Magnetization at corners in two-dimensional Ising models

Abstract

We investigate the corner spin magnetization of two-dimensional ferromagnetic Ising models in various wedge geometries. Results are obtained for triangular and square lattices by numerical studies on finite wedges using the star-triangle transformation, as well as analytic calculations using corner transfer matrices and a fermionic representation of the row-to-row transfer matrix. The corner magnetizations vanish at the bulk critical temperature Tc with an exponentβc differing from the bulk exponent. For isotropic systems with free edges we find thatβ c is given simply byβ c =π/2θ, whereθ is the angle at the corner. For apex magnetizations of conical lattices we obtain the strikingly similar resultαa=π/4θ. These formulas apply equally well to anisotropic lattices if the angleθ is interpreted as an effective angle,θeff, determined by the relative strengths of the interactions.