Dynamic Models for Two Nonreciprocally Coupled Fields: A Microscopic Derivation for Zero, One, and Two Conservation Laws

TL;DR

This paper derives microscopic models for two nonreciprocally coupled fields with varying conservation laws, linking statistical mechanics to macroscopic nonreciprocal field theories through mean-field approximations.

Contribution

It provides a systematic derivation of nonreciprocal field equations from microscopic Ising models, covering cases with zero, one, and two conservation laws.

Findings

Derived macroscopic equations from microscopic models.

Mapped out linear instabilities for different dynamics.

Established a microscopic foundation for nonreciprocal field theories.

Abstract

We construct dynamic models governing two nonreciprocally coupled fields for several cases with zero, one, and two conservation laws. Starting from two microscopic nonreciprocally coupled Ising models, and using the mean-field approximation, we obtain closed-form evolution equations for the spatially resolved magnetization in each lattice. Only allowing for single spin-flip dynamics, the macroscopic equations in the thermodynamic limit are closely related to the nonreciprocal Allen-Cahn equations, i.e. conservation laws are absent. Likewise, only accounting for spin-exchange dynamics within each lattice, the thermodynamic limit yields equations similar to the nonreciprocal Cahn-Hilliard model, i.e. with two conservation laws. In the case of spin-exchange dynamics within and between the two lattices, we obtain two nonreciprocally coupled equations that add up to one conservation law. For each of these cases, we systematically map out the linear instabilities that can arise. Moreover, combining the different dynamics gives a large number of further models. Our results provide a microscopic foundation for a broad class of nonreciprocal field theories, establishing a direct link between nonequilibrium statistical mechanics and macroscopic continuum descriptions.