Quantum chaos: an introduction via chains of interacting spins-1/2

TL;DR

This paper introduces quantum chaos by examining eigenvalues and eigenstates of many-body spin-1/2 systems, highlighting spectral level repulsion as a key signature of chaos in deterministic, non-random models.

Contribution

It demonstrates how quantum chaos manifests in one-dimensional spin-1/2 chains without randomness, through spectral statistics and eigenstate analysis, providing accessible computational tools.

Findings

Eigenvalue spectra show level repulsion indicative of chaos

Eigenstate structures correlate with chaotic behavior

Deterministic spin chains can exhibit quantum chaos signatures

Abstract

We introduce aspects of quantum chaos by analyzing the eigenvalues and the eigenstates of quantum many-body systems. The properties of quantum systems whose classical counterparts are chaotic differ from those whose classical counterparts are not chaotic. The spectrum of the first exhibits repulsion of the energy levels. This is one of the main signatures of quantum chaos. We show how level repulsion develops in one-dimensional systems of interacting spins 1/2 which are devoid of random elements and involve only two-body interactions. In addition to the statistics of the eigenvalues, we analyze how the structure of the eigenstates may indicate chaos. The programs used to obtain the data are available online.