We study the critical behavior of a two-dimensional Ising model with nonreciprocal vision-cone interactions, which explicitly violate reciprocity and detailed balance. Extensive Monte Carlo simulations and dynamic renormalization-group analysis show that the asymptotic critical exponents remain fully consistent with the equilibrium Ising universality class over a broad range of nonreciprocal coupling strengths $λ$. In contrast, dimensionless quantities such as the Binder cumulant and the correlation-length ratio display pronounced anisotropic nonequilibrium corrections and systematically deviate from their equilibrium Ising values. The renormalization-group flow further demonstrates that the nonreciprocal perturbation is irrelevant at the Wilson-Fisher fixed point while generating a finite shift of the critical temperature proportional to $λ^2$. Our results demonstrate the remarkable robustness of two-dimensional Ising criticality against this class of directional interactions.
We study the critical behavior of a two-dimensional Ising model with nonreciprocal vision-cone interactions, which explicitly violate reciprocity and detailed balance. Extensive Monte Carlo simulations and dynamic renormalization-group analysis show that the asymptotic critical exponents remain fully consistent with the equilibrium Ising universality class over a broad range of nonreciprocal coupling strengths λ. In contrast, dimensionless quantities such as the Binder cumulant and the correlation-length ratio display pronounced anisotropic nonequilibrium corrections and systematically deviate from their equilibrium Ising values. The renormalization-group flow further demonstrates that the nonreciprocal perturbation is irrelevant at the Wilson-Fisher fixed point while generating a finite shift of the critical temperature proportional to λ2. Our results demonstrate the remarkable robustness of two-dimensional Ising criticality against this class of directional interactions.
The Ising universality class is remarkably robust to non-equilibrium perturbations, which generically flow to zero under renormalization. We show that this robustness fails when an order parameter is coupled nonreciprocally to a conserved diffusive density. Below dc=4, the renormalization group flows to a fast-diffusion fixed point at which the density acts as a long-range multiplicative noise, producing a novel universality class. The non-equilibrium nature of the fixed point is manifest in the large-scale violation of the fluctuation-dissipation relations, reflected in a splitting of the scaling exponents of the two-point correlation and response functions--a measurable hallmark of non-equilibrium critical fluctuations. A two-loop calculation establishes the stability of this fixed point but yields a small correction-to-scaling exponent ω≈0.020 in d=3, implying strong finite-size corrections. An all-orders modified Harris criterion ν>2/(d+z−2) confirms that the BIM fixed point governs criticality in d=3, with Ising universality recovered only at d=2.
Nonequilibrium universality of the nonreciprocally coupled O(n1)×O(n2) model
In this work, we investigate an important class of nonequilibrium dynamics in the form of nonreciprocal interactions. In particular, we study how nonreciprocal coupling between two O(ni) order parameters (with i=1,2) affects the universality at a multicritical point, extending the analysis of [J.T. Young et al., Phys. Rev. X 10, 011039 (2020)], which considered the case n1=n2=1, i.e., a Z2×Z2 model. We show that nonequilibrium fixed points (NEFPs) emerge for a broad range of n1,n2 and exhibit intrinsically nonequilibrium critical phenomena, namely a violation of fluctuation-dissipation relations at all scales and underdamped oscillations near criticality in contrast to the overdamped relaxational dynamics of the corresponding equilibrium models. Furthermore, the NEFPs exhibit an emergent discrete scale invariance in certain physically-relevant regimes of n1,n2, but not others, depending on whether the critical exponent ν is real or complex. The boundary between these two regions is described by an exceptional point in the renormalization group (RG) flow, leading to distinctive features in correlation functions and the phase diagram. Another contrast with the previous work is the number and stability of the NEFPs as well as the underlying topology of the RG flow. Finally, we investigate an extreme form of nonreciprocity where one order parameter is independent of the other order parameter but not vice versa. Unlike the Z2×Z2 model, which becomes non-perturbative in this case, we identify a distinct nonequilibrium universality class whose dependent field similarly violates fluctuation-dissipation relations but does not exhibit discrete scale invariance or underdamped oscillations near criticality.
Nonequilibrium dynamics play an important role in all contexts of physics, both classical and quantum as well as living and nonliving, so it is crucial to develop a foundational understanding of nonequilibrium phase transitions. In this work we investigate an important class of nonequilibrium dynamics in the form of nonreciprocal interactions. In particular we study how nonreciprocal coupling between two O(n_{i}) order parameters (with i=1,2) affects the universality at a multicritical point, extending the analysis of J. T. Young et al. [Phys. Rev. X 10, 011039 (2020)2160-330810.1103/PhysRevX.10.011039], which considered the case n_{1}=n_{2}=1, i.e., a Z_{2}×Z_{2} model. We show that nonequilibrium fixed points (NEFPs) emerge for a broad range of n_{1},n_{2} and exhibit intrinsically nonequilibrium critical phenomena, namely a violation of fluctuation-dissipation relations at all scales and underdamped oscillations near criticality in contrast to the overdamped relaxational dynamics of the corresponding equilibrium models. Furthermore, the NEFPs exhibit an emergent discrete scale invariance in certain physically relevant regimes of n_{1},n_{2}, but not others, depending on whether the critical exponent ν is real or complex. The boundary between these two regions is described by an exceptional point in the renormalization group (RG) flow, leading to distinctive features in correlation functions and the phase diagram. Another contrast with the previous work is the number and stability of the NEFPs as well as the underlying topology of the RG flow. Finally, we investigate an extreme form of nonreciprocity where one order parameter is independent of the other order parameter but not vice versa. Unlike the Z_{2}×Z_{2} model, which becomes nonperturbative in this case, we identify a distinct nonequilibrium universality class whose dependent field similarly violates fluctuation-dissipation relations but does not exhibit discrete scale invariance or underdamped oscillations near criticality.
We perform a comprehensive numerical study of how variations in microscopic coupling strength affect critical behavior in the two-dimensional Ising model. Using extensive Monte Carlo simulations based on the Wolff cluster algorithm, we investigate the clean Ising model on a square lattice for interaction strengths in the range 0.80 ≤ J ≤ 1.20. While the exact solution at J = 1 has long established the model’s critical properties, a systematic numerical assessment of universality under continuous coupling rescaling has not been performed. Here, we address this gap by analyzing systems of linear size L = 20 − 500, allowing for high-precision finite-size scaling of thermodynamic observables. For each coupling, the critical temperature T_C(J) is determined via finite-size extrapolation of the susceptibility maxima, yielding results in quantitative agreement with Onsager’s exact prediction. At the extrapolated critical points, we extract the critical exponents ν, β, γ, and α through detailed scaling analyses. These exponents remain consistent with the exact Ising benchmarks across the full range of interaction strengths, confirming that the universal aspects of the phase transition are preserved. Our findings demonstrate that modifying the microscopic coupling strength rescales the critical temperature without altering the asymptotic critical behavior, thereby providing a clear and quantitative confirmation of the robustness of the two-dimensional Ising universality class.
Abstract We investigate the dynamic phase transition properties of the two-dimensional anisotropic kinetic Ising model driven by a square-wave magnetic field. Spatial anisotropy is introduced through direction-dependent coupling strengths, and the system is analyzed using large-scale Monte Carlo simulations combined with finite-size scaling techniques. Several observables are computed to characterize the critical behavior and identify the transition points. Our results show that the presence of spatial anisotropy does not alter the universal nature of the transition, which remains consistent with that of the equilibrium Ising model. The findings highlight the robustness of the equilibrium Ising universality class even under spatial anisotropy and a time-dependent driving magnetic field.
We study the q-state Potts model on a square lattice with directed nearest-neighbor spin-spin interactions that are inherently non-reciprocal. Both equilibrium and non-equilibrium dynamics are investigated. Analytically, we demonstrate that non-reciprocal interactions do not alter the critical exponents of the model under equilibrium dynamics. In contrast, numerical simulations with selfish non-equilibrium dynamics reveal distinctive behavior. For q=2 (non-reciprocal non-equilibrium Ising model), the critical exponents remain consistent with those of the equilibrium Ising universality class. However, for q=3 and q=4, the critical exponents vary continuously. Remarkably, a super-universal scaling function -- Binder cumulant as a function of ξ2/ξ0, where ξ2 is the second moment correlation length and ξ0 its maximum value -- remains identical to that of the equilibrium q=3,4 Potts models. These findings indicate that non-reciprocal Potts models belong to the superuniversality class of their respective equilibrium counterparts.