We show that spatial resolved dissipation can act on d-dimensional spin systems in the Ising universality class by qualitatively modifying the nature of their critical points. We consider power-law decaying spin losses with a Lindbladian spectrum closing at small momenta as ∝q^{α}, with α a positive tunable exponent directly related to the power-law decay of the spatial profile of losses at long distances, 1/r^{(α+d)}. This yields a class of soft modes asymptotically decoupled from dissipation at small momenta, which are responsible for the emergence of a critical scaling regime ascribable to the nonunitary counterpart of the universality class of long-range interacting Ising models. For α<1 we find a nonequilibrium critical point ruled by a dynamical field theory described by a Langevin model with coexisting inertial (∼∂_{t}^{2}) and frictional (∼∂_{t}) kinetic coefficients, and driven by a gapless Markovian noise with variance ∝q^{α} at small momenta. This effective field theory is beyond the Halperin-Hohenberg description of dynamical criticality, and its critical exponents differ from their unitary long-range counterparts. Our Letter lays out perspectives for a revision of universality in driven open systems by employing dark states tailored by programmable dissipation.
We show that spatial resolved dissipation can act on d-dimensional spin systems in the Ising universality class by qualitatively modifying the nature of their critical points. We consider power-law decaying spin losses with a Lindbladian spectrum closing at small momenta as ∝q^{α}, with α a positive tunable exponent directly related to the power-law decay of the spatial profile of losses at long distances, 1/r^{(α+d)}. This yields a class of soft modes asymptotically decoupled from dissipation at small momenta, which are responsible for the emergence of a critical scaling regime ascribable to the nonunitary counterpart of the universality class of long-range interacting Ising models. For α<1 we find a nonequilibrium critical point ruled by a dynamical field theory described by a Langevin model with coexisting inertial (∼∂{t}^{2}) and frictional (∼∂{t}) kinetic coefficients, and driven by a gapless Markovian noise with variance ∝q^{α} at small momenta. This effective field theory is beyond the Halperin-Hohenberg description of dynamical criticality, and its critical exponents differ from their unitary long-range counterparts. Our Letter lays out perspectives for a revision of universality in driven open systems by employing dark states tailored by programmable dissipation.
Dimensional correspondences have a long history in critical phenomena. Here, we review the effective dimension approach, which relates the scaling exponents of a critical system in d spatial dimensions with power-law decaying interactions r^{d+σ} to a local system, i.e., with finite-range interactions, in an effective fractal dimension d_{eff}. This method simplifies the study of long-range models by leveraging known results from their local counterparts. While the validity of this approximation beyond the mean-field level has been long debated, we demonstrate that the effective dimension approach, while approximate for non-Gaussian fixed points, accurately estimates the critical exponents of long-range models with an accuracy typically larger than 97%. To do so, we review perturbative renormalization-group (RG) results, extend the approximation's validity using functional RG techniques, and compare our findings with precise numerical data from conformal bootstrap for the two-dimensional Ising model with long-range interactions.
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 theoretically study the dynamics of a transverse-field Ising chain with power-law decaying interactions characterized by an exponent α, which can be experimentally realized in ion traps. We focus on two classes of emergent dynamical critical phenomena following a quantum quench from a ferromagnetic initial state: The first one manifests in the time-averaged order parameter, which vanishes at a critical transverse field. We argue that such a transition occurs only for long-range interactions α≤2. The second class corresponds to the emergence of time-periodic singularities in the return probability to the ground-state manifold which is obtained for all values of α and agrees with the order parameter transition for α≤2. We characterize how the two classes of nonequilibrium criticality correspond to each other and give a physical interpretation based on the symmetry of the time-evolved quantum states.
Abstract The equilibrium and nonequilibrium properties of ferromagnetic systems may be affected by the long-range nature of the coupling interaction. Here we study the phase separation process of a one-dimensional Ising model in the presence of a power-law decaying coupling, with , and we focus on the two-time autocorrelation function . We find that it obeys the scaling form , where is the typical domain size at time t , and where can only be of two types. For , when domain walls diffuse freely, falls in the nearest-neighbour (nn) universality class. Conversely, for , when domain walls dynamics is driven, displays a new universal behavior. In particular, the so-called Fisher-Huse exponent, which characterizes the asymptotic behavior of for , is in the nn universality class ( ) and for .
We consider the d-dimensional transverse-field Ising model with power-law interactions $J/{r}^{d+\ensuremath{\sigma}}$ in the presence of a noisy longitudinal field with zero average. We study the longitudinal-magnetization dynamics of an initial paramagnetic state after a sudden switch-on of both the interactions and the noisy field. While the system eventually relaxes to an infinite-temperature state with vanishing magnetization correlations, we find that two-time correlation functions show aging at intermediate times. Moreover, for times shorter than the inverse noise strength $\ensuremath{\kappa}$ and distances longer than $a{(J/\ensuremath{\kappa})}^{2/\ensuremath{\sigma}}$ with a being the lattice spacing, we find a critical scaling regime of correlation and response functions consistent with the model A dynamical universality class with an initial-slip exponent $\ensuremath{\theta}=1$ and dynamical critical exponent $z=\ensuremath{\sigma}/2$. We obtain our results analytically by deriving an effective action for the magnetization field including the noise in a nonperturbative way. The above scaling regime is governed by a nonequilibrium fixed point dominated by the noise fluctuations.
Understanding the dynamical universality classes of systems with long-range interactions remains a key challenge in statistical physics. In this Letter, we analytically and numerically investigate the non-equilibrium critical dynamics of the one-dimensional spin-1/2 Nagle-Kardar model, which is characterized by the competition between short- and long-range interactions and the presence of a tricritical point. We focus on the slowing-down of the magnetization m at criticality under Glauber dynamics. Starting from the corresponding master equation, we perform a coarse-graining procedure to obtain a Fokker-Planck equation for the macroscopic variables. Then, the asymptotic decay of the magnetization is derived using central manifold theory. We find that m decays as t−1/2 along the critical line and as t−1/4 precisely at the tricritical point. This finding confirms that the dynamical critical exponent is z=2 as for mean-field models, proving that the macroscopic critical dynamics of the Nagle-Kardar model falls within the dynamical universality class of purely relaxational, non-conserved order parameters (model A). While Kardar proved that the equilibrium Curie-Weiss theory extends to Ising models where nearest-neighbor interactions are included, we here show that such result is valid also for critical dynamics. Our work provides the semi-analytical solution for the critical dynamics of a model with mixed-range interactions, assigning its universality class.