Canonical density matrices from eigenstates of mixed systems

TL;DR

This paper investigates how canonical density matrices emerge in large quantum systems with mixed phase space, showing that the degree of quantum chaos controls the likelihood of thermal states in eigenstates.

Contribution

It demonstrates a tunable, universal relation between quantum chaos and the emergence of thermal states in eigenstates of finite many-body systems with mixed phase space.

Findings

Probability of canonical states increases with chaos degree

Universal relation between chaos measure and thermalization

Continuous transition from integrability to chaos

Abstract

One key issue of the foundation of statistical mechanics is the emergence of equilibrium ensembles in isolated and closed quantum systems. Recently, it was predicted that in the thermodynamic ($N\rightarrow\infty$) limit of large quantum many-body systems canonical density matrices emerge for small subsystems from almost all pure states. This notion of canonical typicality is assumed to originate from the entanglement between subsystem and environment and the resulting intrinsic quantum complexity of the many-body state. For individual eigenstates it has been shown that local observables show thermal properties provided the eigenstate thermalization hypothesis holds, which requires the system to be quantum chaotic. In the present paper, we study the emergence of thermal states in the regime of a quantum analog of a mixed phase space. Specifically, we study the emergence of the canonical density matrix of an impurity upon reduction from isolated energy eigenstates of a large but finite quantum system the impurity is embedded in. Our system can be tuned by means of a single parameter from quantum integrability to quantum chaos and corresponds in between to a system with mixed quantum phase space. We show that the probability for finding a canonical density matrix when reducing the ensemble of energy eigenstates of the finite many-body system can be quantitatively controlled and tuned by the degree of quantum chaos present. For the transition from quantum integrability to quantum chaos we find a continuous and universal (i.e. size independent) relation between the fraction of canonical eigenstates and the degree of chaoticity as measured by the Brody parameter or the Shannon entropy.