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Progress of Theoretical Physics Supplement

Block-Spin Distribution Functions at Criticality

Macoto Kikuchi

Published May 16, 2013• 0 citations
Publisher

Abstract

The block-spin distribution functions are studied for the two-dimensional Ising model, the three-dimensional Ising model and the two-dimensional three-state Potts model using the Monte Carlo technique. The nearest-neighbour block-spin correlation function for the two-dimensional Ising model is also calculated. These functions which reflect the local properties of the systems are proved to be good probes for the critical phenomena.

DOI
10.1143/ptp.101.557

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The block-spin distribution functions are studied for the two-dimensional Ising model, the three-dimensional Ising model and the two-dimensional three-state Potts model using the Monte Carlo technique. The nearest-neighbour block-spin correlation function for the two-dimensional Ising model is also calculated. These functions which reflect the local properties of the systems are proved to be good probes for the critical phenomena.
The authors have performed extensive Monte Carlo simulations of two-dimensional layered Ising lattice gases with (m-1)-neighbour exclusion in the x direction and pair attraction in the y direction. They studied the specific heat of the models in systems with different rectangular shapes and the spin-spin correlation function in the x direction near to the critical point. For the m=3 model both the thermal and magnetic exponents were estimated to be close to the known values of the three-state Potts model. Estimates for the critical exponents of the m=4 model are consistent with a commensurate-incommensurate transition; however they find no independent evidence for the existence of a floating phase.
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.
Extending the Monte Carlo method to dynamic critical phenomena we investigated the time-dependent correlation functions in the two-dimensional one-spin-flip Ising model and the critical behavior of the associated relaxation times. These relaxation times are the following: ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}}^{\ensuremath{\Delta}T}$, characterizing the approach of the order parameter to equilibrium after a change of temperature $\ensuremath{\Delta}T$ of the system; ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}$ and ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}$ characterizing the slowing down of the order-parameter correlation and autocorrelation functions, respectively; ${\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}$ and ${\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}$, characterizing the slowing down of the energy correlation and autocorrelation functions; and finally ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}$, characterizing the cross-correlation function. We give estimates for the associated exponents ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}}^{\ensuremath{\Delta}T}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}\ensuremath{\approx}1.90\ifmmode\pm\else\textpm\fi{}0.10$, and ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}\ensuremath{\approx}1.60\ifmmode\pm\else\textpm\fi{}0.10$, ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0.95\ifmmode\pm\else\textpm\fi{}0.10$, ${\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0$, which are consistent with the dynamic scaling hypothesis and with exact inequalities. A detailed comparison with recent high-temperature-expansion estimates is performed, and the reliability of the Monte Carlo results is carefully analyzed.
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 $\ensuremath{\nu}$ 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.