Nonreciprocal Blume-Capel model with antisymmetric single-ion anisotropies

We investigate the interplay between nonreciprocal interactions and chemical-potential imbalance in a two-species nonreciprocal Blume-Capel model. Combining a systematic mean-field bifurcation analysis with large-scale Monte Carlo simulations in two and three dimensions, we map the model's dynamical regimes and transitions. Mean-field theory predicts a rich phase structure-disorder, a time-dependent "swap" (limit-cycle) phase, and static ordered states-separated by Hopf, saddle-node on invariant circle, saddle-node of limit cycles, pitchfork, and saddle-node bifurcations. In two dimensions, Monte Carlo simulations reveal that spiral defects destabilize global swapping and, unless vacancies are strongly favored, destroy long-range order. Crucially, a finite single-ion anisotropy Δ_{α}=-Δ_{β} promotes vacancy occupation in the α species and suppresses nonreciprocal dynamics, thereby restoring a robust static ordered phase. Finite-size scaling of susceptibility and Binder cumulants places the disorder → static transition firmly in the two-dimensional Ising universality class. Moreover, within the static ordered phase, we observe a crossover that sharpens into a line of first-order phase transitions; these two regimes are separated by a critical point, which is analogous to the liquid-gas coexistence curve terminating at a critical point. In three dimensions, simulations largely mirror mean-field expectations, though swap → static ordering occurs indirectly via a disordered regime. Our results demonstrate that vacancy energetics provide a simple, experimentally relevant control knob that stabilizes equilibrium-like order in nonreciprocal systems and that defects can generate novel critical behavior.