Exact Solution of the Anisotropic Special Transition in the O(n) Model in Two Dimensions

The effect of surface exchange anisotropies is known to play an important role in magnetic critical and multicritical behavior at surfaces. We give an exact analysis of this problem in d=2 for the O(n) model using the Coulomb gas, conformal invariance, and integrability techniques. We obtain the full set of critical exponents at the anisotropic special transition-where the symmetry on the boundary is broken down to O(n(1))xO(n-n(1))-as a function of n(1). We also obtain the full phase diagram and crossover exponents. Crucial in this analysis is a new solution of the boundary Yang-Baxter equations for loop models. The appearance of the generalization of Schramm-Loewner evolution SLE(kappa,rho) is also discussed.