Reduced Density Matrix after a Quantum Quench

TL;DR

This paper investigates the time evolution of the reduced density matrix after a quantum quench in the transverse-field Ising chain, demonstrating a universal power-law relaxation and the importance of local conservation laws for accurate description.

Contribution

It provides a detailed construction of the generalized Gibbs ensemble in terms of local integrals of motion and analyzes the approach to equilibrium of the reduced density matrix.

Findings

The RDM approaches the GGE RDM as t^{-3/2}

Local conservation laws with range comparable to the subsystem size are sufficient for accurate description

Excluding local conservation laws significantly worsens the GGE approximation

Abstract

We consider the reduced density matrix (RDM) \rho_A(t) for a finite subsystem A after a global quantum quench in the infinite transverse-field Ising chain. It has been recently shown that the infinite time limit of \rho_A(t) is described by the RDM \rho_{GGE,A} of a generalized Gibbs ensemble. Here we present some details on how to construct this ensemble in terms of local integrals of motion, and show its equivalence to the expression in terms of mode occupation numbers widely used in the literature. We then address the question, how \rho_A(t) approaches \rho_{GGE,A} as a function of time. To that end we introduce a distance on the space of density matrices and show that it approaches zero as a universal power-law t^{-3/2} in time. As the RDM completely determines all local observables within A, this provides information on the relaxation of correlation functions of local operators. We then address the issue, of how well a truncated generalized Gibbs ensemble with a finite number of local higher conservation laws describes a given subsystem at late times. We find that taking into account only local conservation laws with a range at most comparable to the subsystem size provides a good description. However, excluding even a single one of the most local conservation laws in general completely spoils this agreement.