A semiclassical matrix model for quantum chaotic transport

It is well-known that, in some regimes, random matrix theory (RMT) offers a good description of the properties of quantum systems whose classical limit displays chaotic dynamics [1]. In particular, RMT has been successfully applied to the calculation of several quantities in the context of electron transport [2], in agreement with numerical simulations and experimental results involving, for example, semiconductor quantum dots [3]. The success of RMT resides in its phenomenological character: instead of trying to predict the behavior of a specific system, it treats the quantum $\mathcal {S}$-matrix as a random variable, and computes the average value of observables in an ensemble of systems. The results are universal, in the sense that they describe generic chaotic systems. The only input is the universality class, corresponding to presence or absence of time-reversal invariance.

We have in mind a chaotic cavity, coupled to two ideal leads supporting, respectively, N₁ and N₂ open channels. Transport can be characterized by a unitary $\mathcal {S}$-matrix of dimension N₁ + N₂, or by an N₂ × N₁ transmission matrix t. The quantities ${\rm Tr}[\mathcal {T}^n]$ are called transport moments or linear statistics, where $\mathcal {T}=t^\dag t$ (the first two such moments are related to the conductance and the shot-noise power). They can be generalized to the so-called non-linear statistics which are related, for example, to conductance fluctuations. Within RMT, the $\mathcal {S}$-matrix is taken to be a random unitary matrix from one of Dyson's circular ensembles [2] (we do not consider the more general Poisson kernel [4], used when direct processes are important).

A natural question is how to derive the universal RMT predictions from a semiclassical description in terms of classical trajectories [5], which is in principle applicable to specific systems. It is now understood that the main ingredient is the constructive interference among trajectories that are almost identical (up to time-reversal), except at small regions that are called 'encounters' [6]. This permitted a diagrammatic formulation of the semiclassical theory, perturbative in the parameter (N₁ + N₂)⁻¹, that reproduced RMT results in a variety of situations [6–14] (it is even possible to go beyond RMT and take into account corrections due to the Ehrenfest time, as discussed for example in [15]).

All semiclassical works up to now have relied on explicit and lengthy calculations based either on group-theoretical manipulations involving factorizations of permutations, or on graph-theoretical manipulations involving trees and other diagrams (the state of the art can be found in [16]). These combinatorial aspects are interesting and have revealed connections to unsuspected areas. However, a more direct demonstration of the RMT-semiclassics equivalence would be desirable. In this work we offer such a demonstration, which uncovers the underlying algebraic structure behind that equivalence. For simplicity, we focus here on systems with broken time-reversal symmetry. More detailed calculations and the treatment of other symmetry classes will be presented elsewhere [17].

We proceed in two steps. First, we postulate a matrix integral which can be expressed diagrammatically, by means of Wick's rule, with exactly the same diagrammatic rules that govern the semiclassical approach. Second, we solve this integral, using the Andréief identity and the machinery of the Weingarten functions, and show that it agrees with the RMT prediction.

A non-increasing sequence of positive integers λ₁, λ₂, ... is called a partition of n if ∑_iλ_i = n and this is denoted by λ⊢n. The number of parts in λ is ℓ(λ). The functions

$$\begin{equation} p_\lambda (\mathcal {T})=\prod _{i=1}^{\ell (\lambda )}{\rm Tr}(\mathcal {T}^{\lambda _i}) \end{equation} \tag{ 1 }$$

can be used to expand a generic statistic. For systems with broken time-reversal symmetry, RMT predicts [18]

$$\begin{equation} \langle p_\lambda (\mathcal {T})\rangle =\frac{1}{n!}\sum _{\mu \vdash n}\frac{[N_1]_\mu [N_2]_\mu }{[N_1+N_2]_\mu }\chi _\mu (1^n)\chi _\mu (\lambda ), \end{equation} \tag{ 2 }$$

where the brackets denote an ensemble average, 1ⁿ represents the partition with n unit parts, χ are the characters of the permutation group S_n, and

$$\begin{equation} [N]_\mu =\prod _{i=1}^{\ell (\mu )}\frac{(N+\mu _i-i)!}{(N-i)!}. \end{equation} \tag{ 3 }$$

In the semiclassical limit, the element t_oi of the transmission matrix may be approximated [5] by a sum over trajectories starting at the incoming channel i and ending at the outgoing channel o. When we write a trace like Tr $\mathcal {T}^n$, we end up with n trajectories associated with the t (direct trajectories) and another set of n trajectories coming from the t† (partner trajectories). The structure of the trace imposes that the direct ones take i_k to o_k (labels must be equal), while the partner ones take i_k to o_{k + 1} (labels are cyclically permuted). If we have a product of traces, to each trace will correspond a cycle in the label permutation (for details, see [13]).

Systematic constructive interference survives an energy average only when each partner trajectory follows closely a direct one for a period of time, and they are allowed to exchange partners at what is called an encounter. The two sets of trajectories are thus nearly equal, differing only in the negligible encounter regions. These trajectory multiplets are represented by a diagram whose edges are the regions where the trajectories (almost) coincide and the internal vertices are the encounters. There are n external vertices of valence one on the left and on the right, corresponding to the incoming and outgoing channels, respectively. The diagrammatic rule is that each internal vertex produces −(N₁ + N₂), while each edge produces (N₁ + N₂)⁻¹. This has been discussed in detail in a number of papers [8–10, 13].

We shall define a matrix integral that mimics these diagrammatic rules. For that, we rely on Wick's rule.

Suppose the matrix integral

$$\begin{equation} \langle f(Z,Z^\dag )\rangle =\frac{1}{\mathcal {Z}}\int {\rm d}Z \,{\rm e}^{-M{\rm Tr}ZZ^\dag }f(Z,Z^\dag ), \end{equation} \tag{ 4 }$$

where $\mathcal {Z}$ is a normalization factor, M is a parameter and dZ denotes the flat measure on the space of N-dimensional matrices, such that each matrix element is independently integrated over the whole complex plane. Note that N is not related to channel numbers; its use is traditional in matrix integrals. Instead, we shall later set M = N₁ + N₂. Because of the Gaussian term, we clearly have

$$\begin{equation} \langle Z_{mj}Z^\dag _{qr}\rangle =\frac{\delta _{mr}\delta _{jq}}{M}. \end{equation} \tag{ 5 }$$

Wick's rule,

$$\begin{equation} \left\langle \prod _{k=1}^n Z_{m_kj_k}Z^\dag _{q_{k}r_k}\right\rangle =\sum _{\sigma \in S_n}\prod _{k=1}^n\big\langle Z_{m_kj_k}Z^\dag _{q_{\sigma (k)}r_{\sigma (k)}}\big\rangle =\frac{1}{M^n}\sum _{\sigma \in S_n} \prod _{k=1}^n\delta _{m_kr_{\sigma (k)}}\delta _{j_{k}q_{\sigma (k)}}, \end{equation} \tag{ 6 }$$

is a well-known result [19]. In words, it says that we must sum over all possible pairings between Z and Z†, the product of the average values of the pairs.

Wick's rule has an interesting diagrammatical representation [20–22]. Each matrix element Z_mj is represented as a pair of arrows, one associated with m and the other with j, with a marked end at the tail. In the matrix elements of Z*, the marked end is the head (see figure 1). When computing an average like 〈p_λ(ZZ†)〉, we shall represent a term like Tr(ZZ†)^q by a vertex of valence 2q with a specific internal structure, such that the incoming m_k is followed by the outgoing m_k, and the incoming j_k is followed by the outgoing j_k. The marked ends are all outside the vertex. An example is shown in figure 1. Lines associated with m labels are drawn solid, lines associated with j labels are drawn dashed.

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Our 'semiclassical matrix integral' is

$$\begin{equation} G=\frac{1}{\mathcal {Z}}\int {\rm d}Z \,{\rm e}^{-M\sum _{q\ge 1} \frac{1}{q}{\rm Tr}[(ZZ^\dag )^q]} \prod _{k=1}^n Z_{i_ko_k}Z^\dag _{o_{\pi (k)}i_k}. \end{equation} \tag{ 7 }$$

It depends on two sets of n indices, i and o, which will later be summed over. It also depends on a permutation π which corresponds to the label permutation, i.e. to the statistics we wish to compute.

We take ${\rm e}^{-M{\rm Tr}ZZ^\dag }$ as part of the integration measure and expand the remaining exponential as a power series in M. This produces all possible products of traces, each trace being accompanied by a factor −M. Each trace is represented by a vertex as in figure 1. These are the internal vertices of the semiclassical diagrams. The matrix elements Z_io ($Z^\dag _{oi}$) are the outgoing (incoming) arrows representing the outgoing (incoming) channels. These are the external vertices of the semiclassical diagrams.

We then integrate term by term using Wick's rule. All possible connections must be made among the vertices, using all marked ends of arrows, and each edge carries a factor 1/M from (5). The correct diagrammatic rule is therefore produced. We show an example in figure 2. The δ-functions in (5) imply that we can associate to each solid line the i index of its incoming channel, and to each dashed line the o index of its outgoing channel.

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The function G will depend only on the cycle type of π, which we denote by λ (this means that the cycles of π have lengths given by the parts of λ). Linear statistics correspond to full cycles, having λ = (n), while non-linear statistics correspond to more general cycle structures. In fact, our matrix integral is almost the same as $\langle p_\lambda (\mathcal {T})\rangle$, where the brackets denote an energy average. It is not exactly the same because, when we apply Wick's rule, some of the resulting diagrams may contain periodic orbits, and this is not allowed in semiclassical diagrams.

We can fix this problem with a little trick. Note that the sum over the index associated to the periodic orbit is free and produces a factor N. Hence, the contribution of a diagram with t periodic orbits will be proportional to N^t. In order to exclude periodic orbits from our semiclassical theory, we simply have to consider the part of the result that is independent of N. Equivalently, we may take the limit N → 0. This is possible since, once it is computed, G is an analytic function of N.

In order to carry out the exact solution of our model (7), we use some well-known facts about symmetric functions and characters that are summarized at the end of the paper.

Let us start with the normalization constant $\mathcal {Z}=\int {\rm d}Z \,{\rm e}^{-M{\rm Tr}ZZ^\dag }$. We introduce the singular value decomposition

$$\begin{equation} Z=UDV^\dag ,\quad DD^\dag =X={\rm diag}(x_1,\ldots ,x_N). \end{equation} \tag{ 8 }$$

The measure dZ can be written as

$$\begin{equation} {\rm d}Z=(\Delta (x))^2\,{\rm d}\vec{x}\,{\rm d}U\,{\rm d}V, \end{equation} \tag{ 9 }$$

where Δ(x) is the Vandermonde

$$\begin{equation} \Delta (x)=\prod _{1\le i<j\le N}(x_j-x_i), \end{equation} \tag{ 10 }$$

${\rm d}\vec{x}={\rm d}x_1\cdots {\rm d}x_N$ and dU, dV are Haar measures on the unitary group [23]¹. The angular integrals are trivial; the remaining integral may be performed with the help of the Andréief identity (easily proved using the Leibniz formula for the determinant),

$$\begin{equation} \int {\rm d}\vec{x}\det (f_i(x_j))\det (g_i(x_j)) = N!\det \int {\rm d}x f_i(x)g_j(x). \end{equation} \tag{ 11 }$$

The result is that

$$\begin{equation} \mathcal {Z}=\frac{N!}{M^{N^2}}\prod _{k=1}^{N-1}k!^2. \end{equation} \tag{ 12 }$$

The same change of variables turns (7) into

$$\begin{equation} \int _0^\infty \frac{{\rm d}\vec{x}}{\mathcal {Z}}(\Delta (x))^2 \det (1-X)^M A_\pi (X), \end{equation} \tag{ 13 }$$

where A_π(X) = ∫dU dVa_π(X, U, V) and

$$\begin{equation} a_\pi =\prod _{k=1}^n\sum _{j_k,m_k}U_{i_kj_k}D_{j_k}V^\dag _{j_ko_k} V_{o_{\pi (k)}m_k}D^\dag _{m_k}U^\dag _{m_ki_k}. \end{equation} \tag{ 14 }$$

We have written only one index for the diagonal matrices, with an obvious meaning. The angular integrals may be expressed in terms of the so-called Weingarten functions Wg [24–26]. We have

$$\begin{equation} A_\pi =\sum _{\sigma \tau \rho \theta \in S_n}{\rm Wg}(\rho \theta ^{-1}){\rm Wg}(\tau \sigma ^{-1})\prod _{k=1}^n\sum _{j_k,m_k}B_kD_{j_k}D^\dag _{m_k}, \end{equation} \tag{ 15 }$$

with

$$\begin{equation} B_k= \delta _{i_ki_{\sigma (k)}}\delta _{j_km_{\tau (k)}}\delta _{o_{k}o_{\rho (\pi (k))}} \delta _{m_kj_{\theta (k)}}. \end{equation} \tag{ 16 }$$

We now sum over i and o, over the left and right channels, respectively, to get

$$\begin{equation} \sum _{k=1}^n\sum _{i_k=1}^{N_1}\sum _{o_k=1}^{N_2}\delta _{i_ki_{\sigma (k)}} \delta _{o_{k}o_{\rho (\pi (k))}}=N_1^{c(\sigma )}N_2^{c(\rho \pi )}=p_\sigma (1^{N_1}) p_{\rho \pi }(1^{N_2}), \end{equation} \tag{ 17 }$$

where c( · ) denotes the number of cycles of a permutation. Also, since $\prod _k\delta _{j_km_{\tau (k)}}\delta _{m_kj_{\theta (k)}}= \prod _{k}\delta _{j_km_{\tau (k)}}\delta _{m_km_{\tau \theta (k)}}$, the sum over the j is simple, and the sum over the m gives

$$\begin{equation} \sum _{k=1}^n\sum _{m_k}\delta _{m_km_{\tau \theta (k)}}x_{m_k}=p_{\tau \theta }(x). \end{equation} \tag{ 18 }$$

The Weingarten function depends only on the conjugacy class of its argument and has the character expansion

$$\begin{equation} {\rm Wg}(\lambda )=\sum _{\mu \vdash n} w_\mu \chi _\mu (\lambda ),\quad w_\mu =\frac{1}{n!}\frac{\chi _\mu (1^n)}{[N]_\mu }. \end{equation} \tag{ 19 }$$

Using the orthogonality relations for characters then implies

$$\begin{equation} A_\pi (x)=\sum _{\mu \vdash n}\frac{[N_1]_\mu [N_2]_\mu }{([N]_\mu )^2}\chi _\mu (\pi )s_\mu (x), \end{equation} \tag{ 20 }$$

where s_μ(x) is a Schur function.

Going back to (13), the integral over the eigenvalues is

$$\begin{equation} E=\frac{1}{\mathcal {Z}}\int _0^1{\rm d}\vec{x}(\Delta (x))^2 \,{\rm det}(1-X)^Ms_\mu (x). \end{equation} \tag{ 21 }$$

The determinantal form of the Schur function, together with the Andréief identity, yields

$$\begin{equation} E=\frac{N!M!^N}{\mathcal {Z}}\,{\rm det}\left[\frac{(N+\mu _j-j+i-1)!}{(N+\mu _j-j+i+M)!}\right]. \end{equation} \tag{ 22 }$$

Standard manipulations with determinants then lead to

$$\begin{equation} E=\frac{M^{N^2}([N]_\mu )^2 \chi _\mu (1^n)}{n!} \prod _{i=1}^N\frac{(M+N-i)!}{(\mu _i-i+2N+M)!}. \end{equation} \tag{ 23 }$$

Combining (23) with (20), we get

$$\begin{equation} G=\frac{1}{n!} \sum _{\mu \vdash n}[N_1]_\mu [N_2]_\mu \chi _\mu (1^n)\chi _\mu (\pi )g(N,M,\mu ), \end{equation} \tag{ 24 }$$

where

$$\begin{equation} g(N,M,\mu )=M^{N^2}\prod _{i=1}^N\frac{(M+N-i)!}{(\mu _i-i+2N+M)!}. \end{equation} \tag{ 25 }$$

The last step is the limit N → 0, which is quite simple: lim_{N → 0}g(N, M, μ) = 1/[M]_μ. Identifying M with total number of channels, M = N₁ + N₂, we finally obtain

$$\begin{equation} \lim _{N\rightarrow 0}G(\lambda )=\frac{1}{n!}\sum _{\mu \vdash n}\frac{[N_1]_\mu [N_2]_\mu }{[N_1+N_2]_\mu }\chi _\mu (1^n)\chi _\mu (\lambda ), \end{equation} \tag{ 26 }$$

where again λ is the cycle type of π. Therefore, the RMT prediction (2) is recovered.

We have introduced a new approach to the semiclassical calculation of quantum transport properties in chaotic systems, which consists in building an auxiliary matrix integral that has the correct diagrammatic rules and can be exactly solved. As an illustration, we derived the counting statistics for broken time-reversal symmetry and arbitrary numbers of channels. Counting statistics for other symmetry classes will be reported elsewhere [17].

We believe this method will open the way to semiclassical calculations that were previously unavailable, and may even provide results beyond random matrix theory. For example, it may be adapted to treat problems where the semiclassical trajectories have different energies, as is necessary in calculations involving time delay [12] or the proximity gap in Andreev billiards [27].

We also believe that similar ideas can be applied to closed systems, allowing the calculation of all spectral correlation functions. This would provide justification for the celebrated Bohigas–Giannoni–Schmit conjecture [28] beyond the currently understood form factor [29]. In fact, for broken time-reversal symmetry and short times, this program has already been carried out [30].

I gratefully acknowledge interesting conversations with Gregory Berkolaiko and Sebastian Müller, as well as the financial support from CNPq and from grant 2012/00699-1, São Paulo Research Foundation (FAPESP).

For completeness, we present a summary of facts about symmetric functions and characters, which have been used in this paper. They can be found in many standard references.

Let x denote a set of N variables. The power sum symmetric polynomials

$$\begin{equation} p_\lambda (x)=\prod _{i=1}^{\ell (\lambda )} \sum _{j=1}^N x_j^{\lambda _i} \end{equation} \tag{ A.1 }$$

with λ⊢n form a basis for the vector space of homogeneous symmetric polynomials of degree n. It can be defined for matrix argument as in (1). Another basis for this space are Schur functions s_λ(x), which can be written as a ratio of determinants,

$$\begin{equation} s_\lambda (x)=\frac{{\rm det}\big(x_i^{\lambda _j-j+N}\big)}{\,{\rm det}\big(x_i^{j-1}\big)}=\frac{{\rm det}\big(x_i^{\lambda _j-j+N}\big)}{\Delta (x)}. \end{equation} \tag{ A.2 }$$

It is known that if all N variables in a Schur function are equal to 1, it reduces to

$$\begin{equation} s_\lambda (1^N)=\prod _{1\le i<j\le N}\frac{\lambda _i-\lambda _j-i+j}{j-i}=\frac{\chi _\lambda (1^n)[N]_\lambda }{n!}. \end{equation} \tag{ A.3 }$$

Let S_n be the group of all n! permutations of n symbols. The above families of functions are related by

$$\begin{equation} p_\lambda (x)=\sum _{\mu \vdash n}\chi _\mu (\lambda )s_\mu (x),\quad s_\lambda (x)=\sum _{\mu \vdash n}\frac{\chi _\lambda (\mu )}{z_\mu }p_\mu (x) , \end{equation} \tag{ A.4 }$$

where z_λ is the order of the centralizer of the conjugacy class in S_n containing all permutations of cycle type λ. Relation (A.4) can also be written as a sum over permutations as

$$\begin{equation} s_\lambda (x)=\frac{1}{n!}\sum _{\pi \in S_n}\chi _\lambda (\pi )p_\pi (x). \end{equation} \tag{ A.5 }$$

The quantity χ_μ(π) is the character of permutation π in the irreducible representation labeled by partition μ. It depends only on the conjugacy class of π. These quantities satisfy the orthogonality relations

$$\begin{equation} \sum _{\mu \vdash n}\chi _\mu (\lambda )\chi _\mu (\omega )=z_\lambda \delta _{\lambda ,\omega }, \quad \sum _{\lambda \vdash n}\frac{1}{z_\lambda } \chi _\mu (\lambda ) \chi _\omega (\lambda )= \delta _{\mu ,\omega }. \end{equation} \tag{ A.6 }$$

The latter is generalized as a sum over permutations as

$$\begin{equation} \sum _{\pi \in S_n}\chi _\mu (\pi ) \chi _\omega (\pi \sigma )= n!\delta _{\mu ,\omega }\frac{\chi _\mu (\sigma )}{\chi _\mu (1^n)}. \end{equation} \tag{ A.7 }$$