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arXiv:2411.12680

Nonequilibrium universality of the nonreciprocally coupled $\mathbf{O(n_1) \times O(n_2)}$ model

Jeremy T. Young, Alexey Vyacheslavovich Gorshkov, Mohammad F. Maghrebi

Published Nov 19, 2024• 1 citation
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Abstract

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)], which considered the case $n_1 = n_2 = 1$, i.e., a $\mathbb{Z}_2 \times \mathbb{Z}_2$ 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 $\mathbb{Z}_2 \times \mathbb{Z}_2$ 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.

DOI
10.48550/arxiv.2411.12680

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Nonequilibrium universality of the nonreciprocally coupled O(n1)×O(n2)\mathbf{O(n_1) \times O(n_2)}O(n1​)×O(n2​) model
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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.
Driven-dissipative systems are expected to give rise to nonequilibrium phenomena that are absent in their equilibrium counterparts.However, phase transitions in these systems generically exhibit an effectively classical equilibrium behavior in spite of their nonequilibrium origin.In this paper, we show that multicritical points in such systems lead to a rich and genuinely nonequilibrium behavior.Specifically, we investigate a driven-dissipative model of interacting bosons that possesses two distinct phase transitions: one from a high-to a low-density phase-reminiscent of a liquid-gas transition-and another to an antiferromagnetic phase.Each phase transition is described by the Ising universality class characterized by an (emergent or microscopic) Z 2 symmetry.However, they coalesce at a multicritical point, giving rise to a nonequilibrium model of coupled Ising-like order parameters described by a Z 2 × Z 2 symmetry.Using a dynamical renormalization-group approach, we show that a pair of nonequilibrium fixed points (NEFPs) emerge that govern the long-distance critical behavior of the system.We elucidate various exotic features of these NEFPs.In particular, we show that a generic continuous scale invariance at criticality is reduced to a discrete scale invariance.This further results in complex-valued critical exponents and spiraling phase boundaries, and it is also accompanied by a complex Liouvillian gap even close to the phase transition.As direct evidence of the nonequilibrium nature of the NEFPs, we show that the fluctuation-dissipation relation is violated at all scales, leading to an effective temperature that becomes "hotter" and "hotter" at longer and longer wavelengths.Finally, we argue that this nonequilibrium behavior can be observed in cavity arrays with cross-Kerr nonlinearities.
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=4d_c=4dc​=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≈0.020ω\approx0.020ω≈0.020 in d=3d=3d=3, implying strong finite-size corrections. An all-orders modified Harris criterion ν>2/(d+z−2)ν>2/(d+z-2)ν>2/(d+z−2) confirms that the BIM fixed point governs criticality in d=3d=3d=3, with Ising universality recovered only at d=2d=2d=2.
We now begin our exposition of universal dynamic scaling behavior that emerges under non-equilibrium conditions. We are mainly concerned with the relaxational models A and B here, but also touch on other dynamic universality classes. First, we consider non-equilibrium critical relaxation from a disordered initial state, and compute the associated universal scaling exponents within the dynamic renormalization group framework. Related phenomena in the early-time regime, prior to reaching temporally translationally invariant asymptotics, are ‘aging’ and a non-equilibrium fluctuation-dissipation ratio, both of which have enjoyed prominence in the literature on ‘glassy’ kinetics. In this context, we derive the critical initial slip exponent for the order parameter, and briefly discuss interesting persistence properties. Second, we explore the long-time scaling laws in phase ordering kinetics and coarsening for the relaxational models, following a fast temperature ‘quench’ into the ordered phase, in systems with either non-conserved or conserved order parameters. Aside from a few explicit computations in the spherical model limit (wherein the number of order parameter components n → ∞), we largely employ phenomenological considerations and scaling theory, stressing the importance of topological defects for energy dissipation during coarsening. Next we address the question how violations of Einstein's relation that links relaxation coefficients and noise correlations might affect the asymptotic dynamic critical scaling behavior. Whereas in systems with non-conserved order parameter typically the equilibrium scaling laws are recovered in the vicinity of the critical point, in contrast genuinely novel universality classes may emerge in the case of a conserved parameter, provided it is driven out of equilibrium in a spatially anisotropic fashion.
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 λλλ. 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 λ2λ^2λ2. Our results demonstrate the remarkable robustness of two-dimensional Ising criticality against this class of directional interactions.
We investigate the critical dynamics of the n-component relaxational models C and D, which incorporate the coupling of a nonconserved and conserved order parameter S, respectively, to the conserved energy density rho, under nonequilibrium conditions by means of the dynamical renormalization group. Detailed balance violations can be implemented isotropically by allowing for different effective temperatures for the heat baths coupling to the slow modes. In the case of model D with conserved order parameter, the energy density fluctuations can be integrated out, leaving no trace of the nonequilibrium perturbations in the asymptotic regime. For model C with scalar order parameter, in equilibrium governed by strong dynamic scaling (z(S)=z(rho)), we find no genuine nonequilibrium fixed point either. The nonequilibrium critical dynamics of model C with n=1 thus follows the behavior of other systems with nonconserved order parameter wherein detailed balance becomes effectively restored at the phase transition. For n> or =4, the energy density generally decouples from the order parameter. However, for n=2 and n=3, in the weak dynamic scaling regime (z(S)< or =z(rho)) entire lines of genuine nonequilibrium model C fixed points emerge to one-loop order, which are characterized by continuously varying static and dynamic critical exponents. Similarly, the nonequilibrium model C with spatially anisotropic noise and n<4 allows for continuously varying exponents, yet with strong dynamic scaling. Subjecting model D to anisotropic nonequilibrium perturbations leads to genuinely different critical behavior with softening only in subsectors of momentum space and correspondingly anisotropic scaling exponents. Similar to the two-temperature model B (randomly driven diffusive systems) the effective theory at criticality can be cast into an equilibrium model D dynamics, albeit incorporating long-range interactions of the uniaxial dipolar or ferroelastic type.