Power-law approach to steady state in open lattices of non-interacting electrons

TL;DR

This paper investigates how open quantum systems with many excitations approach their non-equilibrium steady states, revealing a universal power-law relaxation behavior due to the density of near-zero decay rates.

Contribution

It introduces a power-law framework for understanding the relaxation dynamics in open fermionic lattices with closing spectral gaps in the thermodynamic limit.

Findings

Relaxation follows a universal $ au^{-3/2}$ power law.

Power-law behavior is independent of spatial dimension.

The approach is exemplified in non-interacting fermion lattices with Markovian leads.

Abstract

We address the question of how a non-equilibrium steady state (NESS) is reached in the Linbdladian dynamics of an open quantum system. We develop an expansion of the density matrix in terms of the NESS-excitations, each of which has its own (exponential) decay rate. However, when the decay rates tend to zero for many NESS-excitations (the spectral gap of the Liouvillian is closed in the thermodynamic limit), the long-time dynamics of the system can exhibit a power-law behaviour. This relaxation to NESS expectation values is determined by the density of states close to zero spectral gap and the value of the operator in these states. We illustrate this main idea on the example of the lattice of non-interacting fermions coupled to Markovian leads at infinite bias voltage. The current comes towards its NESS value starting from a typical initial state as $\tau^{-3/2}$. This behaviour is universal and independent of the space dimension.