Probing Thermalization through Spectral Analysis with Matrix Product Operators.

Yilun Yang,Sofyan Iblisdir,Mari Carmen Bañuls,J Ignacio Cirac

doi:10.1103/physrevlett.124.100602

Abstract

We combine matrix product operator techniques with Chebyshev polynomial expansions and present a method that is able to explore spectral properties of quantum many-body Hamiltonians. In particular, we show how this method can be used to probe thermalization of large spin chains without explicitly simulating their time evolution, as well as to compute full and local densities of states. The performance is illustrated with the examples of the Ising and PXP spin chains. For the nonintegrable Ising chain, our findings corroborate the presence of thermalization for several initial states, well beyond what direct time-dependent simulations have been able to achieve so far.

Highlights

We combine matrix product operator techniques with Chebyshev polynomial expansions and present a method that is able to explore spectral properties of quantum many-body Hamiltonians
The performance is illustrated with the examples of the Ising and PXP spin chains
The study of one-dimensional quantum many-body systems has motivated the emergence of a number of techniques, based on tensor network states (TNS)

Summary

Probing Thermalization through Spectral Analysis with Matrix Product Operators

The study of one-dimensional quantum many-body systems has motivated the emergence of a number of techniques, based on tensor network states (TNS) They use matrix product states (MPS) and matrix product density operators (MPDO) [1,2,3,4,5] to approximate the ground states, low-lying excitations, thermal states, as well as time evolution. We combine TNS and the kernel polynomial method (KPM) [15] in a general scheme that provides access to the full density of states (DOS) of a given many-body Hamiltonian [16], and to energy functions that are intimately related to the out-ofequilibrium dynamics, including the local density of states (LDOS) With these functions it is possible to probe the eigenstate thermalization hypothesis (ETH) [17,18] across the spectrum, and to verify the thermalization of initial states without explicitly simulating the time evolution. We will aim an approximation to gMðE; OÞ ≡ tr1⁄2OδMðE − HÞ; ð2Þ

Published by the American Physical Society

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