Statistical Mechanics of the Non-reciprocal Ising model near Zero Temperature*

We study the kinetic Ising model with non-reciprocal interactions between spins, the maximum entropy distribution of binary variables conditioned on a past state with constraints on means and lagged correlations. As a maximum entropy model, the non-reciprocal Ising model offers a coarse-grained representation of biological systems, in particular the inherently non-reciprocal interactions of a neural system. We consider two non-reciprocally coupled 1D Ising chains in the low temperature regime. In this limit, the dynamics becomes deterministic, corresponding to a cellular automata update rule. Adjusting the coupling strength can alter the update rule, causing a dynamical phase transition in the absence of noise analogous to a quantum critical point of a thermodynamic system. Notably, as non-reciprocity increases through the transition at zero temperature, the system becomes spatially decoupled, inducing disorder at zero temperature. Close to the phase transition, we demonstrate the low temperature effective theory of the system is given by the driven Sine-Gordon equation. Previous studies have shown that the introduction of non-reciprocity can lead to the emergence of relevant sound modes in the long wavelength dynamics of certain systems, altering critical phenomena. A similar effect is evident in our model, where the dynamical critical exponent 'z' deviates from the expected value of 'z=2' in the reciprocal kinetic Ising model.