Nonreciprocal Ising model

Systems with nonreciprocal interactions generically display time-dependent states. These are routinely observed in finite systems, from neuroscience to active matter, in which globally ordered oscillations exist. However, the stability of these uniform nonreciprocal phases in noisy spatially-extended systems, their fate in the thermodynamic limit, and the critical behavior of the corresponding phase transitions are not fully understood. Here, we address these questions by introducing a nonreciprocal generalization of the Ising model and study its phase transitions by means of numerical and analytical approaches. While the mean-field equations predict three stable homogeneous phases (disordered, ordered and a time-dependent swap phase), our large scale numerical simulations reveal a more complex picture. Static order is destroyed in any finite dimension due to the growth of rare droplets unless the symmetry between the two spin types is broken triggering a stabilizing droplet-capture mechanism. The swap phase is destroyed by fluctuations in two dimensions through the proliferation of spiral defects but stabilized in three dimensions where nonreciprocity changes the critical exponents from Ising to XY, thus giving rise to a robust spatially-distributed clock.