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Physical Review E

Bulk and surface phase transitions in the three-dimensional O(4) spin model

Youjin Deng

Published May 19, 2006• 38 citations
Publisher

Abstract

We investigate the $O(4)$ 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 ${K}_{c}=0.935\phantom{\rule{0.2em}{0ex}}856(2)$, and determine the bulk thermal magnetic renormalization exponents as ${y}_{t}=1.337\phantom{\rule{0.2em}{0ex}}5(15)$ and ${y}_{h}=2.482\phantom{\rule{0.2em}{0ex}}0(2)$, respectively. The universal ratio $Q={⟨{m}^{2}⟩}^{2}∕⟨{m}^{4}⟩$ 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 $O(4)$ model with two open surfaces on which the coupling strength ${K}_{1}$ can be varied. At the ordinary transitions, the surface magnetic exponent is determined as ${y}_{h1}^{(\mathrm{o})}=1.020\phantom{\rule{0.2em}{0ex}}2(12)$. 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 ${y}_{t1}^{(\mathrm{s})}$ is rather close to zero, and we cannot exclude that the corresponding surface transition is Kosterlitz-Thouless-like. The surface magnetic exponent is ${y}_{h1}^{(\mathrm{s})}=1.816(2)$.

DOI
10.1103/physreve.73.056116

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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 \ensuremathκ=K1∕K\ensuremath−1=1.258(20)\ensuremath{\kappa}={K}_{1}∕K\ensuremath{-}1=1.258(20)\ensuremathκ=K1​∕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).
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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.
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We study the critical behavior at the ordinary surface universality class of the three-dimensional O(N)\mathrm{O}(N)O(N) model, bounded by a two-dimensional surface. Using high-precision Monte Carlo simulations of an improved lattice model, where the leading bulk scaling correction is suppressed, and finite-size scaling analysis of the fourth cumulant of the surface magnetization, we obtain precise estimates of the scaling dimension of the surface field operator for N=2,3,4N=2,3,4N=2,3,4. We also determine the fixed-point values of two renormalization-group invariant observables, which characterize the finite-size scaling behavior at the ordinary 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.
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