Eigenstate thermalization hypothesis, time operator, and extremely quick relaxation of fidelity

Abstract

The eigenstate thermalization hypothesis (ETH) states that for nonintegrable systems, each energy eigenstate accurately gives microcanonical expectation values for a class of observables. In this paper, we explore the ETH in terms of the time-energy uncertainty and an intrinsic thermal nature shared by the majority of quasi eigenstates of operationally defined 'time operator': First, we show that the energy eigenstates are superposition of uncountably many quasi eigenstates of suitably defined 'time operator'. Majority of such quasi eigenstates are thermal for thermodynamic isolated quantum manybody systems and approximately orthogonal in terms of extremely short relaxation time of the fidelity. In this manner, our scenario provides a theoretical explanation of ETH.

Variational estimation of entanglement entropy for bipartite many-body systems

“…Here, J stands for the nearest neighbor interaction strength. We use this model as its thermal nature for relatively small system size N has been studied well [10,23] both for integrable γ j = 0 and nonintegrable γ j = 0 cases. We calculated the bipartite purity and von Neumann entropy of an energy eigenstate, where we regard the left-most N s sites as part A and remaining N − N s sites as part B.…”

“…Recently, considerable attention has been paid to the bipartite entanglement in the context of the equilibration and thermalization of isolated quantum many-body systems [1][2][3][4]. In particular, for the canonical typicality [1,5] and eigenstate thermmalization hypothesis (ETH) [6][7][8][9][10][11], the bipartite entanglement well quantifies the closeness of the subsystem's density matrix to the canonical state by restricting to local observables. For a randomly sampled typical pure state, Page analytically calculated the average bipartite entanglement entropy [12].…”

Recurrence times of the small-size Lieb-Liniger Bose gas in the weak- and strong-coupling regimes

“…It is of fundamental interest to study isolated quantum systems in the view of foundations of quantum statistical mechanics, i.e., equilibration and thermalization [1][2][3], typicality [4][5][6][7], and ergodic theorems [8]. It has been experimentally established that an isolated quantum system can equilibrate with respect to local (or few-body) observables, although the quantum state remains pure even after equilibration.…”

“…It should be noted that recurrence after equilibration (or "collapse and revival") in a small system is a purely quantum phenomenon which has no classical counterpart (see Ref. [3]). Since compatibility of equilibration and recurrence is a heart of the emergence of irreversibility from reversible microscopic dynamics, the long-term dynamics of small isolated quantum systems is of fundamental interest.…”