Critical behaviour at an edge for the SAW and Ising model

Abstract

The authors have studied the behaviour of two- and three-dimensional self avoiding walks confined to a wedge of wedge angle alpha . Series have been obtained and analysed for the (angular dependent) critical exponents characterising various edge susceptibilities. In terms of a general scaling form for the edge free energy, fe approximately t(d-2) nu psi e(ht-y0 nu , h1t-y1 nu , h2t-y2 nu ), they find for the two-dimensional case the following scaling indices: y0=91/48, y1=3/8, y2( alpha )=-5 pi /8 alpha . They argue that these results are exact, from which follow all exponents for the bulk, surface and edge problem. In three dimensions the authors obtain y0 approximately=2.488, y1=0.65+or-0.02, y2( alpha )= alpha +b pi / alpha where a=0.51+or-0.04, b=-0.847+or-0.017, which, for y2, is precisely of the functional form given by mean-field theory, y2( alpha )=1/2- pi /2. They argue that a=1/2 for all three-dimensional O(N) models. This simple angular dependence of y2 is different from that suggested by Cardy's one-loop epsilon -expansion, (1983). For the square lattice, they have also studied the case in which the wedge is rotated through an angle of pi /4, and find that the various exponents are unchanged. For the three-dimensional Ising model in a wedge, analogy with the SAW results, plus mean-field results in conjunction with RG and series work yield y0 approximately=2.485, y1=0.71+or-0.02 and y2=a+b pi / alpha with a=1/2 and b=-0.79+or-0.02.