Numerical method to evaluate the dynamical critical exponent

Abstract

A finite-size scaling approach is used to show numerically that dynamical scaling occurs for short and long times independently of the initial conditions. Its main idea is to construct particular quantities scaling as L 0 in the thermodynamic limit L→ϱ, L being the linear size of the system. These are the quantities for which the dynamic scaling occurs for short and long times. This approach is applied to obtain the critical dynamical behavior of two-and three-dimensional ferromagnetic Ising models, subjected to Glauber dynamics. ͓S0163-1829͑97͒00302-0͔Although the static critical properties of classical spin systems are well described by renormalization group theories ͑theoretical framework and calculation procedures͒, 1 the dynamic critical properties 2 are not as well understood. In particular, the value of the dynamic critical exponent z is still an open question even for the two-dimensional Ising model 3-6 when dynamics with local flips of spins are considered. The determination of the exponent z for classical models in different lattice dimensions has been done by using several approaches: field-theoretical dynamical renormalization group methods, 2,7 Monte Carlo simulations, 8-10 renormalization group methods, 11-14 damage spreading, 6,15,16 nonequilibrium relaxation, 17,18 and series expansion. 4,19 For the Ising model the various methods obtain in two dimensions 2.10ϽzϽ2.52 and in three dimensions 1.95ϽzϽ2.35. Usually, some of these methods obtain the z exponent from longtime behavior. This limit is hard to be reached because of the critical slowing down that always appears, except for clusters algorithms. 20 Besides, it is also very hard to obtain good statistics in these procedures. Critical slowing down and poor statistics are among the reasons why different calculations produce so many different values for z.Our objective in this work is to present a finite-size dynamical scaling approach which overcomes the usual difficulties pointed out above. The method is then applied to twoand three-dimensional kinetic Ising models with single spin flips. The method can also be applied to systems with more complex ordering behavior subjected to different dynamics.Recently, a method has been proposed 21 to evaluate the z exponent from short-time behavior. It is based on the scaling relation for the dynamics at early times. 22 In this time regime the magnetization initially grows, characterizing a new universal stage of the relaxation of the magnetization, the so called ''critical initial slip.'' However, it turns out that the initial condition ͑zero magnetization and very short correlation length͒ is essential to obtain the dynamical exponent. This happens because the critical initial slip sets right in after a microscopic time scale and eventually crosses over the long-time regime. The characteristic time associated with the critical initial slip is t 0 ϭm 0Ϫz/x , where m 0 is the initial magnetization and x is a new exponent. If m 0 ϭ0, we have that t 0 →ϱ and the early time scaling overlaps with the expected l...

Universal short-time quantum critical dynamics of finite-size systems

“…In the presence of finite system size, C(τ, L) shows power-law increase with an exponent θz for fixed τ /L z . When M 0 = M sat and g varies, let us consider a quantity Q related to the average sign of the order parameter, defined as [62,63]…”

Finite-Time and Finite-Size Scaling of the Kuramoto Oscillators

“…As the system size becomes larger, the time scale at criticality often diverges together with the correlation length, which gives rise to the problem of the critical slowing down. In order to circumvent this, the finite-size dynamical scaling approach [3,4] has been introduced. The basic assumption in this approach of dynamical scaling is that one can extract critical behavior by observing the temporal relaxation of the system in early times even before reaching equilibrium.…”

“…If we further assume that Q is chosen in such a way that its anomalous dimension is null [4], the scaling form of Q is written as Qðt; L; KÞ ¼ fðtL −z ; ðK − K c ÞL 1=ν Þ. In words, the first scaling variable tL −z describes the competition of the two time scales, the finite observation time t and the relaxation time τ, while the second scaling variable ðK − K c ÞL 1=ν is for the competition of the two length scales, the finite system size L and the correlation length ξ.…”

“…In order to carry out FTFSS, we use the initial condition θ i ðt ¼ 0Þ ¼ 0 for all oscillators and measure the key quantity QðtÞ defined by [4,14,15] QðtÞ ≡ sgn…”

“…It is to be emphasized that the value of QðtÞ is not directly related to the order parameter, but it detects how much initial information at t ¼ 0 is lost as the system approaches equilibrium which is characterized by Q ¼ 0. The advantage of using Q lies in that it does not have the anomalous dimension [4], which allows us to use the simple FTFSS form in Eq. (1).…”