Relaxation of the Ising spin system coupled to a bosonic bath and the time dependent mean field equation

TL;DR

This paper derives a time-dependent mean field equation for the Ising model coupled to a bosonic bath, analyzing relaxation dynamics and critical behavior, bridging finite size effects and thermodynamic limit.

Contribution

It introduces a new master equation based on the Redfield equation for the Ising model coupled to a bosonic bath, and derives a mean field equation for relaxation at finite temperature.

Findings

Relaxation time dependence on system size and temperature.

Critical behavior analysis around the phase transition.

Finite size effects captured by the master equation.

Abstract

The Ising model doesn't have a strictly defined dynamics, only a spectrum. There are different ways to equip it with a time dependence e.g. the Glauber or the Kawasaki dynamics, which are both stochastic, but it means there is a master equation which can also describes their dynamics. We present a Gluber-type master equation derived from the Redfield equation, where the spin system is coupled to a bosonic bath. We derive a time dependent mean field equation which describes the relaxation of the spin system at finite temperature. Using the fully connected, uniform Ising model the relaxation time will be studied, and the critical behaviour around the critical temperature. The master equation shows the finite size effects, and the mean field equation the thermodynamic limit.