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Physical review. B, Condensed matter

Critical parameters for thed=3 Ising model in a film geometry

C. Ruge, Friedrich Wagner

Published Aug 1, 1995• 29 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.52.4209

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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 \ensuremathβ1{\mathrm{\ensuremath{\beta}}}_{1}\ensuremathβ1​ have been determined using finite-size scaling. The errors of the resulting values \ensuremathβ1m{\mathrm{\ensuremath{\beta}}}_{1}^{\mathit{m}}\ensuremathβ1m​=0.2375(15) and \ensuremathβ10{\mathrm{\ensuremath{\beta}}}_{1}^{0}\ensuremathβ10​=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 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.
With use of a novel Monte Carlo method suited for the study of surface phenomena, the authors have investigated a three-dimensional Ising model with modified surface exchange. The results verify the scaling behavior of the magnetization profile and confirm and extend renormalization-group predictions. The surface-layer magnetization exponent is ${\ensuremath{\beta}}_{1}=0.78\ifmmode\pm\else\textpm\fi{}0.02$ except at the "special" (multicritical) transition where ${{\ensuremath{\beta}}_{1}}^{m}=0.175\ifmmode\pm\else\textpm\fi{}0.025$ and the crossover exponent is $\ensuremath{\phi}=0.56\ifmmode\pm\else\textpm\fi{}0.04$.
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 ${\mathrm{\ensuremath{\beta}}}_{\mathit{i}}^{\mathit{m}}$=0.237(5).
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 study the special point in the phase diagram of a semi-infinite system where the bulk transition is in the three-dimensional Ising universality class. To this end we performed a finite-size-scaling study of the improved Blume-Capel model on the simple cubic lattice with two different types of surface interaction. In order to check for the effect of leading bulk corrections we also simulated the spin-1/2 Ising model on the simple cubic lattice. We accurately estimated the surface enhancement coupling at the special point of these models. We find yt1=0.718(2){y}_{{t}_{1}}=0.718(2)yt1​​=0.718(2) and yh1=1.6465(6){y}_{{h}_{1}}=1.6465(6)yh1​​=1.6465(6) for the surface renormalization group exponents of the special transition. These results are compared with previous ones obtained by using field-theoretic methods and Monte Carlo simulations of the spin-1/2 Ising model. Furthermore, we study the behavior of the surface transition near the special point, and finally we discuss films with special boundary conditions at one surface and fixed ones at the other.