Explicit dynamic finite-size scaling in the time-dependent correlation functions for the three-dimensional classical
XY model is explored. The dynamic scaling method, proposed by us previously, is utilized for finding the time-scaling variables of the model. Time-dependent correlation functions of the model for different numbers of spins could be collapsed into universal curves. Compared to infinite-range spin models, the width of the scaling region is found to be much narrower.
Using spin-dynamics techniques we have performed large-scale computer simulations of the dynamic behavior of the classical three component XY model (i.e., the anisotropic limit of an easy-plane Heisenberg ferromagnet), on square lattices of size up to
1922, for several temperatures below, at, and above
TKT. The temporal evolution of spin configurations was determined numerically from coupled equations of motion for individual spins by a fourth-order predictor-corrector method, with initial spin configurations generated by a hybrid Monte Carlo algorithm. The neutron-scattering function S(q,\ensuremath{\omega}) was calculated from the resultant space-time displaced spin-spin correlation function. Pronounced spin-wave peaks were found both in the in-plane and the out-of-plane scattering function over a wide range of temperatures. The in-plane scattering function
Sxx also has a large number of clear but weak additional peaks, which we interpret to come from two-spin-wave scattering. In addition, we observed a small central peak in
Sxx, even at temperatures well below the phase transition. We used dynamic finite-size scaling theory to extract the dynamic critical exponent z. We find z=1.00(4) for all T\ensuremath{\le}
TKT, in excellent agreement with theoretical predictions, although the shape of S(q,\ensuremath{\omega}) is not well described by current theory. \textcopyright{} 1996 The American Physical Society.
As a further test of the dynamic scaling method proposed by us previously, dynamic finite-size effect in an infinite-range classical antiferromagnetic spin model, which is an anisotropic version of two-sublattice Lieb-Mattis model, is studied. Application of our dynamic scaling method to the model yields time scaling exponents similar to those of the infinite-range ferromagnetic classical spin van der Waals model. With the hereby obtained time scaling variables, we can successfully collapse the time-dependent correlation functions of the model into universal curves.
With a large-scale Monte Carlo simulation, non-equilibrium dynamics of the two-dimensional fully frustrated XY model is investigated. We tackle the Kosterlitz-Thouless phase transition. Starting from an ordered initial state, we study the dynamic evolution of the magnetization as well as a specifically defined Binder cumulant. From the dynamic scaling ansatz, we extract the correlating time of the dynamics and the spatial correlation length of the equilibrium state. The dynamic exponent z is determined with relatively high accuracy. Especially, we suggest and demonstrate how one may directly measure the dynamic exponent z above TKT from the scaling fit of the Binder cumulant. These results indicate that the dynamic exponent z fluctuates around z=2, and this is consistent with that observed at temperatures below the transition temperature TKT.
A phenomenological dynamic scaling method is proposed and utilized to interpret finite-size effects in the time-dependent correlation function for the classical spin van der Waals model. We claim that the equations of motion for the classical spin van der Waals model are well defined only within a certain characteristic time. The order of magnitude for the size-dependent maximum time interval of the time-dependent correlation function, analytically obtained in the large N limit via the method of Laplace by Dekeyser and Lee, is found to be related to our characteristic time. As a by-product of this work, time scaling variables are found with which we can completely collapse the time-dependent correlation functions of the classical spin van der Waals model into one curve. \textcopyright{} 1996 The American Physical Society.
The thermodynamics of the two-dimensional (2D) XY model is formulated by a transfer-matrix method and analyzed by a density-matrix renormalization group. The finite-size scaling and the beta function of the model are studied by the Roomany-Wyld renormalization-group theory. It is found that the 2D XY model has an essential finite-size effect and the Berezinskii-Kosterlitz-Thouless transition with the critical temperature
TBKT=0.892 appears in a finite system of 2000--3000 spins as a massless to massive transition with the effective critical temperature ${T}_{c}=1.07\ifmmode\pm\else\textpm\fi{}0.01.$