The symmetric two-layer Ising model (TLIM) is studied by the corner transfer matrix renormalisation group method. The critical points and critical exponents are calculated. It is found that the TLIM belongs to the same universality class as the Ising model. The shift exponent is calculated to be 1.773, which is consistent with the theoretical prediction 1.75 with 1.3% deviation.
Dynamic relaxation of the XY model and fully frustrated XY model quenched from an initial ordered state to the critical temperature or below is investigated with Monte Carlo methods. Universal power law scaling behaviour is observed. The dynamic critical exponent $z$ and the static exponent $\eta$ are extracted from the time-dependent Binder cumulant and magnetization. The results are competitive to those measured with traditional methods
With Monte Carlo simulations we investigate the nonequilibrium critical dynamic behavior of the two-dimensional random-bond Ising model. Based on the short-time dynamic scaling form, we estimate all the static and dynamic exponents from dynamic processes starting with both disordered and ordered states. Corrections to scaling are carefully considered.
Using Monte Carlo methods, the short-time dynamic scaling behaviour of two-dimensional critical XY systems is investigated. Our results for the XY model show that there exists universal scaling behaviour already in the short-time regime, but the values of the dynamic exponent z differ for different initial conditions. For the fully frustrated XY model, power law scaling behaviour is also observed in the short-time regime. However, a violation of the standard scaling relation between the exponents is detected. PACS: 64.60.Ht, 02.70.Lq, 75.10.Hk, 64.60.Fr Keywords: dynamic critical phenomena, Monte Carlo methods, classical spin systemsRecently much progress has been made in critical dynamics. It was discovered that universal scaling behaviour may emerge already in the macroscopic shorttime regime [1,2,3,4,5,6]. Extensive Monte Carlo simulations show that the short-time dynamic scaling is not only conceptually interesting but also practically important, e.g. it leads to new ways for the determination of the critical exponents and the critical temperature [6,7,8,9,10,11,3].For critical systems with second order phase transitions, comprehensive understanding has been achieved. For a relaxational dynamic process of model A starting from an ordered state, the scaling form is given, e.g. for the k-th moment of the magnetization at the critical temperature, by [6,3] where t is the time variable, L is the lattice size, η and z represent the standard static and dynamic critical exponents. This scaling form looks similar to that in the long-time regime but it is now assumed to hold also in the macroscopic short-time regime after a microscopic time scale t mic . For a relaxation process starting from a disordered state with small or zero initial magnetization, the scaling form for the k-th moment at the critical temperature is found to beHere it is important that a new independent critical exponent x 0 has been introduced to describe the dependence of the scaling behaviour on the initial magnetization. The exponent x 0 is the scaling dimension of the global magnetization M(t) and also of the magnetization density. Therefore, even if m 0 = 0, x 0 still enters observables related to the initial conditions. For example, for sufficiently large lattice size the auto-correlation has a power law behaviourwith s i being a spin variable. The exponent λ is related to x 0 by [12]In general, for arbitrary initial magnetization m 0 between 0 and 1 a generalized scaling form can be written down. Renormalization group calculations for the O(N) vector model show that the static exponents η and the dynamic exponent z in the scaling forms (2) and (3) take the same values as in equilibrium or in the long-time regime of the dynamic evolution where they are defined. This is also confirmed by Monte Carlo simulations for various critical magnetic systems with second order phase 1 transitions. Furthermore, numerical results indicate in good accuracy that the values of the exponents η and z are independent of the initial conditions, i.e. ...
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