Dynamical phase transitions in the nonreciprocal Ising model

TL;DR

This paper investigates phase transitions in a nonreciprocal Ising model, revealing the existence of stable time-dependent phases in 3D and critical behavior akin to the 3D XY model, with implications for understanding nonreciprocal many-body systems.

Contribution

It introduces a minimal nonreciprocal Ising model and characterizes its phase diagram, including the stability of time-dependent phases and critical phenomena, extending understanding of nonreciprocal interactions.

Findings

Swap phase is destabilized by defects in 2D.

Swap phase is stable and exhibits time crystal properties in 3D.

Transition from disorder to swap follows 3D XY critical exponents.

Abstract

Nonreciprocal interactions in many-body systems lead to time-dependent states, commonly observed in biological, chemical, and ecological systems. The stability of these states in the thermodynamic limit and the critical behavior of the phase transition from static to time-dependent states are not yet fully understood. To address these questions, we study a minimalistic system endowed with nonreciprocal interactions: an Ising model with two spin species having opposing goals. The mean-field equation predicts three stable phases: disorder, static order, and a time-dependent swap phase. Large scale numerical simulations support the following: (i) in 2D, the swap phase is destabilized by defects; (ii) in 3D, the swap phase is stable, and has the properties of a time crystal; (iii) the transition from disorder to swap in 3D is characterized by the critical exponents of the 3D XY model, and corresponds to the breaking of a continuous symmetry, time translation invariance; (iv) when the two species have fully anti-symmetric couplings, the static-order phase is unstable in any finite dimension due to droplet growth; (v) in the general case of asymmetric couplings, static order can be restored by a droplet-capture mechanism preventing the droplets from growing indefinitely. We provide details on the full phase diagram which includes first- and second-order-like phase transitions and study how the system coarsens into swap and static-order states.