (37)

in full agreement with (25) with N1 = 1.In Fig. 1 we plot the average and variance of conductance as

functions of γ when N1 = 1 and N2 = 5, comparing the exactintegration of formula (35) and our approximate expansions(25) and (31). The approximation is excellent for the average,while for the variance the quality deteriorates for γ close to 1.

VI. AVERAGE SHOT NOISE UP TO SECOND ORDER

Another important quantity that can be measured in thetransport context is shot noise [53]. This is related tofluctuations of the electric current as a time series. Since it isevaluated at zero temperature, it is of quantum nature, arisingfrom the granularity of electric charge. In terms of reflection

FIG. 2. Average shot noise as a function of γ for N1 = 1 andN2 = 5. Solid and dashed lines are, respectively, the exact result andour approximation.

eigenvalues, shot noise is given by

p(R) =N1∑i=1

Ri(1 − Ri) = s1(R) − s2(R) + s11(R). (38)

Its average value is known for arbitrary channel numbers inthe ideal case [16]. Using the present approach, we include γ

effects up to second order. The result is

〈p〉γ ≈ N21 N2

2

N (N2 − 1)+ N2

2 (N1 − N2)2Trγ

(N2 − 1)(N2 − 4)

+ N22 [A2(Trγ )2 − B2Trγ 2]

(N2 − 1)(N2 − 4)(N2 − 9), (39)

where

A2 = N(5N2

2 − 4N1N2 + N21

) + N1 − 5N2 (40)

and

B2 = N2N22 + 4N2

2 − 14N1N2 + 3N21 + 3. (41)

This is in agreement with the result available for large channelnumbers [40], which is exact in γ . The asymptotic limit withN1 = N2 can easily be obtained as

limN1=N2→∞

〈p〉γN1

≈ 1

8+ (trγ )2 − trγ 2

16. (42)

We compare approximation (39) against the exact result forN1 = 1 and N2 = 5 in Fig. 2 [the exact result is obtained bynumerical integration of G(1 − G) times the density (35)]. Theapproximation is not able to account for the fact that the noisevanishes at γ = 1 (since all particles are surely reflected), butit can be very good for moderate γ .

VII. CONCLUSION

In summary, combining the theory of symmetric functionsand generalized Selberg integrals, we presented a systematicperturbation theory in the opacity matrix γ for the jpd ofreflection eigenvalues in chaotic cavities with β = 2 andsupporting one ideal and one nonideal lead. This jpd is foundto be given by the standard Jacobi ensemble (1), valid forthe ideal case, times a correction that can be systematicallyexpanded in γ [see (15)]. Using this result, we computed the

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STATISTICS OF QUANTUM TRANSPORT IN WEAKLY . . . PHYSICAL REVIEW E 88, 052912 (2013)

average and variance of conductance, as well as average shotnoise, up to the second order in γ and moments of conductanceto leading order.

Our results are valid for arbitrary N1,N2, in contrastto previously available results which are exact in γ butperturbative in N1,N2 and often limited to the leading-orderterm as N → ∞. Comparison with numerics for N1 = 1showed that our perturbative expressions are generally ratheraccurate for moderate γ and have the advantage of a completeanalytical tractability.

Naturally, it would be interesting to extend this calculationto higher orders in γ . However, the expressions become quite

cumbersome. This may be related to the asymmetric roleof the parameters N1 and N2. Therefore it would be evenmore desirable to be able to consider both leads as nonideal.Extension to other symmetry classes is another challengingopen problem.

ACKNOWLEDGMENTS

This work has been partly supported by Grant No.2011/07362-0, Sao Paulo Research Foundation (FAPESP), andby the project Labex PALM-RANDMAT.

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