Finite-size scaling and critical exponents in critical relaxation

Nonequilibrium critical dynamics of the two-dimensional Ashkin-Teller model at the Baxter line

We investigate the short-time universal behavior of the two-dimensional Ashkin-Teller model at the Baxter line by performing time-dependent Monte Carlo simulations. First, as preparatory results, we obtain the critical parameters by searching the optimal power-law decay of the magnetization. Thus, the dynamic critical exponents θ_{m} and θ_{p}, related to the magnetic and electric order parameters, as well as the persistence exponent θ_{g}, are estimated using heat-bath Monte Carlo simulations. In addition, we estimate the dynamic exponent z and the static critical exponents β and ν for both order parameters. We propose a refined method to estimate the static exponents that considers two different averages: one that combines an internal average using several seeds with another, which is taken over temporal variations in the power laws. Moreover, we also performed the bootstrapping method for a complementary analysis. Our results show that the ratio β/ν exhibits universal behavior along the critical line corroborating the conjecture for both magnetization and polarization.

Linear relaxation in large two-dimensional Ising models

Critical dynamics in two-dimension Ising lattices up to 2048×2048 is simulated on field-programmable-gate-array- based computing devices. Linear relaxation times are measured from extremely long Monte Carlo simulations. The longest simulation has 7.1×10(16) spin updates, which would take over 37 years to simulate on a general purpose computer. The linear relaxation time of the Ising lattices is found to follow the dynamic scaling law for correlation lengths as long as 2048. The dynamic exponent z of the system is found to be 2.179(12), which is consistent with previous studies of Ising lattices with shorter correlation lengths. It is also found that Monte Carlo simulations of critical dynamics in Ising lattices larger than 512×512 are very sensitive to the statistical correlations between pseudorandom numbers, making it even more difficult to study such large systems.

Generalized dynamic scaling for quantum critical relaxation in imaginary time

We study the imaginary-time relaxation critical dynamics of a quantum system with a vanishing initial correlation length and an arbitrary initial order parameter M0. We find that in quantum critical dynamics, the behavior of M0 under scale transformations deviates from a simple power law, which was proposed for very small M0 previously. A universal characteristic function is then suggested to describe the rescaled initial magnetization, similar to classical critical dynamics. This characteristic function is shown to be able to describe the quantum critical dynamics in both short- and long-time stages of the evolution. The one-dimensional transverse-field Ising model is employed to numerically determine the specific form of the characteristic function. We demonstrate that it is applicable as long as the system is in the vicinity of the quantum critical point. The universality of the characteristic function is confirmed by numerical simulations of models belonging to the same universality class.

Time-dependent Monte Carlo simulations of critical and Lifshitz points of the axial-next-nearest-neighbor Ising model

In this work, we study the critical behavior of second-order points, specifically the Lifshitz point (LP) of a three-dimensional Ising model with axial competing interactions [the axial-next-nearest-neighbor Ising (ANNNI) model], using time-dependent Monte Carlo simulations. We use a recently developed technique that helps us localize the critical temperature corresponding to the best power law for magnetization decay over time: <M>(m(0)=1)~t(-β/νz), which is expected of simulations starting from initially ordered states. We obtain original results for the dynamic critical exponent z, evaluated from the behavior of the ratio F(2)(t)=<M(2)>(m(0)=0)/<M>(2)(m(0)=1)~t(3/z), along the critical line up to the LP. We explore all the critical exponents of the LP in detail, including the dynamic critical exponent θ that characterizes the initial slip of magnetization and the global persistence exponent θ(g) associated with the probability P(t) that magnetization keeps its signal up to time t. Our estimates for the dynamic critical exponents at the Lifshitz point are z=2.34(2) and θ(g)=0.336(4), values that are very different from those of the three-dimensional Ising model (the ANNNI model without the next-nearest-neighbor interactions at the z axis, i.e., J(2)=0), i.e., z≈2.07 and θ(g)≈0.38. We also present estimates for the static critical exponents β and ν, obtained from extended time-dependent scaling relations. Our results for static exponents are in good agreement with previous works.

The two-dimensional site-diluted Ising model: a short-time-dynamics approach

The site-diluted Ising ferromagnet is investigated on a square lattice, within short-time-dynamics numerical simulations, for different site concentrations. The dynamical exponents θ and z are obtained and it is shown that these exponents do depend strongly on the disorder, exhibiting a clear breakdown of universality, characterized by relative variations of nearly 100% in the range of site concentrations investigated. In what concerns the static exponents β and ν, universality is preserved within the error bars.

Nonequilibrium critical relaxation of the order parameter and energy in the two-dimensional ferromagnetic Potts model

The static and dynamic critical properties of the ferromagnetic q -state Potts models on a square lattice with q=2 and 3 are numerically studied via the nonequilibrium relaxation method. The relaxation behavior of both the order parameter and energy as well as that of the second moments are investigated, from which static and dynamic critical exponents can be obtained. We find that the static exponents thus obtained from the relaxation of the order parameter and energy together with the second moments of the order parameter exhibit a close agreement with the exact exponents, especially for the case of the q=2 (Ising) model, when care is taken in the choice of the initial states for the relaxation of the second moments. As for the case of q=3 , the estimates for the static exponents become less accurate, but still exhibit reasonable agreement with the exactly known static exponents. The dynamic critical exponent for the q=2 (Ising) model is estimated from the relaxation of the second moments of the order parameter with mixed initial conditions to give z(q=2) approximately 2.1668(19) .

Criticality of a contact process with coupled diffusive and non-diffusive fields

We investigate the critical behavior of a model with two coupled critical densities, one of which is diffusive. The model simulates the propagation of an epidemic process in a population, which uses the underlying lattice to leave a track of the recent disease history. We determine the critical density of the population above which the system reaches an active stationary state with a finite density of active particles. We also perform a scaling analysis to determine the order parameter, the correlation length, and critical relaxation exponents. We show that the model does not belong to the usual directed percolation universality class and is compatible with the class of directed percolation with diffusive and conserved fields.

Short-time dynamics of polypeptides

The authors study the short-time dynamics of helix-forming polypeptide chains using an all-atom representation of the molecules and an implicit solvation model to approximate the interaction with the surrounding solvent. The results confirm earlier observations that the helix-coil transition in proteins can be described by a set of critical exponents. The high statistics of the simulations allows the authors to determine the exponent values with increased precision and support universality of the helix-coil transition in homopolymers and (helical) proteins.

Short-time dynamics of the helix-coil transition in polypeptides

We study the critical relaxation of the helix-coil transition in all-atom models of polyalanine chains. We show that at the critical temperature the decay of a completely helical conformation can be described by scaling relations that allow us estimating the pertinent critical exponents. The present approach opens a new way for characterizing transitions in proteins and may lead to a better understanding of their folding mechanism. An application of the technique to the 34-residue human parathyroid fragment PTH(1-34) supports universality of the helix-coil transition in homopolymers and (helical) proteins.

Continuous majority-vote model

We introduce a kinetic irreversible XY model and investigate its dynamic critical behavior through short-time Monte Carlo simulations on square lattices with periodic boundary conditions, starting from an ordered state. We find evidence that this system exhibits a Kosterlitz-Thouless-like phase for low values of the noise parameter. We present results for the correlation function exponent eta for several noise values. We also find that the dynamic critical exponent z is in agreement with the value expected for local update Monte Carlo rules.

Short-time dynamics for the spin-3/2 Blume-Capel model

We employed Monte Carlo simulations and short-time dynamic scaling to determine the static and dynamic critical exponents for the generalized two-dimensional Blume-Capel model of spin-3/2. We showed that the critical behavior at the second-order phase-transition line between the paramagnetic and ferromagnetic phases is in the same universality class of the two-dimensional Ising model. However, at the double critical end point, which is present in the phase diagram of the model, the critical exponent beta , associated to the order parameter, is different from that of the Ising model.

Universality and scaling study of the critical behavior of the two-dimensional Blume-Capel model in short-time dynamics

In this paper we study the short-time behavior of the Blume-Capel model at the tricritical point as well as along the second order critical line. Dynamic and static exponents are estimated by exploring scaling relations for the magnetization and its moments at an early stage of the dynamic evolution. Our estimates for the dynamic exponents, at the tricritical point, are z=2.215(2) and theta=-0.53(2).

Short-time dynamics of a metamagnetic model

We studied a layered metamagnetic Ising model with competing ferromagnetic and antiferromagnetic interactions on a square lattice. The model is formed of ferromagnetic chains coupled by an antiferromagnetic interaction. Using Monte Carlo simulations we have determined the phase diagram of the model, which exhibits a tricritical point. By exploring the short-time scaling dynamics, we have found the dynamic and static critical exponents along the continuous transition line between the antiferromagnetic and paramagnetic phases.

Short-time dynamics and magnetic critical behavior of the two-dimensional random-bond potts model

The critical behavior in the short-time dynamics for the random-bond Potts ferromagnet in two dimensions is investigated by short-time dynamic Monte Carlo simulations. The numerical calculations show that this dynamic approach can be applied efficiently to study the scaling characteristic, which is used to estimate the critical exponents straight theta,beta/nu, and z, for quenched disordered systems from the power-law behavior of the kth moments of magnetization.

Short-time critical dynamics for the transverse ising model

We have analyzed the short-time critical behavior for the one-dimensional quantum transverse Ising model through Monte Carlo simulations at zero temperature. We used the scaling relation for the dynamics at the early time stages in order to obtain the static critical exponents (beta,nu) and the dynamical critical exponent z for this model. While the values found for the static exponents are in agreement with the exact ones, here, the dynamical critical exponent is found for a quantum spin model.

Critical behavior of a two-species reaction-diffusion problem

We present a Monte Carlo study in dimension d=1 of the two-species reaction-diffusion process A+B-->2B and B-->A. Below a critical value rho(c) of the conserved total density rho the system falls into an absorbing state without B particles. Above rho(c) the steady state B particle density rho(st)(B) is the order parameter. This system is related to directed percolation but in a different universality class identified by Kree et al. [Phys. Rev. A 39, 2214 (1989)]. We present an algorithm that enables us to simulate simultaneously the full range of densities rho between zero and some maximum density. From finite-size scaling we obtain the steady state exponents beta=0.435(10), nu=2.21(5), and eta=-0.606(4) for the order parameter, the correlation length, and the critical correlation function, respectively. Independent simulation indicates that the critical initial increase exponent takes the value straight theta(')=0.30(2), in agreement with the theoretical relation straight theta(')=-eta/2 due to Van Wijland et al. [Physica A 251, 179 (1998)].

Universal behavior in an irreversible model with C(3v) symmetry

We analyze the critical behavior of a two-dimensional irreversible cellular automaton whose dynamic rules are invariant under the symmetry operations of the point group C3v. We study the dynamical phase transition that takes place in the model and obtain the static and short-time critical exponents by the use of Monte Carlo simulations. Our results indicate that the present model is in the same universality class as the three-state Potts model.

Monte Carlo simulation of an antiferromagnetic Ising model at two competing temperatures

We consider a two-dimensional antiferromagnet Ising system interacting with a heat bath at temperature T. The dynamics of the system is simulated by two competing stochastic processes: the two-spin-exchange Kawasaki kinetics at temperature T>0 and the one-spin-flip Glauber dynamics at T(G)-->0(-), which mimics the increase of the energy of the system. These two processes have probabilities 1-p and p, respectively. Monte Carlo simulations were employed to determine the phase diagram for the stationary states of the model and the corresponding critical exponents. Contrary to the ferromagnetic case, the phase diagram obtained does not exhibit the phenomenon of self-organization: for any nonzero value of the competing parameter p, and for any value of T, the only stationary phase which remains is the ferromagnetic one. At the phase transition between the antiferromagnetic and paramagnetic phases, at p=0, the values found for the critical exponents agree with those of the corresponding equilibrium Ising model.