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 $\mathrm{O}(n)$ 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 ${K}_{\mathrm{c}}(n=2)=0.454\phantom{\rule{0.2em}{0ex}}1659$ (10) and ${K}_{\mathrm{c}}(n=3)=0.693\phantom{\rule{0.2em}{0ex}}003$ (2). We simulate the three models with open surfaces and determine the surface magnetic exponents at the ordinary transition to be ${y}_{h1}^{(\mathrm{o})}=0.7374\phantom{\rule{0.2em}{0ex}}(15)$, 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 ${\ensuremath{\kappa}}_{\mathrm{c}}(n=1)=0.502\phantom{\rule{0.2em}{0ex}}14$ (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 ${y}_{t1}^{(\mathrm{s})}=0.715$ (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 $\ensuremath{-}1∕2$ occur for both models. Also for the Heisenberg model we find substantial evidence for the existence of a special surface transition.