Solving the kinetic Ising model with nonreciprocity.

Abstract

Nonreciprocal interactions are a generic feature of nonequilibrium systems. We define a nonreciprocal generalization of the kinetic Ising model in one spatial dimension. We solve the model exactly using two different approaches for infinite, semi-infinite, and finite systems with either periodic or open boundary conditions. The exact solution allows us to explore a range of novel phenomena tied to nonreciprocity like nonreciprocity induced frustration and wave phenomena with interesting parity-dependence for finite systems of size N. We study dynamical questions like the approach to equilibrium with various boundary conditions. We find different regimes, separated by Nth-order exceptional points, which can be classified as overdamped, underdamped, or critically damped phases. Despite these different regimes, long-time order is only present at zero temperature. Additionally, we explore the low-energy behavior of the system in various limits, including the aging and spatiotemporal Porod regimes, demonstrating that nonreciprocity induces unique scaling behavior at zero temperature. Lastly, we present general results for systems where spins interact with no more than two spins, outlining the conditions under which long-time order may exist.