Approximating the long time average of the density operator: Diagonal ensemble

TL;DR

This paper introduces a tensor network-based method to approximate the long-time average of the density operator in isolated quantum systems, avoiding real-time simulation and demonstrating convergence to thermal values.

Contribution

It adapts a filtering scheme to efficiently approximate the diagonal ensemble, providing a new tool for studying long-time quantum dynamics without full time evolution.

Findings

Local observables converge polynomially to thermal values

Method performs well on non-integrable spin chains

Avoids computationally intensive real-time evolution

Abstract

For an isolated generic quantum system out of equilibrium, the long time average of observables is given by the diagonal ensemble, i.e. the mixed state with the same probability for energy eigenstates as the initial state but without coherences between different energies. In this work we present a method to approximate the diagonal ensemble using tensor networks. Instead of simulating the real time evolution, we adapt a filtering scheme introduced earlier in [Phys. Rev. B 101, 144305 (2020)] to this problem. We analyze the performance of the method on a non-integrable spin chain, for which we observe that local observables converge towards thermal values polynomially with the inverse width of the filter.