Semiclassical field theory approach to quantum chaos

Physical review. A, 1989

A combination of semiclassical arguments and random-matrix theory is used to analyze transition strengths in quantum systems whose associated classical systems are chaotic. The mean behavior is found semiclassically while the local fluctuations are characterized by a Porter-Thomas distribution.

2002

We study the statistics of wave functions in a ballistic chaotic system. The statistical ensemble is generated by adding weak smooth random potential, which allows us to apply the ballistic σ-model approach. We analyze conditions of applicability of the σ-model, emphasizing the role played by the single-particle mean free path and the Lyapunov exponent due to the random potential. In particular, we present a resolution of the puzzle of repetitions of periodic orbits counted differently by the σ-model and by the trace formula.

Nuclear Physics B, 1993

We demonstrate a striking connection between the universal parametric correlations of the spectra of quantum chaotic systems and a class of integrable quantum hamiltonians. We begin by deriving a non-perturbative expression for the universal rn-point correlation function of the spectra of random matrix ensembles in terms of a non-linear supermatrix a-model. These results are shown to coincide with those from previous studies of weakly disordered metallic systems. We then introduce a continuous matrix model which describes the quantum mechanics of the Sutherland hamiltonian describing particles interacting through an inverse-square pairwise potential. We demonstrate that a field theoretic approach can be employed to determine exact analytical expressions for correlations of the quantum hamiltonian. The results, which are expressed in terms of a non-linear a-model, are shown to coincide with those for analogous correlation functions of random matrix ensembles after an appropriate change of variables. We also discuss possible generalizations of the matrix model to higher dimensions. These results reveal a common mathematical structure which underlies branches of theoretical physics ranging from continuous matrix models to strongly interacting quantum hamiltonians, and universalities in the spectra of quantum chaotic systems. 0550-3213/93/s 06.00

Zeitschrift f�r Physik B Condensed Matter, 1994

We calculate the k-level correlation function for the eigenphases of the N-dimensional Floquet operator of kicked dynamics whose classical counterpart is chaotic in the limit N ~ c~ by applying standard equilibrium statistical mechanics to a fictitious N-particle system which "is constructed from the eigenvalue equation of the Floquet operator and show that they become equal to those of random matrices from the circular ensemble associated with the appropriate universality class.

This thesis provides a comprehensive exploration of quantum chaos theory, a field dedicated to understanding the quantum mechanical behavior of systems that exhibit chaotic dynamics in their classical limit. Unlike classical chaos, characterized by exponential sensitivity to initial conditions and continuous energy spectra, quantum systems are governed by the linear Schrödinger equation and possess discrete energy spectra in bounded systems. This fundamental difference necessitates a distinct approach to defining and identifying quantum chaos, focusing on statistical signatures rather than direct trajectory-based measures. The report delves into the historical development of the field, from its classical roots to the pioneering quantum insights that shaped its trajectory. It meticulously examines the theoretical frameworks, including Random Matrix Theory and semiclassical approximations like the Gutzwiller Trace Formula, which serve as crucial bridges between classical and quantum descriptions. Key quantum signatures such as spectral statistics (Wigner-Dyson and Poisson distributions), quantum scarring, quantum ergodicity, and information scrambling (probed by Out-of-Time-Ordered Correlators) are discussed in detail, along with their experimental observations across diverse physical systems, from atomic and nuclear physics to condensed matter and quantum optics. The thesis also explores the profound applications of quantum chaos in emerging technologies like quantum computing and its deep connections to fundamental physics, including black hole dynamics and quantum gravity. Finally, it addresses the persistent challenges and outlines promising future directions, emphasizing the ongoing quest for a rigorous definition, the impact of noise in open quantum systems, and the burgeoning field of many-body quantum chaos, all of which continue to shape the frontiers of this fascinating interdisciplinary domain.

Physical Review E, 2005

We argue semiclassically, on the basis of Gutzwiller's periodic-orbit theory, that full classical chaos is paralleled by quantum energy spectra with universal spectral statistics, in agreement with random-matrix theory. For dynamics from all three Wigner-Dyson symmetry classes, we calculate the small-time spectral form factor K(τ ) as power series in the time τ . Each term τ n of that series is provided by specific families of pairs of periodic orbits. The contributing pairs are classified in terms of close self-encounters in phase space. The frequency of occurrence of self-encounters is calculated by invoking ergodicity. Combinatorial rules for building pairs involve non-trivial properties of permutations. We show our series to be equivalent to perturbative implementations of the non-linear sigma models for the Wigner-Dyson ensembles of random matrices and for disordered systems; our families of orbit pairs are one-to-one with Feynman diagrams known from the sigma model.

Journal of Physics A: Mathematical and General, 1997

The autocorrelation function of spectral determinants is proposed as a convenient tool for the characterization of spectral statistics in general, and for the study of the intimate link between quantum chaos and random matrix theory, in particular. For this purpose, the correlation functions of spectral determinants are evaluated for various random matrix ensembles, and are compared with the corresponding semiclassical expressions. The method is demonstrated by applying it to the spectra of the quantized Sinai billiards in two and three dimensions. 03.65.Sq, 05.45.+b Typeset using REVT E X † Present Address:

There is a newly emerging understanding that in the chaotic domain of isolated finite interacting many particle systems smoothed densities define the statistical description of these systems and these densities follow from embedded (two-body) random matrix ensembles and their various deformations. These ensembles predict that the smoothed form of matrix elements of a transition operator between the chaotic eigenstates weighted by the densities at the two ends (i.e. the bivariate strength density) will be a bivariate Gaussian with the bivariate correlation coefficient arising out of the non-commutability of the hamiltonian and the transition operator involved. The ensemble theory extends to systems with a mean-field and a chaos generating two-body interaction (as in nuclei, atoms and diffusive quantum dots). These developments in many-body quantum chaos are described with special reference to one-body transition operators.

Physical Review E, 2007

We study the level statistics of an interacting multi-qubit system, namely the kicked Ising spin chain, in the regime of quantum chaos. Long range quasi-energy level statistics show effects analogous to the ones observed in semi-classical systems due to the presence of classical periodic orbits, while short range level statistics display perfect statistical agreement with random matrix theory. Even though our system possesses no classical limit, our result suggest existence of an important non-universal system specific behavior at short time scale, which clearly goes beyond finite size effects in random matrix theory.

The concept of structural invariance previously introduced by the authors is used to argue that the connection between random matrix theory and quantum systems with a chaotic classical counterpart is in fact largely exact in the semiclassical limit, holding for all correlation functions and all energy ranges. This goes considerably further than the usual results obtained through periodic orbit theory.These results hold for eigenvalues of bounded time-independent systems as well as for eigenphases of periodically kicked systems and scattering systems.