Surface and bulk transitions in three-dimensional O(n) models

Abstract

Using Monte Carlo methods and finite-size scaling, we investigate surface criticality in the O$(n)$ models on the simple-cubic lattice with $n=1$, 2, and 3, i.e. the Ising, XY, and Heisenberg models. For the critical couplings we find $K_{\mathrm{c}}(n=2)=0.4541655(10)$ and $K_{\mathrm{c}}(n=3)=0.693002(2)$. We simulate the three models with open surfaces and determine the surface magnetic exponents at the ordinary transition to be $y_{h1}^{(\mathrm{o})}=0.7374(15)$, $0.781(2)$, and $0.813(2)$ for $n=1$, 2, and 3, respectively. Then we vary the surface coupling $K_1$ and locate the so-called special transition at ${\kappa }_{\mathrm{c}}(n=1)=0.50214(8)$ and ${\kappa }_{\mathrm{c}}(n=2)=0.6222(3)$, where $\kappa =K_1/K-1$. The corresponding surface thermal and magnetic exponents are $y_{t1}^{(\mathrm{s})}=0.715(1)$ and $y_{h1}^{(\mathrm{s})}=1.636(1)$ for the Ising model, and $y_{t1}^{(\mathrm{s})}=0.608(4)$ and$y_{h1}^{(\mathrm{s})}=1.675(1)$ for the XY model. Finite-size corrections with an exponent close to -1/2 occur for both models. Also for the Heisenberg model we find substantial evidence for the existence of a special surface transition.

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