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Physical review. B, Condensed matter

Study of cell distribution functions of the three-dimensional Ising model

Kimmo K. Kaski, Kurt Binder, J. D. Gunton

Published Apr 1, 1984• 98 citations
Publisher

Abstract

The single and joint distribution functions of the nearest-neighbor cell magnetization of the three-dimensional Ising model have been studied by Monte Carlo methods. Near the critical point, both distribution functions tend, for large $L$, towards scaled universal forms. Estimates of the order parameter and susceptibility are obtained from the distribution functions. The critical temperature is estimated by a method independent of the estimate of the critical exponents. The critical exponent $\ensuremath{\nu}$ is obtained with the use of finite-size scaling arguments. The joint distribution function shows a double-well structure for $T<{T}_{c}$ and can be represented by a coarse-grained effective Hamiltonian of two cell variables. We have determined the dependence of this functional on the coarse-graining cell size for several temperatures. In particular, we have calculated the dependence of the "spinodal curve" on the (scaled) coarse-graining size. The relevance of this work to the kinetics of first-order phase transitions is also discussed.

DOI
10.1103/physrevb.29.3996

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The single and joint distribution functions of the nearest-neighbor cell magnetization of the three-dimensional Ising model have been studied by Monte Carlo methods. Near the critical point, both distribution functions tend, for large LLL, towards scaled universal forms. Estimates of the order parameter and susceptibility are obtained from the distribution functions. The critical temperature is estimated by a method independent of the estimate of the critical exponents. The critical exponent \ensuremathν\ensuremath{\nu}\ensuremathν is obtained with the use of finite-size scaling arguments. The joint distribution function shows a double-well structure for T<TcT<{T}_{c}T<Tc​ and can be represented by a coarse-grained effective Hamiltonian of two cell variables. We have determined the dependence of this functional on the coarse-graining cell size for several temperatures. In particular, we have calculated the dependence of the "spinodal curve" on the (scaled) coarse-graining size. The relevance of this work to the kinetics of first-order phase transitions is also discussed.
The distribution function PL(s){P}_{L}(s)PL​(s) of the local order parameter sss in finite blocks of size Ld{L}^{d}Ld is studied for Ising models for dimensionalities d=2,3,and4d=2, 3, \mathrm{and} 4d=2,3,and4 by Monte Carlo methods. A real-space renormalization group based on phenomenological scaling yields fairly accurate results for rather small LLL (e.g., the standard exponents $\ensuremath{\beta}$ and $\ensuremath{\nu}$ for d=3d=3d=3 are found as $\frac{2\ensuremath{\beta}}{\ensuremath{\nu}}=1.03\ifmmode\pm\else\textpm\fi{}0.01$, $\frac{1}{\ensuremath{\nu}}=1.60\ifmmode\pm\else\textpm\fi{}0.05$). The method can easily be generalized to arbitrary Hamiltonians, including spin dimensionalities n>1n>1n>1.
In extensive Monte Carlo simulations the phase transition of the random-field Ising model in three dimensions is investigated. The values of the critical exponents are determined via finite-size scaling. For a Gaussian distribution of the random fields it is found that the correlation length \ensuremath{\xi} diverges with an exponent \ensuremath{\nu}=1.1\ifmmode\pm\else\textpm\fi{}0.2 at the critical temperature and that \ensuremath{\chi}\ensuremath{\sim}${\ensuremath{\xi}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$ with \ensuremath{\eta}=0.50\ifmmode\pm\else\textpm\fi{}0.05 for the connected susceptibility and ${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$\ensuremath{\sim}${\ensuremath{\xi}}^{4\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$ with \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=1.03\ifmmode\pm\else\textpm\fi{}0.05 for the disconnected susceptibility. Together with the amplitude ratio A=${\mathrm{lim}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{T}\mathit{c}}$${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$/${\mathrm{\ensuremath{\chi}}}^{2}$(hr{\mathit{h}}_{\mathit{r}}hr​/T)2{)}^{2})2 being close to one this gives further support for a two-exponent scaling scenario implying \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\eta}. The magnetization behaves discontinuously at the transition, i.e., \ensuremath{\beta}=0. However, no divergence for the specific heat and in particular no latent heat is found. Also the probability distribution of the magnetization does not show a multipeak structure that would be characteristic for the phase-coexistence at first-order phase-transition points.
Critical finite-size scaling functions for the order-parameter distribution of the two- and three-dimensional Ising model are investigated. Within a recently introduced classification theory of phase transitions the universal part of critical finite-size scaling functions has been derived by employing a scaling limit which differs from the traditional finite-size scaling limit. In this paper the analytical predictions are compared with Monte Carlo simulation results. We find good agreement between the analytical expression and the simulation results. The agreement is consistent with the possibility that the functional form of the critical finite-size scaling function for the order-parameter distribution is determined uniquely by only a few universal parameters, most notably the equation of state exponent.
The finite-size scaling (FSS) functions for anisotropic three-dimensional (3D) Ising models of size L 1 × L 1 × aL 1 ( a : anisotropy parameter) are studied by Monte Carlo simulations. We study the a dependence of FSS functions of the Binder parameter g and the magnetization distribution function p ( m ). We have shown that the FSS functions for p ( m ) at the critical temperature change from a two-peak structure to a single-peak one by increasing or decreasing a from 1. We also study the FSS near the critical temperature of the layered square-lattice Ising model, when the systems have a large two-dimensional (2D) anisotropy. We have found the 3D and 2D FSS behaviour depending on the parameter which is fixed; a unified view of 3D and 2D FSS behaviour has been obtained for the anisotropic 3D Ising models.
We present the first Monte Carlo estimate of the edge critical exponent ${\ensuremath{\Delta}}_{2}$ for the three-dimensional nearest-neighbor ferromagnetic Ising model at the ordinary edge transition. The finite-size dependence of the spin-spin correlation function is used to obtain the edge-bulk exponent ${\ensuremath{\eta}}_{0}$,2=1.704\ifmmode\pm\else\textpm\fi{}0.1, and this gives ${\ensuremath{\Delta}}_{2}$/${\ensuremath{\nu}}_{b}$=-1.185\ifmmode\pm\else\textpm\fi{}0.1. This last value is in agreement with the \ensuremath{\epsilon}-expansion estimate of -29/24=-1.208. . . .
The original collective variables method for calculating the thermodynamic characteristics of a three-dimensional Ising-like system is developed in the approximation of the sextic distribution for modes of spin-density oscillations $({\ensuremath{\rho}}^{6}$ model), taking into account the first confluent correction. The calculations within the framework of this higher non-Gaussian approximation are performed in the temperature regions above and below the critical value of Tc{T}_{c}Tc​ and are illustrated by an example of a simple cubic lattice and an exponentially decreasing interaction potential. The collective variables method makes it possible to obtain universal and nonuniversal quantities by using a unified approach. The dependence of the critical region size and phase-transition temperature on the microscopic parameters is investigated. A microscopic analog of the Landau free energy is calculated. Explicit expressions for the thermodynamic characteristics are obtained as functions of temperature and microscopic parameters of the system. It is shown that each of the leading critical amplitudes and correction-to-scaling amplitudes can be represented as a product of a universal factor not depending on the microscopic parameters of the system and a nonuniversal factor depending on these parameters. The thermodynamic characteristics near Tc{T}_{c}Tc​ and their amplitudes are given for various values of the effective radius of the exponentially decreasing interaction potential (including the values corresponding to the nearest-neighbor interaction and the interactions between the nearest and next-nearest neighbors and between the nearest, next-nearest, and third neighbors). The results of calculations and their comparison with other authors' data show that the ${\ensuremath{\rho}}^{6}$ model provides a better quantitative description of the critical behavior of a three-dimensional Ising ferromagnet than the ${\ensuremath{\rho}}^{4}$ model.