We investigate scaling and universality in nonequilibrium spin correlation functions in the presence of uncorrelated noise. In the absence of noise, spin correlation functions exhibit a crossover from monotonic decay at fast sweep velocities to oscillatory behavior at slow sweeps. We show that, under a stochastically driven field, the critical sweep velocity at which the spin correlation functions undergo an abrupt change decreases with increasing noise strength and scales linearly with the square of the noise intensity. Remarkably, when the noise intensity and sweep velocity are comparable, the excitation probability becomes locked to pk = 1/2 over a finite momentum window, signaling the emergence of noise-induced maximally mixed modes. This gives rise to a highly oscillatory region in the dynamical phase diagram, whose threshold sweep velocity increases with noise and likewise exhibits quadratic scaling with the noise strength. Finally, we identify a universal scaling function under which all boundary sweep-velocity curves collapse onto a single universal curve.
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.
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.