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

TL;DR

This study uses Monte Carlo simulations and finite-size scaling to analyze surface critical phenomena in three-dimensional O(n) models, identifying critical couplings, surface exponents, and the existence of special surface transitions.

Contribution

It provides the first detailed numerical determination of surface critical exponents and transition points for the 3D O(n) models with n=1, 2, and 3, including the Ising, XY, and Heisenberg models.

Findings

Critical couplings for n=2 and 3 are precisely determined.

Surface magnetic exponents at the ordinary transition are calculated.

Evidence for the existence of a special surface transition in the Heisenberg model.

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 $K_{\rm c}(n=2)=0.454 1655 (10)$ and $K_{\rm c}(n=3)= 0.693 002 (2)$. We simulate the three models with open surfaces and determine the surface magnetic exponents at the ordinary transition to be $y_{h1}^{\rm (o)}=0.7374 (15)$, $0.781 (2)$, 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 $\kappa_{\rm c} (n=1)=0.50214 (8)$ and $\kappa_{\rm c} (n=2)=0.6222 (3)$, where $\kappa=K_1/K-1$. The corresponding surface thermal and magnetic exponents are $y_{t1}^{\rm (s)} =0.715 (1)$ and $y_{h1}^{\rm (s)} =1.636 (1)$ for the Ising model, and $y_{t1}^{\rm (s)} =0.608 (4)$ and$y_{h1}^{\rm (s)} =1.675 (1)$ 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.