Kinetic Random-Field Nonreciprocal Ising Model

Abstract

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.

Figures (12)

• Figure 1: Representative phase portraits illustrating two regimes: top row, $\tilde{h}<\tilde{h}_c$ (parameters: $\tilde{J}=0.5$ for the disordered fixed point and $\tilde{J}=2.0$ for the swap state, with $\tilde{K}=1$ and $\tilde{h}=0.5$); bottom row, $\tilde{h}>\tilde{h}_c$ (parameters: $\tilde{J}=0.5$ disordered and $\tilde{J}=3.0$ swap, with $\tilde{K}=1$ and $\tilde{h}=1.5$). These panels show that the system crosses from a supercritical Hopf to an SNLC bifurcation as disorder increases past $\tilde{h}_c$. These results are obtained from the mean-field theory.
• Figure 2: Hysteresis of the (a) synchronisation amplitude $R$ and (b) rotation measure $S$ for $\tilde{h}>\tilde{h}_c$ from mean-field theory calculations. Here we fix $\tilde{h} = 1.2$ and $\tilde{K} = 0.3$, for which we observe a clear signature of a discontinuous transition; EFT gives similar trends.
• Figure 3: Dependence of the swap phase on nonreciprocity $\tilde{K}$ and interaction strength $\tilde{J}$: when $\tilde{h}>\tilde{h}_c$, a finite threshold $\tilde{K}_c$ is required to sustain oscillations, and the swap phase vanishes for $\tilde{K}<\tilde{K}_c$ (illustrated for $\tilde{h}=1.5$). The results shown here are obtained within the mean-field approximation.
• Figure 4: Plotted at $\tilde{K}=0.35$. (a) Order parameter $R$ vs $\tilde{J}$ at $L=18$: as the random-field strength $\tilde{h}$ is increased, the onset of the swap phase sharpens, indicating a change from a continuous to a discontinuous transition. (b) Example time-series of the rotation measure $S$ in the coexistence region demonstrating abrupt switches between the disordered and swap phases. $t$ is measured in units of Monte Carlo sweeps (plotted at the first-order phase transition point and $\tilde{h}=2.4$).
• Figure 5: Plotted at $\tilde{K}=0.35$. (a) Binder cumulant $U_{18}$ at $\tilde{h} = 2.0$ below the estimated $\tilde{h}_c$. (b) Binder cumulant $U_{18}$ at $\tilde{h} =2.4$ above the estimated $\tilde{h}_c$: a pronounced dip appears at the point of first-order phase transition.