We consider the Wigner--Smith time-delay matrix of a chaotic cavity with broken time-reversal symmetry, a finite number $M$ of equivalent open channels, and a uniform tunnel barrier of reflection probability $R$. Exact random-matrix results for nonideal coupling are known at the level of the joint matrix distribution, whereas explicit finite-channel formulas for Schur moments are much less developed. We show that every Schur moment of the nonideal Wigner--Smith matrix factorizes into its ideal-coupling inverse-Laguerre value and a barrier factor given by a circular unitary ensemble (CUE) average of a Schur polynomial. A dual-Cauchy generating function reduces the latter to a Toeplitz determinant with a rational symbol, placing the barrier problem within the classical theory of exact CUE characteristic-polynomial ratios.
For one-row partitions, specializing the exact CUE ratio formula gives a closed generating function for all barrier coefficients and therefore all finite delay-time moments in the allowed range $n\le M$. The exact ratio formula decomposes into a sector visible to the semiclassical expansion and an additional contribution proportional to $R^{M+1}$, which is invisible to every algebraic order in $1/M$. For the second moment, this contribution exactly reproduces the exponentially small term in the known random-matrix expression for $\langleτ_W^2\rangle$ that is absent from the semiclassical expansion. We also derive an exact finite sum for all one-column partitions and an exact complement symmetry for Schur averages inside the $M\times M$ square. The new contribution therefore lies not in the underlying CUE ratio identities, which are classical, but in combining them with the exact nonideal Wigner--Smith mapping and extracting the resulting finite-channel time-delay formulas and corrections to semiclassical Schur-moment conjectures.
We consider the Wigner--Smith time-delay matrix of a chaotic cavity with broken time-reversal symmetry, a finite number M of equivalent open channels, and a uniform tunnel barrier of reflection probability R. Exact random-matrix results for nonideal coupling are known at the level of the joint matrix distribution, whereas explicit finite-channel formulas for Schur moments are much less developed. We show that every Schur moment of the nonideal Wigner--Smith matrix factorizes into its ideal-coupling inverse-Laguerre value and a barrier factor given by a circular unitary ensemble (CUE) average of a Schur polynomial. A dual-Cauchy generating function reduces the latter to a Toeplitz determinant with a rational symbol, placing the barrier problem within the classical theory of exact CUE characteristic-polynomial ratios.
For one-row partitions, specializing the exact CUE ratio formula gives a closed generating function for all barrier coefficients and therefore all finite delay-time moments in the allowed range n≤M. The exact ratio formula decomposes into a sector visible to the semiclassical expansion and an additional contribution proportional to RM+1, which is invisible to every algebraic order in 1/M. For the second moment, this contribution exactly reproduces the exponentially small term in the known random-matrix expression for ⟨τW2⟩ that is absent from the semiclassical expansion. We also derive an exact finite sum for all one-column partitions and an exact complement symmetry for Schur averages inside the M×M square. The new contribution therefore lies not in the underlying CUE ratio identities, which are classical, but in combining them with the exact nonideal Wigner--Smith mapping and extracting the resulting finite-channel time-delay formulas and corrections to semiclassical Schur-moment conjectures.
Using a semiclassical approach formulated in terms of a matrix integral, we consider the description of quantum chaotic transport when a tunnel barrier is present in one of the leads. For this description, we obtain the transport moments as a power series in the reflection probability of the barrier, whose coefficients are rational functions of the number of open channels M. Our results are therefore valid in the quantum regime and not only when M ≫ 1. The expressions we arrive at are not identical to the Random Matrix Theory ones corresponding to the same problem, but are in fact much simpler. Both theories agree as far as we can test.
We show the emergence of random matrix theory (RMT) spectral correlations in the chaotic phase of generic, periodically kicked, interacting, quantum many-body systems by analytically calculating the spectral form factor (SFF), K ( t ) , up to two leading orders in time, t . We explicitly consider the presence or absence of time-reversal ( T ) symmetry to investigate all three Dyson symmetry classes, namely, the circular orthogonal ensemble (COE) for T -invariant systems with T 2 = 1 , the circular unitary ensemble (CUE) in the absence of T symmetry, and the circular symplectic ensemble (CSE) for T -invariant systems with T 2 = − 1 . Our derivation applies the random phase approximation to perform ensemble averages, and we give its justification for generic many-body interactions. We express the SFF as a sum of contributions of certain diagrams. While the number of diagrams that contribute to the SFF in the first and second orders in t for the COE and CUE classes is finite, the number of such diagrams is exponentially large in t for the CSE class. Thus, we have developed a new diagrammatic technique that uses reduced diagrams to incorporate contributions from many diagrams. For the COE class, we show that beyond Thouless time t * , the SFF takes the form K ( t ) ≃ 2 t − 2 t 2 / N up to second order in time, where N is the Hilbert space dimension. For the CUE class, we show that K ( t ) ≃ t beyond t * , and there is no universal term in the second order, unlike the COE case. For the CSE class, we show that K ( t ) ≃ 2 t + 2 t 2 / N up to two orders in time beyond t * . The above findings of the SFF up to second order in time are in agreement with the RMT predictions for these classes. In all three cases, the scaling of t * with system size L is determined by the eigenvalues of a doubly stochastic matrix M . For strongly interacting fermionic chains, M is S U ( 2 ) invariant in all three cases, leading to t * ∝ L 2 in the presence of U ( 1 ) symmetry. In the absence of U ( 1 ) symmetry, we find t *
We study the complex eigenvalue statistics of the asymmetric quantum baker map with partial projective openings. The classical asymmetric baker map, with its discontinuity at q=2/3, is fully chaotic, has no reflection symmetry, and provides a clean setting with tunable escape rate and fractal repeller dimension. We consider three distinct opening geometries in position space: localized (contiguous channels), random, and uniform (equispaced channels), all controlled by a tunable amplitude reflectivity parameter ρ that interpolates between the fully open (ρ=0) and the closed (ρ=1) limits. We use the partially truncated circular unitary ensemble (PTCUE) as the random matrix theory benchmark. The main focus is on the joint distribution of the complex spacing ratio z, defined as the ratio of the distances from an eigenvalue to its nearest and next-nearest neighbors in the complex plane. We find a smooth crossover from a quasi-1D spectral regime, where eigenvalues cluster near the unit circle and the phase distribution of z is peaked, to a two-dimensional Ginibre-like regime, where the distribution becomes nearly uniform and level repulsion is fully developed. Both the number of open channels M and the reflectivity ρ modulate this crossover, and ρ provides an additional continuous control even at fixed opening size. All three opening models converge to PTCUE statistics at large M, while differences are most pronounced for the localized model at small M. No evidence of an abrupt transition is found. This crossover which suggests a universal behavior, has deep consequences for open quantum and wave-chaotic experiments.
We discuss the semiclassical approximation to transport problems in quantum chaotic systems. The figures of merit are moments of the transmission matrix and of the time delay matrix. After reviewing a few results obtained by treating these matrices are random matrices, we show how expressions for their elements in terms of sums over trajectories lead to diagrammatic formulations that correspond to perturbative calculations. This semiclassical approach agrees with random matrix theory when it should, and allows further elements to be incorporated, like tunnel barriers, superconductors, absorption effects. We also discuss how this approach can be encoded in matrix integrals, resulting in a powerful and versatile theory that is amenable to algebraic solutions.
We derive semiclassical periodic orbit expansions for a correlation function of the Wigner time delay. We consider the Fourier transform of the two-point correlation function, the form factor K(τ,x,y,M), that depends on the number of open channels M, a non-symmetry breaking parameter x, and a symmetry breaking parameter y. Several terms in the Taylor expansion about τ=0, which depend on all parameters, are shown to be identical to those obtained from Random Matrix Theory.
The complex spacing ratio is the complex displacement from a reference eigenvalue to its nearest neighbor divided by the corresponding displacement to its next-to-nearest neighbor. Its statistics provide a useful diagnostic of spectral correlations and nonintegrability in open quantum systems. Here, starting from the exact joint eigenvalue probability densities, we derive finite-N complex-spacing-ratio distributions for the Gaussian ensembles of non-Hermitian random matrices in classes AI† and AII†, realized by complex symmetric and complex self-dual random matrices, respectively. In class AII†, we obtain an exact algebraic expression for arbitrary N and explicitly evaluate the distributions and representative moments for N=3,4,5,6. In class AI†, although the joint density retains a noncompact integral over nonunitary eigenvector degrees of freedom, we analytically derive a normalized one-dimensional integral representation for N=3 and determine the asymptotic behavior, including a logarithmic correction to the cubic level repulsion and a nonanalytic contribution to the angular density. We further confirm these analytical results through direct numerical diagonalization of non-Hermitian random matrices.