Kinetic random-field nonreciprocal Ising model

We introduce and analyze the kinetic random-field nonreciprocal Ising model, which combines bimodal (double-delta) diffusive disorder with pairwise nonreciprocal interactions between two species. By employing mean-field and effective-field theories alongside kinetic Monte Carlo simulations (3D Glauber dynamics), we identify a nonequilibrium tricritical (Bautin) point separating continuous Hopf-type transitions from discontinuous saddle-node-of-limit-cycle (SNLC) transitions. For weak disorder below a critical value, collective oscillations (the "swap" phase) emerge via a supercritical Hopf bifurcation; above this value, the transition becomes first-order (SNLC), exhibiting hysteresis and characteristic Binder-cumulant signatures. Finite-size scaling of the susceptibility confirms the distinct critical and discontinuous behaviors in the Hopf and SNLC regimes, respectively (yielding effective exponents ≈1.96 and ≈3.0). In the first-order regime, the swap phase requires a threshold nonreciprocity that increases with disorder strength. The above conclusions also hold qualitatively for a random field sampled from a double-Gaussian distribution. Finally, we identify a droplet-induced swap phase at higher disorder that cycles through eight metastable states, driven by droplet nucleation in the dynamical free-energy landscape. These findings reveal how disorder and nonreciprocity generate rich nonequilibrium criticality relevant to driven and active systems.