Date of Original Version
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 Kc(n=2)=0.4541659 (10) and Kc(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), 0.781 (2), and 0.813 (2) for n=1, 2, and 3, respectively. Then we vary the surface coupling K1 and locate the so-called special transition at κc(n=1)=0.50214 (8) and κc(n=2)=0.6222 (3), where κ=K1 ∕ K−1. The corresponding surface thermal and magnetic exponents are y(s)t1=0.715 (1) and y(s)h1=1.636 (1) for the Ising model, and y(s)t1=0.608 (4) and y(s)h1=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.
Youjin Deng, Henk W. J. Blöte, and M. P. Nightingale. (2005). "Surface and bulk transitions in three-dimensional O(n) models." Physical Review E, 72(1), 016128. Available at: http://dx.doi.org/10.1103/PhysRevE.72.016128