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Physical review. B, Condensed matter

Finite-size scaling in the three-dimensional Ising model

Michael N. Barber, Robert B. Pearson, D. Toussaint, John L. Richardson

Published Aug 1, 1985• 134 citations
Publisher

Abstract

We have performed extensive studies of finite-size scaling in the three-dimensional Ising model, using a specially designed Ising processor. We show results for the susceptibility, the energy and specific heat, and the connected part of the fourth moment of the magnetization. Corrections to finite-size scaling are found to be small only for lattices of linear dimension greater than about L=24. This effect severely limits the accuracy to which critical parameters can be estimated. We estimate ${K}_{c}$=0.221 650\ifmmode\pm\else\textpm\fi{}0.000 005 for the critical coupling, while exponent estimates are consistent with those obtained by other methods.

DOI
10.1103/physrevb.32.1720

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We have performed extensive studies of finite-size scaling in the three-dimensional Ising model, using a specially designed Ising processor. We show results for the susceptibility, the energy and specific heat, and the connected part of the fourth moment of the magnetization. Corrections to finite-size scaling are found to be small only for lattices of linear dimension greater than about L=24. This effect severely limits the accuracy to which critical parameters can be estimated. We estimate Kc{K}_{c}Kc​=0.221 650\ifmmode\pm\else\textpm\fi{}0.000 005 for the critical coupling, while exponent estimates are consistent with those obtained by other methods.
The four-dimensional Ising model is studied to probe the possibility of observing in Monte Carlo simulations the logarithmic corrections to the mean-field theory near criticality. The finite-size-scaling behavior for the correlation length is proposed. The scaling forms of the finite-size renormalized coupling, susceptibility, and fourth field derivative at the renormalized tree-level approximation are derived. These results are used to analyze simulation data of the simple hypercubic lattices of sizes 4\ensuremath{\le}L\ensuremath{\le}14, near criticality. Our simulation results are in agreement with the presence of logarithmic corrections and recent field-theoretical calculations of the specific heat. The approach to the nonscattering theory is observed and is consistent with the predicted logarithmic finite-size dependence.
We study the spin-1/2 Ising model and the Blume-Capel model at various values of the parameter DDD on the simple cubic lattice. To this end we perform Monte Carlo simulations using a hybrid of the local Metropolis, the single cluster and the wall cluster algorithm. Using finite size scaling we determine the value ${D}^{\ensuremath{\ast}}=0.656(20)$ of the parameter DDD, where leading corrections to scaling vanish. We find $\ensuremath{\omega}=0.832(6)$ for the exponent of leading corrections to scaling. In order to compute accurate estimates of critical exponents, we construct improved observables that have a small amplitude of the leading correction for any model. Analyzing data obtained for D=0.641D=0.641D=0.641 and 0.655 on lattices of a linear size up to L=360L=360L=360 we obtain $\ensuremath{\nu}=0.63002(10)$ and $\ensuremath{\eta}=0.03627(10)$. We compare our results with those obtained from previous Monte Carlo simulations and high-temperature series expansions of lattice models, by using field-theoretic methods and experiments.
We compute an improved action for the Ising universality class in three dimensions that has suppressed leading corrections to scaling. It is obtained by tuning models with two coupling constants. We studied three different models: the $\ifmmode\pm\else\textpm\fi{}1$ Ising model with nearest-neighbor and body diagonal interaction, the spin-1 model with states $0,\ifmmode\pm\else\textpm\fi{}1$, and nearest-neighbor interaction, and ${\ensuremath{\varphi}}^{4}$ theory on the lattice (Landau-Ginzburg model). The remarkable finite-size scaling properties of the suitably tuned spin-1 model are compared in detail with those of the standard Ising model. Great care is taken to estimate the systematic errors from residual corrections to scaling. Our best estimates for the critical exponents are $\ensuremath{\nu}=0.6298(5)$ and $\ensuremath{\eta}=0.0366(8)$, where the given error estimates take into account the statistical and systematic uncertainties.
INTRODUCTION Finite-size scaling (FSS) is a well established technique for the extraction of critical exponents from finite volume analyses [1]. Such exponents characterise critical phenomena at a second-order phase transition. The simplest model exhibiting such a transition is the Ising model in two dimensions, which, despite a long history and extensive study, still o#ers new results and insights. Here, we study the model under the special boundary conditions of Brascamp and Kunz [2] to extract new information and to help resolve some hitherto puzzling features of FSS. Let CL (#) be the specific heat at inverse temperature # for a system of linear extent L. FSS of the specific heat is characterized by the location of its peak, #L , its height CL (# L ) and its value at the infinite-volume critical point CL (# c ). The peak position, #L , is a pseudocritical point which typically approaches # c as L ## as |# # c L -# , (1) where # is the shift exponent. In two dimensions,
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.
The transfer matrices for the ferromagnetic Ising model on the two- and three-dimensional lattices with finite size are formulated using the screw method introduced by Kramers and Wannier, and the largest and next largest eigenvalues of matrices are obtained by a computer. Then the specific heats and magnetic susceptibilities are calculated for each system with various sizes. By use of the finite-size scaling and Roomany-Wyld approximations, the critical exponents of the infinite-size systems are evaluated. They are in quite good agreement with the reliable ones.