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arXiv:nlin/0608059 · Nonlinearity

Semiclassical expansion of parametric correlation functions of the quantum time delay

Jack Kuipers, Martin Sieber

Published Mar 6, 2007• 22 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/0951-7715/20/4/006

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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)K(τ,x,y,M)K(τ,x,y,M), that depends on the number of open channels MMM, a non-symmetry breaking parameter xxx, and a symmetry breaking parameter yyy. Several terms in the Taylor expansion about τ=0τ=0τ=0, which depend on all parameters, are shown to be identical to those obtained from Random Matrix Theory.
The Wigner delay time is addressed semiclassically using Miller's S -matrix expressed in terms of open orbits. This leads to a very appealing expression, in terms of classical paths, for the energy averaged Wigner time delay in chaotic scattering. The same approach also puts in evidence the semiclassical incapability to correctly assess the time delay higher moments. This limitation suggests that the use of the semiclassical approximation to quantify fluctuations in scattering phenomena, like in mesoscopic physics, has to be considered with great caution.
Motivated by the surge of interest in statistics of non-Hermitian random matrices as a framework for description of universal characteristics of dissipative chaotic quantum many-body systems, we address the problem of characterizing the parametric correlations of spectral densities. Considering parameter-dependent ensemble of complex Ginibre matrices we derive an explicit, closed-form expression for the parametric number covariance in the systems of symmetry class A\mathbf{A}A for eigenvalues in a circular domain containing on average a finite number of eigenvalues in the spectral bulk. This behavior is expected to be universal, as further supported by numerical evidence for the real Ginibre ensemble, non-Hermitian Bernoulli Wigner matrices and bi-unitarily invariant ensembles. We also discuss a relation between parametric correlations of spectral densities and the distribution of the so-called eigenvector non-orthogonality factor, which attracted considerable interest in recent years.
We consider the Wigner--Smith time-delay matrix of a chaotic cavity with broken time-reversal symmetry, a finite number MMM of equivalent open channels, and a uniform tunnel barrier of reflection probability RRR. 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≤Mn\le Mn≤M. The exact ratio formula decomposes into a sector visible to the semiclassical expansion and an additional contribution proportional to RM+1R^{M+1}RM+1, which is invisible to every algebraic order in 1/M1/M1/M. For the second moment, this contribution exactly reproduces the exponentially small term in the known random-matrix expression for ⟨τW2⟩\langleτ_W^2\rangle⟨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×MM\times MM×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.
A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT).Most prominent features of such RMT behavior with respect to a random spectrum, both encompassed in spectral pair correlation function, are statistical suppression of small level spacings (correlation hole) and enhanced stiffness of the spectrum at large spectral ranges.For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry [Proc.R. Soc.London, A400, 229 (1985)] within the so-called diagonal approximation of semiclassical periodic-orbit sums, while the derivation of the full RMT spectral form factor K(t) (Fourier transform of spectral pair correlation function), from semiclassics has been completed by Müller et al. [Phys.Rev. Lett.93, 014103 (2004)].In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains.Such systems display two universal types of behaviour which are termed as the 'many-body localized phase' and 'ergodic phase'.In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder.Here we provide a clear theoretical explanation for these observations.We compute K(t) in the leading two orders in t and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field.In particular, we relate K(t) to partition functions of a class of twisted classical Ising models on a ring of size t, hence the leading order RMT behavior K(t)2t is a consequence of translation and reflection symmetry of the Ising partition function.
The fundamental correspondence between quantum chaotic single-particle systems and random matrix theory is well-understood via periodic orbit theory. In contrast, we show that many-body systems with explicit subsystem structure possess characteristics different from the single-particle theory. We present a periodic orbit theory for many-body systems with well defined semiclassical limit. For this we identify periodic orbit families arising exclusively in the many-body setting and implement a central limit theorem characterizing their correlations. Based on this we demonstrate that spectral correlations in chaotic quantum circuits are characterized by the breaking of individual time translation invariance of periodic orbits in the subsystems into residual synchronous time translations only. This provides a systematic approach to confirming random matrix universality in deterministic many-body systems.
Semiclassical spectra weighted with products of diagonal matrix elements of operators ${A}_{\ensuremath{\alpha}},$ i.e., ${g}_{\ensuremath{\alpha}{\ensuremath{\alpha}}^{\ensuremath{'}}}(E)={\ensuremath{\sum}}_{n}〈n|{A}_{\ensuremath{\alpha}}|n〉〈n|{A}_{{\ensuremath{\alpha}}^{\ensuremath{'}}}|n〉/(E\ensuremath{-}{E}_{n}),$ are obtained by harmonic inversion of a cross-correlation signal constructed of classical periodic orbits. The method provides highly resolved semiclassical spectra even in situations of nearly degenerate states, and opens the way to reducing the required signal lengths to shorter than the Heisenberg time. This implies a significant reduction of the number of orbits required for periodic orbit quantization by harmonic inversion.