Hierarchy of relaxation timescales in local random Liouvillians

TL;DR

This paper investigates how the locality of dissipative quantum dynamics influences relaxation timescales, revealing a hierarchy of decay modes and spectral structures in local random Liouvillians.

Contribution

It demonstrates that local Liouvillians exhibit a structured spectrum with multiple decay timescales, extending understanding of open quantum system relaxation behavior.

Findings

Spectrum forms multiple dense clusters with lemon-shaped support.

Eigenvectors correspond to n-body decay modes.

Hierarchy of relaxation timescales for n-body observables.

Abstract

To characterize the generic behavior of open quantum systems, we consider random, purely dissipative Liouvillians with a notion of locality. We find that the positivity of the map implies a sharp separation of the relaxation timescales according to the locality of observables. Specifically, we analyze a spin-1/2 system of size $\ell$ with up to $n$-body Lindblad operators, which are $n$-local in the complexity-theory sense. Without locality ($n=\ell$), the complex Liouvillian spectrum densely covers a "lemon"-shaped support, in agreement with recent findings [Phys. Rev. Lett. 123, 140403;arXiv:1905.02155]. However, for local Liouvillians ($n<\ell$), we find that the spectrum is composed of several dense clusters with random matrix spacing statistics, each featuring a lemon-shaped support wherein all eigenvectors correspond to $n$-body decay modes. This implies a hierarchy of relaxation timescales of $n$-body observables, which we verify to be robust in the thermodynamic limit.