A method to discriminate between localized and chaotic quantum systems

TL;DR

This paper introduces a new method using Krylov basis dynamics to distinguish localized from chaotic quantum systems, validated through models like Anderson localization and many-body spin systems.

Contribution

It develops qualitative criteria based on Krylov basis properties to identify localization versus thermalization in quantum systems, linking dynamics to spectral statistics and eigenstate properties.

Findings

Criteria successfully differentiate localized and delocalized regimes.

Delocalized systems exhibit Wigner-Dyson level statistics.

Eigenstate thermalization is linked to Krylov basis coupling strengths.

Abstract

We study whether a generic isolated quantum system initially set out of equilibrium can be considered as localized close to its initial state. Our approach considers the time evolution in the Krylov basis, which maps the dynamics onto that of a particle moving in a one-dimensional lattice where both the energy in the lattice sites and the tunneling from one lattice site to the next are inhomogeneous. By tying the dynamical propagation in the Krylov basis to that in the basis of microstates, we infer qualitative criteria to distinguish systems that remain localized close to their initial state from systems that undergo quantum thermalization. These criteria are system-dependent and involve the expectation values and standard deviations of both the coupling strengths between Krylov states and their energy. We verify their validity by inspecting two cases: Anderson localization as a function of dimension and the out-of-equilibrium dynamics of a many-body dipolar spin system. We finally investigate the Wigner surmise and the eigenstate thermalization hypothesis, which have both been proposed to characterize quantum chaotic systems. We show that when the average value of the non-diagonal terms in the Lanczos matrix is large compared to their fluctuations and to the fluctuations of the energy expectation values, which typically corresponds to delocalized quantum systems according to our criteria, there can be level repulsion of eigen-energies (also known as spectral rigidity), similar to that of the Wigner-Dyson statistics of energy levels; and we also demonstrate that in the same regime, the expectation values of typical local observables only weakly vary as a function of eigenstates, an important condition for the eigenstate thermalization hypothesis. Our demonstration assumes that, in the chaotic regime, the observable is sufficiently diagonal in the basis of Krylov states.