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Physical Review Letters

New method for determination of critical parameters

C. Ruge, S. Dunkelmann, Friedrich Wagner

Published Oct 26, 1992• 43 citations
Publisher

Abstract

Our method uses topological properties of the large clusters in the single-cluster algorithm of Wolff for Monte Carlo simulations of spin systems. We have applied this method to the d=3 Ising model with film geometry near the special transition. As a test for the method we determined the bulk critical temperature and the critical indices in fair agreement with the results obtained by other methods. The new results refer to the critical surface coupling ${\mathit{J}}_{1\mathit{c}}$/J=1.5004(20) and the surface exponents, where \ensuremath{\varphi}=0.461(15) and ${\mathrm{\ensuremath{\beta}}}_{\mathit{i}}^{\mathit{m}}$=0.237(5).

DOI
10.1103/physrevlett.69.2465

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Extended scaling in high dimensions

Bertrand Berche, Christophe Chatelain, C Dhall +3 more

We apply and test the recently proposed 'extended scaling' scheme in an analysis of the magnetic susceptibility of Ising systems above the upper critical dimension. The data are obtained by Monte Carlo simulations using both the conventional Wolff cluster algorithm and the Prokof'ev–Svistunov worm algorithm. As already observed for other models, extended scaling is shown to extend the high-temperature critical scaling regime over a range of temperatures much wider than that achieved conventionally. It allows for an accurate determination of leading and sub-leading scaling indices, critical temperatures and amplitudes of the confluent corrections.

arXiv:0807.2546•Journal of Statistical Mechanics Theory and Experiment•Nov 13, 2008•18 citations
Our method uses topological properties of the large clusters in the single-cluster algorithm of Wolff for Monte Carlo simulations of spin systems. We have applied this method to the d=3 Ising model with film geometry near the special transition. As a test for the method we determined the bulk critical temperature and the critical indices in fair agreement with the results obtained by other methods. The new results refer to the critical surface coupling J1c{\mathit{J}}_{1\mathit{c}}J1c​/J=1.5004(20) and the surface exponents, where \ensuremath{\varphi}=0.461(15) and \ensuremathβim{\mathrm{\ensuremath{\beta}}}_{\mathit{i}}^{\mathit{m}}\ensuremathβim​=0.237(5).
We simulated the critical behavior of the d=3 Ising model with surfaces with a cluster Monte Carlo method at the multicritical point and at the ordinary transition. The surface exponents ${\mathrm{\ensuremath{\beta}}}_{1}$ have been determined using finite-size scaling. The errors of the resulting values ${\mathrm{\ensuremath{\beta}}}_{1}^{\mathit{m}}$=0.2375(15) and ${\mathrm{\ensuremath{\beta}}}_{1}^{0}$=0.807(4) are an order of magnitude smaller than in previous determinations. Whereas the values of the exponents agree with the prediction of the \ensuremath{\epsilon} expansion, the universal ratios of susceptibility and magnetization amplitudes at the special transition are at variance with those predictions.
We investigate the O(4)O(4)O(4) spin model on the simple-cubic lattice by means of the Wolff cluster algorithm. Using the toroidal boundary condition, we locate the bulk critical point at coupling Kc=0.935856(2){K}_{c}=0.935\phantom{\rule{0.2em}{0ex}}856(2)Kc​=0.935856(2), and determine the bulk thermal magnetic renormalization exponents as yt=1.3375(15){y}_{t}=1.337\phantom{\rule{0.2em}{0ex}}5(15)yt​=1.3375(15) and yh=2.4820(2){y}_{h}=2.482\phantom{\rule{0.2em}{0ex}}0(2)yh​=2.4820(2), respectively. The universal ratio Q=⟨m2⟩2∕⟨m4⟩Q={⟨{m}^{2}⟩}^{2}∕⟨{m}^{4}⟩Q=⟨m2⟩2∕⟨m4⟩ is also determined as 0.9142(1). The precision of these estimates significantly improves over that of the existing results. Then, we simulate the critical O(4)O(4)O(4) model with two open surfaces on which the coupling strength K1{K}_{1}K1​ can be varied. At the ordinary transitions, the surface magnetic exponent is determined as yh1(o)=1.0202(12){y}_{h1}^{(\mathrm{o})}=1.020\phantom{\rule{0.2em}{0ex}}2(12)yh1(o)​=1.0202(12). Further, we find a so-called special surface transition at $\ensuremath{\kappa}={K}_{1}∕K\ensuremath{-}1=1.258(20)$. At this point, the surface thermal exponent yt1(s){y}_{t1}^{(\mathrm{s})}yt1(s)​ is rather close to zero, and we cannot exclude that the corresponding surface transition is Kosterlitz-Thouless-like. The surface magnetic exponent is yh1(s)=1.816(2){y}_{h1}^{(\mathrm{s})}=1.816(2)yh1(s)​=1.816(2).
We present the results of extensive Monte Carlo simulations of phase transitions and critical behavior at the surface of a simple cubic Ising model. Profiles of the magnetization and internal energy are determined as a function of the distance from the surface, and we extract surface and bulk properties as a function of temperature and surface coupling Js{\mathit{J}}_{\mathit{s}}Js​. The surface-bulk multicritical point is located with improved precision, Js{\mathit{J}}_{\mathit{s}}Js​/J=1.52\ifmmode\pm\else\textpm\fi{}0.02, and crossover behavior is studied. New estimates for critical exponents are extracted, ${\ensuremath{\gamma}}_{1}$=0.78\ifmmode\pm\else\textpm\fi{}0.06, ${\ensuremath{\gamma}}_{1}^{\mathit{m}}$=1.41\ifmmode\pm\else\textpm\fi{}0.14, ${\ensuremath{\gamma}}_{11}^{\mathit{m}}$=0.96\ifmmode\pm\else\textpm\fi{}0.09, and along with critical amplitude ratios these are compared with theoretically predicted values.
We present highly accurate Monte Carlo results for simple cubic Ising lattices containing up to spins. These results were obtained by means of the Cluster Processor, a newly built special-purpose computer for the Wolff cluster simulation of the 3D Ising model. We find that the spontaneous magnetization M(t) is accurately described by , where , in a wide temperature range 0.0005 < t < 0.26. Any corrections to scaling with higher powers of t could not be resolved from our data, which implies that they are very small. The magnetization exponent is determined as . An analysis of the magnetization distribution near criticality yields a new determination of the critical point: , with a standard deviation of .
Using Monte Carlo methods and finite-size scaling, we investigate surface criticality in the O(n)\mathrm{O}(n)O(n) models on the simple-cubic lattice with n=1n=1n=1, 2, and 3, i.e., the Ising, XYXYXY, and Heisenberg models. For the critical couplings we find Kc(n=2)=0.4541659{K}_{\mathrm{c}}(n=2)=0.454\phantom{\rule{0.2em}{0ex}}1659Kc​(n=2)=0.4541659 (10) and Kc(n=3)=0.693003{K}_{\mathrm{c}}(n=3)=0.693\phantom{\rule{0.2em}{0ex}}003Kc​(n=3)=0.693003 (2). We simulate the three models with open surfaces and determine the surface magnetic exponents at the ordinary transition to be yh1(o)=0.7374(15){y}_{h1}^{(\mathrm{o})}=0.7374\phantom{\rule{0.2em}{0ex}}(15)yh1(o)​=0.7374(15), 0.781 (2), and 0.813 (2) for n=1n=1n=1, 2, and 3, respectively. Then we vary the surface coupling K1{K}_{1}K1​ and locate the so-called special transition at ${\ensuremath{\kappa}}_{\mathrm{c}}(n=1)=0.502\phantom{\rule{0.2em}{0ex}}14$ (8) and ${\ensuremath{\kappa}}_{\mathrm{c}}(n=2)=0.6222$ (3), where $\ensuremath{\kappa}={K}_{1}∕K\ensuremath{-}1$. The corresponding surface thermal and magnetic exponents are yt1(s)=0.715{y}_{t1}^{(\mathrm{s})}=0.715yt1(s)​=0.715 (1) and yh1(s)=1.636{y}_{h1}^{(\mathrm{s})}=1.636yh1(s)​=1.636 (1) for the Ising model, and yt1(s)=0.608{y}_{t1}^{(\mathrm{s})}=0.608yt1(s)​=0.608 (4) and yh1(s)=1.675{y}_{h1}^{(\mathrm{s})}=1.675yh1(s)​=1.675 (1) for the XYXYXY model. Finite-size corrections with an exponent close to $\ensuremath{-}1∕2$ occur for both models. Also for the Heisenberg model we find substantial evidence for the existence of a special surface transition.
We apply and test the recently proposed 'extended scaling' scheme in an analysis of the magnetic susceptibility of Ising systems above the upper critical dimension. The data are obtained by Monte Carlo simulations using both the conventional Wolff cluster algorithm and the Prokof'ev–Svistunov worm algorithm. As already observed for other models, extended scaling is shown to extend the high-temperature critical scaling regime over a range of temperatures much wider than that achieved conventionally. It allows for an accurate determination of leading and sub-leading scaling indices, critical temperatures and amplitudes of the confluent corrections.