No abstract

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. ...

With Monte Carlo methods we investigate the dynamic relaxation of the fully frustrated XY model in two dimensions below or at the Kosterlitz-Thouless phase transition temperature. Special attention is drawn to the sublattice structure of the dynamic evolution. Short-time scaling behaviour is found and universality is confirmed. The critical exponent θ is measured for different temperature and with different algorithms. PACS: 64.60. Ht, 75.10.Hk, 02.70.Lq, 82.20.Mj Typeset using REVT E X * Work supported in part by the Deutsche Forschungsgemeinschaft; DFG Schu 95/9-1 † On leave of absence from Sichuan

PACS. 75.10Jm -Quantized spin models. PACS. 75.40Mg -Numerical simulation studies. PACS. 64.60Ht -Dynamic critical phenomena.Abstract. -We present a dynamic Monte Carlo study of the Kosterlitz-Thouless phase transition for the spin-1/2 quantum XY model in two-dimensions. The short-time dynamic scaling behaviour is found and the dynamical exponent θ, z and the static exponent η are determined at the transition temperature.The existence and the nature of the phase transition in the quantum XY model is a long-standing problem. In 1973, Kosterlitz and Thouless explained what is now called the Kosterlitz-Thouless (KT) phase transition in the classical XY model, in terms of topological order, characterized by an exponentially divergent spatial correlation length and susceptibility [1]. General universality arguments suggest that the same KT transition may occur in the quantum XY model [2,3,4]. However, a quantitative determination of the critical exponents and the transition temperature with Monte Carlo methods is very difficult since one suffers from critical slowing down.Due to the exponential divergence of the correlation length at the transition temperature T KT and the fact that the system remains critical below T KT , numerical simulations of critical systems with a KT transition are more difficult than those with a second order phase transition. The situation is even more severe for quantum spin systems with a KT transition [4,5]. A standard approach to the quantum XY model is the quantum Monte Carlo (QMC) method where the Suzuki-Trotter transformation is used to transform the quantum system to a classical one [6,7]. For the 2-dimensional spin-1/2 quantum XY model, Loh, Scalapino and Grant first estimated the KT transition temperature between T KT = 0.4 − 0.5 [3]. The authors of refs. [4,5] improved the results with extensive QMC simulations on lattices up to 128 2 . With a loop-cluster algorithm [9,10] which is often more efficient than the conventional QMC methods, Harada and Kawashima recently measured the helicity modulus for temperatures between T = 0.2 − 0.60, and determined rather accurately the transition temperature T KT = 0.3423(2) on the lattice 64 2 [11]. In this loop-cluster algorithm a loop is formed by spin-pairs on the interacting plaquettes and all spins on the loop are flipped Typeset using EURO-T E X

With Monte Carlo simulations, we investigate short-time critical dynamics of the three-dimensional anti-ferromagnetic 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 auto-correlation, 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 non-zero m s is different.