Quantum chaos at finite temperature in local spin Hamiltonians

TL;DR

This paper demonstrates that finite-temperature eigenstates in local quantum spin Hamiltonians exhibit statistical properties akin to constrained random states, revealing a form of maximal chaos at finite temperature beyond semi-classical limits.

Contribution

It introduces a framework to describe finite-temperature eigenstates using constrained random states, extending the understanding of quantum chaos beyond traditional eigensystem statistics.

Findings

Finite-temperature eigenstates match constrained random state statistics.

Excellent agreement in entanglement entropy and fluctuations in maximally chaotic regimes.

Maximal chaos notions apply at finite temperature in local quantum models.

Abstract

Understanding the emergence of chaos in many-body quantum systems away from semi-classical limits, particularly in spatially local interacting spin Hamiltonians, has been a long-standing problem. In these intrinsically quantum regimes, quantum chaos has been primarily understood through the correspondence between the eigensystem statistics of midspectrum eigenstates and the universal statistics described by random matrix theory (RMT). However, this correspondence no longer holds for finite-temperature eigenstates. Here we show that the statistical properties of finite-temperature eigenstates of quantum chaotic Hamiltonians can be accurately described by pure random states constrained by a local charge, with the average charge density of the constrained random state ensemble playing the same role as the average energy density of the eigenstates. By properly normalizing the energy density using a single Hamiltonian-dependent parameter that quantifies the typical energy per degree of freedom, we find excellent agreement between the entanglement entropy statistics of eigenstates and that of constrained random states. Interestingly, in small pockets of Hamiltonian parameter phase space which we previously identified as `maximally chaotic' [PRX 14, 031014 (2024)], we find excellent agreement not only at the level of the first moment, including O(1) corrections, but also at the level of statistical fluctuations. These results show that notions of maximal chaos -- in terms of how much randomness eigenstates contain -- can still be defined at finite temperature in physical Hamiltonian models away from semi-classical and large-$N$ limits.