Kinetic RF Nonreciprocal Ising Model

• The kinetic random-field nonreciprocal Ising model is a disordered, driven framework where two interacting Ising species evolve via stochastic kinetics with antisymmetric coupling.
• Mean-field, effective-field, and Monte Carlo analyses reveal a transition from continuous (Hopf) to discontinuous (SNLC) oscillatory onset marked by a nonequilibrium Bautin point.
• Stronger random fields induce a droplet-induced cyclic regime with an eight-state dynamic, illustrating how disorder reorganizes phase transitions.

The kinetic random-field nonreciprocal Ising model is a disordered, driven extension of the two-species nonreciprocal Ising model in which quenched/random-field-like disorder and antisymmetric inter-species coupling jointly control nonequilibrium ordering. Each lattice site carries two Ising variables, one for species $A$ and one for species $B$, with ferromagnetic nearest-neighbor interactions within each species, a local nonreciprocal coupling between species, and a bimodal random field. Because the influence of $A$ on $B$ is opposite to the influence of $B$ on $A$, the dynamics is not generated by a single equilibrium Hamiltonian; the model is intrinsically nonequilibrium and supports collective oscillations known as the swap phase, a nonequilibrium tricritical (Bautin) point, discontinuous oscillatory onset through saddle-node-of-limit-cycle bifurcations, and a droplet-induced cyclic regime at stronger disorder (R et al., 4 Sep 2025).

1. Microscopic construction

At each site $i$, the model contains two spins, $\sigma_i^A,\sigma_i^B=\pm1$. Spins of the same species interact ferromagnetically with nearest neighbors, while the two species are coupled locally and nonreciprocally. The antisymmetry is imposed by

$K_{AB}=+K,\qquad K_{BA}=-K,$

with $K>0$. This breaks action-reaction symmetry at the microscopic level: species $B$0 and $B$1 do not derive their local tendencies from a single common Hamiltonian. The disorder is a bimodal random field,

$B$2

with $B$3 the field strength. The control parameters are the ferromagnetic coupling $B$4, the nonreciprocity $B$5, the field amplitude $B$6, and the temperature $B$7 (R et al., 4 Sep 2025).

In the mean-field and effective-field descriptions, the couplings are rescaled as

$B$8

after which the same symbols $B$9 are reused for the scaled variables. The model is termed kinetic because its primary definition is dynamical: it evolves by single-spin-flip stochastic kinetics rather than by equilibrium sampling. It is nonequilibrium for two reasons emphasized in the primary study: the antisymmetric couplings violate detailed balance, and in simulation the random field is updated in time under “competing kinetics,” reinforcing the driven character of the dynamics (R et al., 4 Sep 2025).

The flip dynamics is Glauber-like,

$A$0

and the configuration probability obeys the master equation

$A$1

This formulation already places the model outside equilibrium Ising theory: local “selfish” energies exist, but there is in general no global equilibrium Hamiltonian governing the full two-species dynamics (R et al., 4 Sep 2025).

2. Mean-field and effective-field formulations

From the master equation, the authors derive dynamics for the local magnetizations $A$2. Under homogeneous mean-field factorization and configurational averaging over the bimodal field, the species magnetizations $A$3 satisfy

$A$4

These coupled nonlinear ODEs define the nonequilibrium phase structure through their fixed points and limit cycles. In this framework, the disordered phase corresponds to a stable fixed point at the origin, whereas the swap phase corresponds to a stable limit cycle in the $A$5 plane (R et al., 4 Sep 2025).

To incorporate short-range effects beyond MFT, the paper employs an effective-field theory based on the differential-operator method for a simple cubic lattice with coordination $A$6. The EFT dynamics takes the form

$A$7

where the coefficients $A$8 are generated from differential-operator identities acting on

$A$9

In the paper’s interpretation, EFT neglects intersite correlations between different spins but retains some same-spin/self-correlation structure, thereby improving quantitatively over MFT while preserving the same qualitative phase topology (R et al., 4 Sep 2025).

This analytical hierarchy is central to the model’s interpretation. MFT provides the clearest bifurcation picture; EFT shifts phase boundaries toward the simulation results; and both descriptions organize the nonequilibrium transition structure in terms of fixed-point stability and the birth or destruction of limit cycles. A plausible implication is that the model’s qualitative phenomenology is controlled less by the specific approximation and more by the joint presence of disorder and antisymmetric coupling (R et al., 4 Sep 2025).

3. Dynamical order and simulation protocol

The dynamical order is characterized by a synchronization amplitude $B$0, measuring the magnitude of the magnetization vector $B$1, and by an angular-momentum-like quantity

$B$2

The distinction is essential. The swap phase is not merely a state with nonzero magnetization; it is a state of sustained out-of-phase oscillation in which $B$3 executes a limit cycle and $B$4 remains nonzero on average. Static phases instead have $B$5 apart from finite-size fluctuations (R et al., 4 Sep 2025).

The numerical study uses 3D kinetic Monte Carlo with Glauber updates on a periodic simple cubic lattice. One Monte Carlo sweep is the time unit. The random field $B$6 is redrawn every unit time under competing kinetics. In the first-order regime, long runs are required so that the system can relax to global minima; the simulations used to characterize phase behavior discard $B$7 sweeps for equilibration. For threshold-$B$8 scans reported in the supplement, the parameters are $B$9, total $B$0 sweeps, with $B$1 discarded and $B$2 used for measuring the time-averaged angular momentum $B$3 (R et al., 4 Sep 2025).

A configuration is classified as swap if $B$4. The threshold is chosen to separate persistent coherent rotation from droplet-induced intermittent signals of order $B$5. This operational distinction matters because the model contains both ordinary coherent swap cycles and a high-field droplet-induced cyclic regime, and the two produce different temporal signatures even when both generate rotational bursts in magnetization space (R et al., 4 Sep 2025).

4. Nonequilibrium tricriticality and transition structure

The central phase-diagram result is a disorder-controlled change in the onset mechanism of the swap phase. For weak random field, the disordered fixed point loses stability through a supercritical Hopf bifurcation, so the swap phase emerges continuously as $B$6 increases. In mean field, the tricritical disorder threshold is reported as

$B$7

Below this value, oscillatory onset is continuous and Hopf-like (R et al., 4 Sep 2025).

For stronger disorder, the onset changes qualitatively. When $B$8, the oscillatory phase appears via a saddle-node of limit cycles (SNLC). In this regime a stable and an unstable limit cycle are created together, the jump into the oscillatory state is discontinuous, and bistability and hysteresis appear. The paper identifies the change from supercritical Hopf to SNLC as a nonequilibrium tricritical point, more precisely a Bautin point. In the model’s bifurcation logic, the Bautin point separates continuous and discontinuous oscillatory onset (R et al., 4 Sep 2025).

EFT preserves this topology but shifts the threshold upward to

$B$9

while Monte Carlo shifts it further to

$A$0

for the parameter set discussed in the finite-size-scaling analysis at $A$1. The progression from MFT to EFT to MC is interpreted as the effect of increasingly incorporating correlations, fluctuations, and the consequences of competing-kinetics disorder updates (R et al., 4 Sep 2025).

The Monte Carlo evidence is multi-pronged. The order parameter $A$2 changes from a smooth onset at lower $A$3 to a sharp jump at higher $A$4. The Binder cumulant

$A$5

changes smoothly through the transition for $A$6, but develops a pronounced negative dip or sharp minimum for $A$7, which is the standard finite-size signature of phase coexistence and first-order behavior. Time traces of $A$8 in the coexistence region show abrupt switching between the disordered phase and the swap phase, directly exposing bistability (R et al., 4 Sep 2025).

Finite-size scaling of the susceptibility,

$A$9

further separates the two regimes. At $i$0, below the tricritical window, the effective log-log slope is

$i$1

consistent with continuous behavior. Above the tricritical window the slopes are essentially $i$2: $i$3 The article’s interpretation is that these exponents distinguish Hopf-type criticality from discontinuous SNLC behavior (R et al., 4 Sep 2025).

5. Threshold nonreciprocity and droplet-induced swap

A second major result concerns the existence of a threshold nonreciprocity above the tricritical field. In the strong-disorder regime, the conventional swap phase survives only if the antisymmetric coupling exceeds a minimum value $i$4. In MFT, EFT, and MC alike, this threshold increases monotonically with disorder strength: $i$5 Operationally, $i$6 is the smallest nonreciprocity for which some $i$7 sustains a swap state. The physical interpretation given in the paper is that random fields create local preferred orientations and pin domains, so stronger antisymmetric drive is required to synchronize domains and overcome pinning (R et al., 4 Sep 2025).

In the region

$i$8

ordinary coherent swap oscillations are no longer stable. Instead the model displays a droplet-induced swap phase. For moderate $i$9, the system cycles through eight distinct metastable states; at larger $\sigma_i^A,\sigma_i^B=\pm1$0, the cycle reduces to four states, matching the type of droplet-induced swap reported previously in the zero-field model. The eight-state cycle is identified as new and specific to the random-field case (R et al., 4 Sep 2025).

The mechanism is described through a dynamical free-energy picture based on species-specific “selfish free energies,” not a single global free energy for the coupled system. In this picture, random-field disorder generates additional metastable wells. Droplet fluctuations nucleate escapes from one well, and the antisymmetric coupling biases the subsequent direction of motion through phase space. The result is not equilibration but a cyclic itinerary through metastable configurations. The paper’s phase portraits and simulation description imply that the eight states are visited in a definite orientation, clockwise in the MFT portrait (R et al., 4 Sep 2025).

These droplet-induced cycles differ qualitatively from coherent swap. In the coherent swap phase the motion is a true limit cycle with sustained rotation. In the droplet-induced phase, the dynamics consists of recurrent abrupt jumps between metastable states. This is why the $\sigma_i^A,\sigma_i^B=\pm1$1 threshold matters: intermittent rotational bursts do not by themselves establish the presence of a coherent oscillatory phase (R et al., 4 Sep 2025).

6. Relation to surrounding research and broader significance

The model sits at the intersection of several prior lines of work. The clean two-species nonreciprocal Ising model already exhibited disordered, static ordered, and swap phases, and identified droplet-induced oscillatory behavior as well as 3D XY-like criticality for the disorder-to-swap transition (Avni et al., 2023). Reciprocal kinetic random-field Ising studies had separately established that random fields can generate tricriticality, coexistence regions, and strong frequency dependence in driven settings (Yüksel et al., 2012), and can slow coarsening through pinning with algebraic-to-logarithmic crossover in relaxation (Mandal et al., 2013). A distinct bond-disordered nonreciprocal Ising model showed that quenched nonreciprocity can prevent zero-temperature freezing, sustain rare-region reversals, and generate logarithmic “activated” coarsening (Grodzinski et al., 19 Jun 2026).

Against that background, the kinetic random-field nonreciprocal Ising model combines random-field pinning with antisymmetric drive in a single lattice framework and shows that the combination produces a richer structure than either ingredient alone. Directly established consequences include a disorder-tuned change from continuous to discontinuous oscillatory onset, a nonequilibrium Bautin point, a disorder-dependent threshold nonreciprocity, and a new droplet-induced eight-state cyclic regime (R et al., 4 Sep 2025).

The broader implication stated in the primary work is that driven lattice systems with asymmetric interactions and heterogeneous local biases need not simply lose order under disorder. Instead, disorder can reorganize the entire bifurcation structure, create hysteresis and coexistence, and replace coherent collective oscillation with metastability-driven temporal cycling. In that sense, the kinetic random-field nonreciprocal Ising model functions as a minimal statistical-mechanical framework for studying how asymmetric interactions and local heterogeneity jointly generate nonequilibrium criticality, collective oscillation, and droplet-mediated dynamical phases (R et al., 4 Sep 2025).