We investigate the persistence properties of critical d -dimensional systems relaxing from an initial state with non-vanishing order parameter (e.g., the magnetization in the Ising model), focusing on the dynamics of the global order parameter of a d '-dimensional manifold. The persistence probability P c ( t ) shows three distinct long-time decays depending on the value of the parameter ζ = ( D − 2 + η)/ z which also controls the relaxation of the persistence probability in the case of a disordered initial state (vanishing order parameter) as a function of the codimension D = d − d ' and of the critical exponents η and z . We find that the asymptotic behavior of P c ( t ) is exponential for ζ > 1, stretched exponential for 0 ≤ ζ ≤ 1, and algebraic for ζ < 0. Whereas the exponential and stretched exponential relaxations are not affected by the initial value of the order parameter, we predict and observe a crossover between two different power-law decays when the algebraic relaxation occurs, as in the case d ' = d of the global order parameter. We confirm via Monte Carlo simulations our analytical predictions by studying the magnetization of a line and of a plane of the two- and three-dimensional Ising models, respectively, with Glauber dynamics. The measured exponents of the ultimate algebraic decays are in a rather good agreement with our analytical predictions for the Ising universality class. In spite of this agreement, the expected scaling behavior of the persistence probability as a function of time and of the initial value of the order parameter remains problematic. In this context, the non-equilibrium dynamics of the O( n ) model in the limit and its subtle connection with the spherical model are also discussed in detail. In particular, we show that the correlation functions of the components of the order parameter which are respectively parallel and transverse to its average value within the model correspond to the correlation functions of the local and global order parameters of the spherical model.
We investigate global persistence properties for the non-equilibrium critical dynamics of the randomly diluted Ising model. The disorder-averaged persistence probability c ( t ) of the global magnetization is found to decay algebraically with an exponent θ c that we compute analytically in a dimensional expansion in d = 4 − . Corrections to Markov process are found to occur already at one loop order and θ c is thus a novel exponent characterizing this disordered critical point. Our result is thoroughly compared with Monte Carlo simulations in d = 3, which also include a measurement of the initial slip exponent. Taking carefully into account corrections to scaling, θ c is found to be universal, independent of the dilution factor p along the critical line at T c ( p ), and in good agreement with our one-loop calculation.
We explore the critical properties of the recently discovered finite-time dynamical phase transition in the nonequilibrium relaxation of Ising magnets after a temperature quench. The transition is characterized by a sudden switch in the relaxation dynamics and it occurs at a sharp critical time. While previous works have focused either on mean-field interactions or on investigating the critical time, we analyze the critical fluctuations at the phase transition in the nearest-neighbor Ising model on a square lattice using Monte Carlo simulations. By means of a finite-size scaling analysis, we extract the critical exponents for the transition. In two spatial dimensions, we find that the exponents are consistent with those of the two-dimensional Ising universality class when the system is initially in the vicinity of the critical point. For initial temperatures below the critical one, however, the critical exponents differ from the Ising-exponents, indicating a distinct dynamical critical phenomenon.
We study the dynamics of ordering in ferromagnets via Monte Carlo simulations of the Ising model, employing the Glauber spin-flip mechanism, in space dimensions
d=2 and
3. Results for the persistence probability and the domain growth are discussed for quenches to various temperatures (
Tf) below the critical one (
Tc), from different initial temperatures
Ti≥Tc. In long time limit, for
Ti>Tc, the persistence probability exhibits power-law decay with exponents
θ≃0.22 and
≃0.18 in
d=2 and
3, respectively. For finite
Ti, the early time behavior is a different power-law whose life-time diverges and exponent decreases as
Ti→Tc. The crossover length between the two steps diverges as the equilibrium correlation length.
Ti=Tc is expected to provide a {\it{new universality class}} for which we obtain
θ≃0.035 in
d=2 and
≃0.10 in
d=3. The time dependence of the average domain size
ℓ, however, is observed to be rather insensitive to the choice of
Ti.
The persistence probability P g ( t ) of the global order parameter of a simple ferromagnet undergoing phase ordering kinetics after a quench from a fully disordered state to below the critical temperature, T < T c , is analysed. It is argued that the persistence probability decays algebraically with time in the entire low-temperature phase. For Markov processes, the associated global persistence exponent θ g = (2λ C − d )/(2 z ) is related to the autocorrelation exponent λ C . This relationship is confirmed for phase ordering in the exactly solved 1D Ising model and the d -dimensional spherical model. For the 2D Glauber–Ising model, the temperature-independent estimate θ g = 0.063(2) indicates that the dynamics of the global order parameter is described by a non-Markovian process.
With Monte Carlo simulations, we investigate short-time critical dynamics of the three-dimensional antiferromagnetic Ising model with a globally conserved magnetization m(s) (not the order parameter). From the power law behavior of the staggered magnetization (the order parameter), its second moment and the autocorrelation, we determine all static and dynamic critical exponents as well as the critical temperature. The universality class of m(s)=0 is the same as that without a conserved quantity, but the universality class of nonzero m(s) is different.