No abstract

The universal critical point ratio Q is exploited to determine positions of the critical Ising transition lines on the phase diagram of the Ashkin-Teller model on the square lattice. A leading-order expansion of the ratio Q in the presence of a nonvanishing thermal field is found from finite-size scaling and the corresponding expression is fitted by the accurate perturbative transfer-matrix data calculations for the LϫL square clusters with Lр9. ͓S1063-651X͑97͒04503-0͔ ͑1͒Herein we consider only the nearest neighbor pair interactions on the simple square lattice consisting of NϭL 2 sites with periodic boundary conditions and we assume that J 1 ϭJ 2 ͑isotropic case͒.Wagner ͓3͔ has shown that the AT model is equivalent to the alternated eight vertex model, which has not been solved exactly. Only one critical line in the phase diagram of the isotropic AT model is known exactly thanks to the duality relation found by Fan ͓4͔. For this reason many approximate approaches have been applied for constructing the complete phase diagram: the mean-field approximation ͑MFA͒ ͓5,6͔, mean-field renormalization group ͑MFRG͒ ͓7͔, renormalization group ͑RG͒ ͓8͔, and Monte Carlo renormalization group ͑MCRG͒ ͓9͔. It is the aim of this paper to establish an accurate location of the remaining critical lines.In our approach we exploit finite-size scaling for the ratio of the square of the second moment to the fourth moment of the order parameter M :where ͗•••͘ means thermal average and the index L indicates the linear size of the system (LϫL). In the limit L→ϱ this ratio becomes universal in the critical point ͓10͔and is denoted Q hereafter. Three not exactly known critical lines of the isotropic AT model are believed to belong to the Ising universality class ͓5,11͔. Here it is assumed that these lines correspond to the Ising-like continuous transitions with the order parameter M ϭ͚ iϭ1 N S i i . A scaling formula for Q L can be derived starting from the finite-size scaling relation for the singular part of the free energy for the square Ising model ͓12͔.where A and B are unknown amplitudes, g t , g h are nonlinear scaling fields, and y h is the magnetic critical exponent. The nonlinear scaling fields g t and g h can be expanded in terms of the corresponding linear thermal and magnetic scaling fields t and h. Taking into account the relations between the magnetization moments in Eq. ͑2͒ and the corresponding derivatives of the free energy ͓12͔ we have calculated the scaling expansion for Q L (t,hϭ0) to the leading order in t and up to L 3Ϫ4y h :The zeroth-order term Q L (0) was evaluated previously ͓12͔ and the first-order term is of the form where ␣ i (iϭ1, . . . ,16) are unknown amplitudes. In our work we consider only the first three terms in the expansion ͑5͒, but for some future Monte Carlo applications the higherorder terms in

The spin van der Waals model may be regarded as an infinite-lattice-dimensional limit of the spin-4 anisotropic Heisenberg model. By solution of the generalized Langevin equation, time-dependent behavior is obtained. The geometry of realized Hilbert spaces lends a simple interpretation for critical dynamics.PACS numbers: 05.30.Ch, 05.70.LnCertain statistical mechanical models are exactly solvable at some special limits, providing useful insight into the behavior of these models. For example, when the spin dimensionality in the classical Heisenberg model is made infinitely large, one obtains the spherical model. 1 The static behavior of this model is well understood 2 and has been useful for understanding critical behavior. 1 Consider the spin-^-nearest-neighbor (nn) anisotropic Heisenberg model on a hypercubic lattice of dimensionality Z),

The time-dependent autocorrelation function has been derived for the x component of the total spin for the S = 1/2 constant-coupling anisotropic Heisenberg model, i.e. , the Van der Waals system. For T & T" the time correlation is shown to be Gaussian for both XY-like and Ising-like regimes of the model. For T & T" the correlation function is still Gaussian if the model is XY-like; but it is oscillatory if the model is Isinghke. Critical slow down appears only with the Ising-like system for this time-correlation function.

The renormalization scheme recently proposed by White is applied to the d = 1 anisotropic XY model in a transverse field (AXY). It is found that this scheme offers a distinct improvement over standard techniques as far as the computation of the ground state is concerned. However, compared to the Ising model in a transverse Geld, on account of more complicated symmetries the AXY demands more precautions during the construction of a renormalization-group transformation. The new method predicts deGnitely better the location of the phase transition in the XY-like region than in the Ising-like region, but only in the Ising-like region is there any progress for the critical exponent a.