Lindbladian way for the relaxation time approximation, application to Kibble-Zurek processes due to environment temperature quench, and to Lindbladian perturbation theory

TL;DR

This paper introduces a Lindbladian-based relaxation time approximation (RTA) that models thermalization processes, demonstrating its application to quantum phase transitions and perturbations from equilibrium, providing a unified framework for thermalization in open quantum systems.

Contribution

It presents a Lindbladian formulation of the relaxation time approximation, enabling direct modeling of thermalization and out-of-equilibrium dynamics in quantum systems.

Findings

RTA-Lindblad ansatz relates to equilibrium behavior during slow temperature changes.

The approach predicts the decay of order parameters near phase transitions.

A perturbative expression for expectation values under Lindbladian perturbations is derived.

Abstract

In the present paper, a global Lindbladian ansatz is constructed which leads to thermalization at temperature $T$ to the Gibs state of the investigated system. This ansatz connects every two eigenstates of the Hamiltonian and leads to a simple master equation known in the literature as the relaxation time approximation (RTA). The main message of this paper is that RTA, being a Lindbladian approach itself, can be used as Lindbladian securing thermalization when modeling physical processes, and can be consequently combined with other types of Lindbladians which would drive the system of the equilibrium state. I demonstrate it with two applications. The first application is the slow cooling (or heating) of quantum systems by varying the environment temperature to a critical point. With this RTA-Lindblad ansatz, one can directly relate to the equilibrium behavior of the system, and if an order parameter has the exponent $\Psi$, the remaining value at the phase transition will decrease with $1/\tau^{\Psi}$, where $\tau$ is the overall time of the slow process. In the second application, I investigate the change in the expectation value of a conserved quantity (an operator commuting with the Hamiltonian) due to an extra Lindbladian term which would drive the system out from equilibrium, while the thermalizing RTA-Lindbladian term is also present. I give a closed perturbative expression in the first order for the expectation value in the new steady state using only expectation values calculated in the original thermal equilibrium.