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Communications in Theoretical Physics

Sensitivity of energy eigenstates to perturbation in quantum integrable and chaotic systems

Zao Xu, Yin-Chenguang Lyu, Jiaozi Wang, Wen-ge Wang

Published Jan 1, 2021• 2 citations
Publisher

Abstract

Abstract We study the sensitivity of energy eigenstates to small perturbation in quantum integrable and chaotic systems. It is shown that the distribution of rescaled components of perturbed states in unperturbed basis exhibits qualitative difference in these two types of systems: being close to the Gaussian form in quantum chaotic systems, while, far from the Gaussian form in integrable systems.

DOI
10.1088/1572-9494/abc7b0

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Abstract We study the sensitivity of energy eigenstates to small perturbation in quantum integrable and chaotic systems. It is shown that the distribution of rescaled components of perturbed states in unperturbed basis exhibits qualitative difference in these two types of systems: being close to the Gaussian form in quantum chaotic systems, while, far from the Gaussian form in integrable systems.
Numerical results show that, for quantum autonomous chaotic system, the evolution of initially coherent states are sensitive to perturbation. The overlap of a perturbed state with the unperturbed one decays exponentially, which is followed by fluctuation around N-1, N being the dimension of the Hilbert space. The matrix elements of the evolution operator in interaction picture tend to be a random distribution after sufficiently long time, where the interaction is the perturbation, even when the perturbation is very weak. The difference between a regular system and the chaotic one is shown clearly. In a regular system, the overlap shows strong revival. The distribution of the evolution matrix has only a few dominant terms.
In this paper, a quantity that describes a response of a system's eigenstates to a very small perturbation of physical relevance is studied as a measure for characterizing crossover from integrable to chaotic quantum systems. It is computed from the distribution of very small, rescaled components of perturbed eigenfunctions on the unperturbed basis. Physically, it gives a relative measure to prohibition of level transitions induced by the perturbation. Making use of this measure, numerical simulations in the so-called Lipkin-Meshkov-Glick model show in a clear way that the whole integrability-chaos transition region is divided into three subregions: a nearly integrable regime, a nearly chaotic regime, and a crossover regime.
In the past decades, it was recognized that quantum chaos, which is essential for the emergence of statistical mechanics and thermodynamics, manifests itself in the effective description of the eigenstates of chaotic Hamiltonians through random matrix ensembles and the eigenstate thermalization hypothesis.Standard measures of chaos in quantum many-body systems are level statistics and the spectral form factor.In this work, we show that the norm of the adiabatic gauge potential, the generator of adiabatic deformations between eigenstates, serves as a much more sensitive measure of quantum chaos.We are able to detect transitions from integrable to chaotic behavior at perturbation strengths orders of magnitude smaller than those required for standard measures.Using this alternative probe in two generic classes of spin chains, we show that the chaotic threshold decreases exponentially with system size and that one can immediately detect integrability-breaking (chaotic) perturbations by analyzing infinitesimal perturbations even at the integrable point.In some cases, small integrability breaking is shown to lead to anomalously slow relaxation of the system, exponentially long in system size.
We compare the influence of perturbations of classically regular respectively chaotic Hamiltonians on their eigenfunctions. A generic measure of the perturbation strength is given by the average spreading width, which semiclassically is a phase-space integral. Thus, contrary to common assumption, the spreading width cannot be an indicator of regularity or chaos. The distribution of expansion coefficients of perturbed eigenfunctions in terms of unperturbed ones is markedly different for the two cases and may serve as a quantum signature of chaos referring to states rather than to spectra.
Abstract We study properties of eigenfunctions of perturbed systems, given on the eigenbases of unperturbed, integrable systems. For a given pair of perturbed and unperturbed systems, with respect to the energy of each perturbed state, the unperturbed basis states can be divided into two groups: one in the classically-allowed region and the other in the classically-forbidden region; correspondingly, the eigenfunction of the perturbed state is also divided into two parts. In the semiclassical limit, it is shown that, making use of components of the eigenfunction in its classically-allowed region, its components in the classically-forbidden region can be written in the form of a convergent perturbation expansion, which is valid for all perturbation strengths.
We consider the properties of an observable (such as a single spin component that squares to the identity) when expressed as a matrix in the basis of energy eigenstates, and then truncated to a microcanonical slice of energies of varying width. For a quantum chaotic system, we model the unitary or orthogonal matrix that relates the spin basis to the energy basis as a random matrix selected from the appropriate Haar measure. We find that the spectrum of eigenvalues is given by a centered Jacobi distribution that approaches the Wigner semicircle of a random hermitian matrix for small slices. For slices that contain more than half the states, there is a set of eigenvalues of exactly ±1\pm 1±1. The transition to this qualitatively different behavior at half size is similar to that seen in other quantities such as entanglement entropy. Our results serve as a benchmark model for numerical calculations in realistic physical systems.