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Nonreciprocal Blume-Capel Model with Antisymmetric Single-Ion Anisotropies

Arjun R, Pratyush Prakash Patra, A. V. Anil Kumar

TL;DR

This work analyzes a two-species nonreciprocal Blume-Capel model (NR-BCM) with equal and opposite chemical-potential imbalances $\Delta_A/k_BT=-\Delta_B/k_BT=\Delta$, incorporating a fully antisymmetric onsite coupling. The authors combine mean-field bifurcation analysis with large-scale Monte Carlo simulations in $2$D and $3$D to map dynamical regimes—disorder, a time-dependent swap phase, and static order—and to identify the bifurcations that separate them. In $2$D, fluctuations and defects destroy global swapping and long-range order, but a finite $\Delta$ restores a robust static phase and the disorder-to-static transition falls in the $2$D Ising universality class, with a line of first-order transitions ending at a critical point; in $3$D, the swap phase persists and the route to static order proceeds via an intermediate disordered regime. The results show vacancy energetics provide a simple control knob for stabilizing equilibrium-like order in nonreciprocal systems and highlight the role of defects in generating novel critical behavior.

Abstract

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 destabilise global swapping and, unless vacancies are strongly favoured, 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 to static transition firmly in the 2D 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, analogous to the termination of the liquid-gas coexistence curve. In three dimensions, simulations largely mirror mean-field expectations, though swap to static ordering occurs indirectly via a disordered regime. Our results demonstrate that vacancy energetics provide a simple, experimentally relevant control knob that stabilises equilibrium-like order in nonreciprocal systems and that defects can generate novel critical behaviour.

Nonreciprocal Blume-Capel Model with Antisymmetric Single-Ion Anisotropies

TL;DR

This work analyzes a two-species nonreciprocal Blume-Capel model (NR-BCM) with equal and opposite chemical-potential imbalances , incorporating a fully antisymmetric onsite coupling. The authors combine mean-field bifurcation analysis with large-scale Monte Carlo simulations in D and D to map dynamical regimes—disorder, a time-dependent swap phase, and static order—and to identify the bifurcations that separate them. In D, fluctuations and defects destroy global swapping and long-range order, but a finite restores a robust static phase and the disorder-to-static transition falls in the D Ising universality class, with a line of first-order transitions ending at a critical point; in D, the swap phase persists and the route to static order proceeds via an intermediate disordered regime. The results show vacancy energetics provide a simple control knob for stabilizing equilibrium-like order in nonreciprocal systems and highlight the role of defects in generating novel critical behavior.

Abstract

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 destabilise global swapping and, unless vacancies are strongly favoured, destroy long-range order. Crucially, a finite single-ion anisotropy promotes vacancy occupation in the species and suppresses nonreciprocal dynamics, thereby restoring a robust static ordered phase. Finite-size scaling of susceptibility and Binder cumulants places the disorder to static transition firmly in the 2D 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, analogous to the termination of the liquid-gas coexistence curve. In three dimensions, simulations largely mirror mean-field expectations, though swap to static ordering occurs indirectly via a disordered regime. Our results demonstrate that vacancy energetics provide a simple, experimentally relevant control knob that stabilises equilibrium-like order in nonreciprocal systems and that defects can generate novel critical behaviour.

Paper Structure

This paper contains 6 sections, 16 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Mean-field analysis
  4. Monte Carlo Simulations
  5. Conclusion
  6. Acknowledgments

Figures (15)

  • Figure 1: The figures show colour maps of $R$ and $S$. In the first row, $\tilde{K} = 0$ and it shows the colour map of (a) $R$ and (b) $S$. The second row corresponds to $\tilde{K} = 0.7$. (c) The colour map of $R$ for nonzero $\tilde{K}$ is topologically different than for $\tilde{K} = 0$. (d) Colour map of $S$ has nonzero values in the parameter space showing the presence of the swap phase.
  • Figure 2: Phase portraits of the NR-BCM at zero nonreciprocity ($\tilde{K} = 0$) showing the bifurcations which take place as we vary the control parameters. (a) $\to$ (b): supercritical pitchfork bifurcation at the origin. (c) $\to$ (d): supercritical pitchfork bifurcation at the nonzero fixed points. (e) $\to$ (f) saddle-node bifurcation of the saddle point and the stable fixed point corresponding to the full order.
  • Figure 3: Phase portraits of the NR-BCM at nonzero nonreciprocity illustrating the bifurcations which take place as we vary the control parameters. (a) $\to$ (b): supercritical Hopf bifurcation at the origin. (c) $\to$ (d): saddle-node on invariant circle bifurcation. (e) $\to$ (f) saddle-node bifurcation of the saddle point and the stable fixed point with a higher magnitude of $R$.
  • Figure 4: (a) Shows the eigenvalue flow of the trivial fixed point as we traverse from region II to III' at $\tilde{J} = 1.8, \tilde{K}= 0.7$. The flow indicates that for some value of $\Delta$, a subcritical pitchfork bifurcation occurs at the origin. (b) Shows the eigenvalue flow of the nonzero fixed points, which were created from the subcritical pitchfork bifurcation at the origin. As the magnitude of $\Delta$ increases, the eigenvalue crosses the imaginary line, corresponding to a subcritical Hopf bifurcation.
  • Figure 5: (a) Phase portrait illustrating the stable limit cycles at the nonzero fixed point. (b) Shows the colour map of $R(\langle M_A\rangle_t, \langle M_B\rangle_t)S$. Limit cycles centred at nonzero fixed points are evident from the map. Plotted at $\tilde{K} = 1.5$, $\Delta_B/k_BT = 0.7$ and scaling $\Delta_A/k_Bt = \Delta$
  • ...and 10 more figures