Stationary correlations for a far-from-equilibrium spin chain

TL;DR

This paper derives exact equations for correlations in a non-equilibrium 1D Ising model with different temperatures on sublattices, showing that long-range correlations decay exponentially and resemble equilibrium Ising behavior.

Contribution

It provides an exact solution for the stationary state and correlation functions of a far-from-equilibrium spin chain with broken detailed balance.

Findings

Correlations decay exponentially with distance.

Long-distance behavior is Ising-like despite non-equilibrium conditions.

Exact steady-state representation confirms simulation and RG predictions.

Abstract

A kinetic one-dimensional Ising model on a ring evolves according to a generalization of Glauber rates, such that spins at even (odd) lattice sites experience a temperature $T_{e}$ ($T_{o}$). Detailed balance is violated so that the spin chain settles into a non-equilibrium stationary state, characterized by multiple interactions of increasing range and spin order. We derive the equations of motion for arbitrary correlation functions and solve them to obtain an exact representation of the steady state. Two nontrivial amplitudes reflect the sublattice symmetries; otherwise, correlations decay exponentially, modulo the periodicity of the ring. In the long chain limit, they factorize into products of two-point functions, in precise analogy to the equilibrium Ising chain. The exact solution confirms the expectation, based on simulations and renormalization group arguments, that the long-time, long-distance behavior of this two-temperature model is Ising-like, in spite of the apparent complexity of the stationary distribution.