Kinetic random-field nonreciprocal Ising model

We introduce and analyse the kinetic random-field nonreciprocal Ising model, which incorporates bimodal (double-delta) diffusive disorder along with pairwise nonreciprocal interactions between two different species. Using mean-field and effective-field theory, in combination with kinetic Monte Carlo simulations (3D Glauber dynamics), we identify a nonequilibrium tricritical (Bautin) point separating Hopf-type transitions (continuous) from saddle-node-of-limit-cycle (SNLC) transitions (discontinuous). For a weak random field which is less than a critical value, the onset of collective oscillations (the "swap" phase) occurs via a supercritical Hopf bifurcation, whereas for fields greater than the critical value, the transition is first-order (SNLC), exhibiting hysteresis and Binder-cumulant signatures. The finite-size scaling of the susceptibility is consistent with the distinct critical and discontinuous behaviour shown in the Hopf and SNLC regimes, respectively (effective exponents $\approx1.96$ in the Hopf regime and $\approx3.0$ in the SNLC regime). Additionally, in the first-order regime, the swap phase is sustained only above a threshold nonreciprocity, and this threshold increases monotonically with the disorder strength. We further identify a new droplet-induced swap phase in the larger field-strength region, which cycles eight different metastable states. A dynamical free-energy picture rationalises droplet nucleation as the mechanism for these cyclic jumps. Together, these results demonstrate how disorder and nonreciprocity combined generate rich nonequilibrium criticality, with implications for driven and active systems.