We present extensive numerical simulations of a family of non-equilibrium Potts models with absorbing states that allows for a variety of scenarios, depending on the number of spin states and the range of the spin-spin interactions. These scenarios encompass a voter critical point, a discontinuous transition as well as the presence of both a symmetry-breaking phase transition and an absorbing phase transition. While we also investigate standard steady-state quantities, our emphasis is on time-dependent quantities that provide insights into the transient properties of the models.
A nonequilibrium Potts-like model with $q$ absorbing states is studied using Monte Carlo simulations. In two dimensions and $q=3$ the model exhibits a discontinuous transition. For the three-dimensional case and $q=2$ the model exhibits a continuous, transition with $β=1$ (mean-field). Simulations are inconclusive, however, in the two-dimensional case for $q=2$. We suggest that in this case the model is close to or at the crossing point of lines separating three different types of phase transitions. The proposed phase diagram in the $(q,d)$ plane is very similar to that of the equilibrium Pott…
We study the non-equilibrium steady states that emerge when two interacting three-dimensional Potts blocks slide on each other. As at equilibrium the Potts model exhibits different types of phase transitions for different numbers $q$ of spin states, we consider the following three cases: $q=2$ (i.e. the Ising case), $q=3$, and $q=9$, which at equilibrium yield respectively a second order phase transition, a weak first order transition and a strong first order transition. In our study we focus on the anisotropic character of the steady states that result from the relative motion and discuss the…
We describe some of the recent results obtained for models with absorbing states. First, we present the nonequilibrium absorbing-state Potts model and discuss some of the factors that might affect the critical behaviour of such models. In particular we show that in two dimensions the further neighbour interactions might split the voter critical point into two critical points. We also describe some of the results obtained in the context of synchronization of chaotic dynamical systems. Moreover, we discuss the relation of the synchronization transition with some interfacial models.
We study a one-dimensional, nonequilibrium Potts-like model which has $q$ symmetric absorbing states. For $q=2$, as expected, the model belongs to the parity conserving universality class. For $q=3$ the critical behaviour depends on the dynamics of the model. Under a certain dynamics it remains generically in the active phase, which is also the feature of some other models with three absorbing states. However, a modified dynamics induces a parity conserving phase transition. Relations with branching-annihilating random walk models are discussed in order to explain such a behaviour.
We comment on the article by Fujimoto (1997 J. Phys. A: Math. Gen., Vol. 30, 3779), where the exact equilibrium crystal shape (ECS) in the critical Q-state Potts model on the square lattice was calculated, and its equivalence with ECS in the Ising model was established. We confirm these results, giving their alternative derivation applying the transformation properties of the one-particle dispersion relation in the six-vertex model. It is shown, that this dispersion relation is identical with that in the Ising model on the square lattice.
We derive an exact renormalization group recursion relation for the Loschmidt amplitude of the quantum $Q$-state clock model and the quantum $Q$-state Potts model in one dimension. The renormalization group flow is discussed in detail. The fixed-points of the renormalization group flow are found to be complex in general. These fixed-points control the dynamical phases of the two models, giving rise to non-analyticities in its Loschmidt rate function, for both the pure and the disordered system. For the quench protocols studied, dynamical quantum phase transitions are found to occur in the cloc…
Using the theory of large deviations, we analyze the phase transition structure of the Curie-Weiss-Potts spin model, which is a mean-field approximation to the Potts model. This analysis is carried out both for the canonical ensemble and the microcanonical ensemble. Besides giving explicit formulas for the microcanonical entropy and for the equilibrium macrostates with respect to the two ensembles, we analyze ensemble equivalence and nonequivalence at the level of equilibrium macrostates, relating these to concavity and support properties of the microcanonical entropy. The Curie-Weiss-Potts mo…
Spinodal and Equilibrium Global Phase Diagram of the d=3 Merged Potts-Cubic-Clock Model: First-Order Equilibrium and Second-Order Spinodal Boundaries with Hidden Topologies from Renormalization-Group Theory
Recently some two-dimensional models with double symmetric absorbing states were shown to share the same critical behaviour that was called the voter universality class. We show, that for an absorbing-states Potts model with finite but further than nearest neighbour range of interactions the critical point is splitted into two critical points: one of the Ising type, and the other of the directed percolation universality class. Similar splitting takes place in the three-dimensional nearest-neighbour model.
We consider a non-equilibrium three-state model whose dynamics is Markovian and displays the same symmetry as the three-state Potts model, i.e., the transition rates are invariant under the permutation of the states. Unlike the Potts model, detailed balance is in general not satisfied. The aging and the stationary properties of the model defined on a square lattice are obtained by means of large-scale Monte Carlo simulations. We show that the phase diagram presents a critical line, belonging to the three-state Potts universality class, that ends at a point whose universality class is that of t…
We study the quench dynamics of the $q$ Potts model on different bi/tri-dimensional lattice topologies. In particular we are interested in instantaneous quenches from $T_i \rightarrow \infty$ to $T \leq T_s$, where $T_s$ is the (pseudo)-spinodal temperature. The goal is to explain why, in the large-$q$ limit, the low-temperature dynamics freezes on some lattices while, on others, the equilibrium configuration is easily reached. The cubic ($3d$) and the triangular ($2d$) lattices are analysed in detail. We show that the dynamics blocks when lattices have acyclic \textit{unitary structures} whil…
We study the metastable equilibrium properties of the Potts model with heat-bath transition rates using a novel expansion. The method is especially powerful for large number of state spin variables and it is notably accurate in a rather wide range of temperatures around the phase transition.
We consider magnetic friction between two systems of $q$-state Potts spins which are moving along their boundaries with a relative constant velocity $v$. Due to the interaction between the surface spins there is a permanent energy flow and the system is in a steady state which is far from equilibrium. The problem is treated analytically in the limit $v=\infty$ (in one dimension, as well as in two dimensions for large-$q$ values) and for $v$ and $q$ finite by Monte Carlo simulations in two dimensions. Exotic nonequilibrium phase transitions take place, the properties of which depend on the type…
This work is a contribution to the study of universality in out-of-equilibrium lattice models undergoing a second-order phase transition at equilibrium. The experimental protocol that we have chosen is the following: the system is prepared in its high-temperature phase and then quenched at the critical temperature $T_c$. We investigated by mean of Monte Carlo simulations two quantities that are believed to take universal values: the exponent $λ/z$ obtained from the decay of autocorrelation functions and the asymptotic value $X_\infty$ of the fluctuation-dissipation ratio $X(t,s)$. This protoco…
We investigate the non-equilibrium dynamics of the 2D Potts model on the square lattice after a quench below the discontinuous transition point. By means of numerical simulations of systems with q =12,24 and 48 we observe the onset of a stationary regime below the temperature-driven transition, in a temperature interval decreasing with the system size and increasing with q. These results obtained dynamically agree with those obtained from the analytical continuation of the free energy (Meunier and Morel, 2000), from which metastability in the 2D Potts model results to be a finite size effect.
Understanding nonequilibrium systems and the consequences of irreversibility for the system's behavior as compared to the equilibrium case, is a fundamental question in statistical physics. Here, we investigate two types of nonequilbrium phase transitions, a second-order and an infinite-order phase transition, in a prototypical q-state vector Potts model which is driven out of equilibrium by coupling the spins to heat baths at two different temperatures. We discuss the behavior of the quantities that are typically considered in the vicinity of (equilibrium) phase transitions, like the specific…
As is known, at the Gibbs-Boltzmann equilibrium, the mean-field $q$-state Potts model with a ferromagnetic coupling has only a first order phase transition when $q\geq 3$, while there is no phase transition for an antiferromagnetic coupling. The same equilibrium is asymptotically reached when one considers the continuous time evolution according to a Glauber dynamics. In this paper we show that, when we consider instead the Potts model evolving according to a discrete-time dynamics, the Gibbs-Boltzmann equilibrium is reached only when the coupling is ferromagnetic while, when the coupling is a…
We studied the non-equilibrium dynamics of the q-state Potts model in the square lattice, after a quench to sub-critical temperatures. By means of a continuous time Monte Carlo algorithm (non-conserved order parameter dynamics) we analyzed the long term behavior of the energy and relaxation time for a wide range of quench temperatures and system sizes. For q>4 we found the existence of different dynamical regimes, according to quench temperature range. At low (but finite) temperatures and very long times the Lifshitz-Allen-Cahn domain growth behavior is interrupted with finite probability when…
Based on quasi-stationary distribution ideas, a general finite size scaling theory is proposed for discontinuous nonequilibrium phase transitions into absorbing states. Analogously to the equilibrium case, we show that quantities such as, response functions, cumulants, and equal area probability distributions, all scale with the volume, thus allowing proper estimates for the thermodynamic limit. To illustrate these results, five very distinct lattice models displaying nonequilibrium transitions -- to single and infinitely many absorbing states -- are investigated. The innate difficulties in an…
The critical behavior of two-dimensional stochastic lattice gas models with C3v symmetry is analyzed. We study the cumulants of the order parameter for the three state (equilibrium) Potts model and for two irreversible models whose dynamic rules are invariant under the symmetry operations of the point group C3v. By means of extensive numerical analysis of the phase transition we show that irreversibility does not affect the critical behavior of the systems. In particular we find that the Binder reduced fourth order cumulant takes a universal value U* which is the same for the three state Potts…
We have studied the ordering of the q-colours Potts model in two dimensions on a square lattice. On the basis of our observations we propose that if q is large enough the system is not able to break global and local null magnetisation symmetries at zero temperature: when q<4 the system forms domains with a size proportional to the system size while for q>4 it relaxes towards a non-equilibrium phase with energy larger than the ground state energy, in agreement with the previous findings of De Oliveira et al. (M. J. de Oliveira, A. Petri, T. Tome, Europhys. Lett., 65, 20 (2004)).
We investigate the critical properties of a one dimensional stochastic lattice model with n (permutation symmetric) absorbing states. We analyze the cases with $n \leq 4$ by means of the non-hermitian density matrix renormalization group. For n=1 and n=2 we find that the model is respectively in the directed percolation and parity conserving universality class, consistent with previous studies. For n=3 and n=4, the model is in the active phase in the whole parameter space and the critical point is shifted to the limit of one infinite reaction rate. We show that in this limit the dynamics of th…
Using large-scale numerical simulations we studied the kinetics of the 2d q-Potts model for q > 4 after a shallow subcritical quench from a high-temperature homogeneous configuration. This protocol drives the system across a first-order phase transition. The initial state is metastable after the quench and, for final temperatures close to the critical one, the system escapes from it via a multi-nucleation process. The ensuing relaxation towards equilibrium proceeds through coarsening with competition between the equivalent ground states. This process has been analyzed for different choices of …
We present an exact renormalization group analysis of the Loschmidt amplitude of the quantum $N$-state Potts chain with random quench-disordered nearest neighbor bonds, under the extreme dynamical quantum quench. We prove that the phase transition of the Loschmidt rate function remains sharp in general. For typical bond distributions, the phase transition is found to be a linear-cusp, as in the pure model. For some special discrete bond distributions, however, the rate function exhibits logarithmic divergences. These singularities are due to the competition between the non-critical dynamical p…
The properties of the absorbing states of non-equilibrium models belonging to the conserved directed percolation universality class are studied. We find that at the critical point the absorbing states are hyperuniform, exhibiting anomalously small density fluctuations. The exponent characterizing the fluctuations is measured numerically, a scaling relation to other known exponents is suggested, and a new correlation length relating to this ordering is proposed. These results may have relevance to photonic band-gap materials.
Recently, a non-trivial relation between the quasi-particle spectrum and entanglement entropy production was discovered in non-integrable quenches in the paramagnetic Ising quantum spin chain. Here we study the dynamics of analogous quenches in the quantum Potts spin chain. Tuning the parameters of the system, we observe a sudden increase in the entanglement production rate, which is shown to be related to the appearance of new quasiparticle excitations in the post-quench spectrum. Our results demonstrate the generality of the effect and support its interpretation as the non-equilibrium versio…
We present a general scaling theory for the surface critical behavior of non-equilibrium systems with phase transitions into absorbing states. The theory allows for two independent surface exponents which satisfy generalized hyperscaling relations. As an application we study a generalized version of directed percolation with two absorbing states. We find two distinct surface universality classes associated with inactive and reflective walls. Our results indicate that the exponents associated with these two surface universality classes are closely connected.
Conformal Field Theories (CFTs) have been used extensively to understand the physics of critical lattice models at equilibrium. However, the applicability of CFT calculations to the behavior of the lattice systems in the out-of-equilibrium setting is not entirely understood. In this work, we compare the CFT results of the evolution of the entanglement spectrum after a quantum quench with numerical calculations of the entanglement spectrum of the three-state Potts model using matrix product state simulations. Our results lead us to conjecture that CFT does not describe the entanglement spectrum…
One of the most impressive features of continuous phase transitions is the concept of universality, that allows to group the great variety of different critical phenomena into a small number of universality classes. All systems belonging to a given universality class have the same critical exponents, and certain scaling functions become identical near the critical point. It is the aim of this work to demonstrate the usefulness of universal scaling functions for the analysis of non-equilibrium phase transitions. In order to limit the coverage of this article, we focus on a particular class of n…
Using inverse Ising inference we show that the absorbing states of the one-dimensional Manna model can be described by an equilibrium model with an emergent interaction displaying short-ranged repulsion and long-ranged attraction. As the model approaches the critical point the interaction becomes purely repulsive, decaying as $r^{-α}$ and we conjecture the exact value $α=1/2$, suggesting density fluctuations decay as $r^{-3/2}$. We present a simple Gaussian field theory for the long distance behaviour of critical absorbing states and discuss implications for the Manna universality class.
We construct a three-color Potts-like model for the graphene zigzag edge reconstructed with Stone-Wales carbon rings, in order to study its thermal equilibrium properties. We consider two cases which have different ground-states: the edge with non-passivated dangling carbon bonds and the edge fully passivated with hydrogen. We study the concentration of defects perturbing the ground-state configuration as a function of the temperature. The defect concentration is found to be exponentially dependent on the effective parameters that describe the model at all temperatures. Moreover, we analytical…
We investigate the phase transitions of the $q$-state Brownian Potts model in two dimensions (2d) comprising Potts spins that diffuse like Brownian particles and interact ferromagnetically with other spins within a fixed distance. With extensive Monte Carlo simulations we find a continuous phase transition from a paramagnetic to a ferromagnetic phase even for $q>4$. This is in sharp contrast to the existence of a discontinuous phase transition in the equilibrium $q$-state Potts model in 2d with $q>4$. We present detailed numerical evidence for a continuous phase transition and argue that diffu…
We investigate the phase diagram of a three-component system of particles on a one-dimensional filled lattice, or equivalently of a one-dimensional three-state Potts model, with reflection asymmetric mean field interactions. The three types of particles are designated as $A$, $B$, and $C$. The system is described by a grand canonical ensemble with temperature $T$ and chemical potentials $Tλ_A$, $Tλ_B$, and $Tλ_C$. We find that for $λ_A=λ_B=λ_C$ the system undergoes a phase transition from a uniform density to a continuum of phases at a critical temperature $\hat T_c=(2π/\sqrt3)^{-1}$. For othe…
We elucidate a long-standing puzzle about the non-equilibrium universality classes describing self-organized criticality in sandpile models. We show that depinning transitions of linear interfaces in random media and absorbing phase transitions (with a conserved non-diffusive field) are two equivalent languages to describe sandpile criticality. This is so despite the fact that local roughening properties can be radically different in the two pictures, as explained here. Experimental implications of our work as well as promising paths for future theoretical investigations are also discussed.
We investigate the p-state mean-field model of the Potts glass ($2\le p \le 4$) below the continuous phase transition to a glassy phase. We find that apart from a solution with a first hierarchical level of replica-symmetry breaking (1RSB), locally stable close to the transition point, there is a continuous full replica-symmetry breaking (FRSB) solution. The latter is marginally stable and has a higher free energy than the former. We argue that the true equilibrium is reached only by FRSB, being globally thermodynamically homogeneous, whereas 1RSB is only locally homogeneous.
A framework is presented for carrying out simulations of equilibrium systems in the microcanonical ensemble using annealing in an energy ceiling. The framework encompasses an equilibrium version of simulated annealing, population annealing and hybrid algorithms that interpolate between these extremes. These equilibrium, microcanonical annealing algorithms are applied to the thermal first-order transition in the 20-state, two-dimensional Potts model. All of these algorithms are observed to perform well at the first-order transition though for the system sizes studied here, equilibrium simulated…
We present an extension of the density-functional theory (DFT) formalism for lattice gases to systems with internal degrees of freedom. In order to test approximations commonly used in DFT approaches, we investigate the statics and dynamics of occupation (density) profiles in the one-dimensional Potts model. In particular, by taking the exact functional for this model we can directly evaluate the quality of the local equilibrium approximation used in time-dependent density-functional theory (TDFT). Excellent agreement is found in comparison with Monte Carlo simulations. Finally, principal limi…
Reexamining algebraic curves found in the eight-vertex model, we propose an asymptotic form of the correlation functions for off-critical systems possessing rotational and mirror symmetries of the square lattice, i.e., the $C_{4v}$ symmetry. In comparison with the use of the Ornstein-Zernike form, it is efficient to investigate the correlation length with its directional dependence (or anisotropy). We investigate the $Q$-state Potts model on the square lattice. Monte Carlo (MC) simulations are performed using the infinite-size algorithm by Evertz and von der Linden. Fitting the MC data with th…
Stochastic processes with absorbing states feature remarkable examples of non-equilibrium universal phenomena. While a broad understanding has been progressively established in the classical regime, relatively little is known about the behavior of these non-equilibrium systems in the presence of quantum fluctuations. Here we theoretically address such a scenario in an open quantum spin model which in its classical limit undergoes a directed percolation phase transition. By mapping the problem to a non-equilibrium field theory, we show that the introduction of quantum fluctuations stemming from…
Critical behaviour of a nearly critical system, subjected to vivid turbulent mixing, is studied by means of the field theoretic renormalization group. Namely, relaxational stochastic dynamics of a non-conserved order parameter of the Ashkin-Teller-Potts model, coupled to a random velocity field with prescribed statistics, is considered. The mixing is modelled by Kraichnan's rapid-change ensemble: time-decorrelated Gaussian velocity field with the power-like spectrum $\propto k^{-d-ξ}$. It is shown that, depending on the symmetry group of the underlying Potts model, the degree of compressibilit…
We introduce a discrete-time quantum dynamics on a two-dimensional lattice that describes the evolution of a $1+1$-dimensional spin system. The underlying quantum map is constructed such that the reduced state at each time step is separable. We show that for long times this state becomes stationary and displays a continuous phase transition in the density of excited spins. This phenomenon can be understood through a connection to the so-called Domany-Kinzel automaton, which implements a classical non-equilibrium process that features a transition to an absorbing state. Near the transition dens…
The effect of quenched disorder on non-equilibrium phase transitions in the directed percolation universality class is studied by a strong disorder renormalization group approach and by density matrix renormalization group calculations. We show that for sufficiently strong disorder the critical behaviour is controlled by a strong disorder fixed point and in one dimension the critical exponents are conjectured to be exact: β=(3-\sqrt{5})/2 and ν_\perp=2. For disorder strengths outside the attractive region of this fixed point, disorder dependent critical exponents are detected. Existing numeric…
In this paper we study the critical behaviour of the fully-connected p-colours Potts model at the dynamical transition. In the framework of Mode Coupling Theory (MCT), the time autocorrelation function displays a two step relaxation, with two exponents governing the approach to the plateau and the exit from it. Exploiting a relation between statics and equilibrium dynamics which has been recently introduced, we are able to compute the critical slowing down exponents at the dynamical transition with arbitrary precision and for any value of the number of colours p. When available, we compare our…
At non-equilibrium phase transitions into absorbing (trapped) states, it is well known that the directed percolation (DP) critical scaling is shared by two classes of models with a single (S) absorbing state and with infinitely many (IM) absorbing states. We study the crossover behavior in one dimension, arising from a considerable reduction of the number of absorbing states (typically from the IM-type to the S-type DP models), by following two different (excitatory or inhibitory) routes which make the auxiliary field density abruptly jump at the crossover. Along the excitatory route, the syst…
The model of competition between densities of two different species, called predator and prey, is studied on a one dimensional periodic lattice, where each site can be in one of the four states say, empty, or occupied by a single predator, or occupied by a single prey, or by both. Along with the pairwise death of predators and growth of preys, we introduce an interaction where the predators can eat one of the neighboring prey and reproduce a new predator there instantly. The model shows a non-equilibrium phase transition into a unusual absorbing state where predators are absent and the lattice…
We study off-equilibrium behaviors at first-order transitions (FOTs) driven by a time dependence of the temperature across the transition point Tc, such as the linear behavior T(t)/Tc = 1 - t/ts where ts is a time scale. In particular, we investigate the possibility of nontrivial off-equilibrium scaling behaviors in the regime of slow changes, corresponding to large ts, analogous to those arising at continuous transitions, which lead to the so-called Kibble-Zurek mechanism. We consider the 2D Potts models which provide an ideal theoretical laboratory to investigate issues related to FOTs drive…
We present a detailed study of the Equilibriumlike invaded cluster algorithm (EIC), recently proposed as an extension of the invaded cluster (IC) algorithm, designed to drive the system to criticality while still preserving the equilibrium ensemble. We perform extensive simulations on two special cases of the Potts model and examine the precision of critical exponents by including the leading corrections. We show that both thermal and magnetic critical exponents can be obtained with high accuracy compared to the best available results. The choice of the auxiliary parameters of the algorithm is…
