Crossover finite-size scaling at first-order transitions

Abstract

In a recent paper we developed a method which allows one to control rigorously the finite-size behavior in long cylinders near first-order phase transitions at low temperature. Here we apply this method to asymmetric transitions with two competing phases, and to theq-state Potts model as a typical model of a temperature-driven transition, whereq low-temperature phases compete with one high-temperature phase. We obtain the finite-size scaling of the firstN eigenvalues (whereN is the number of competing phases) of the transfer matrix in a periodic box of volumeL × ... ×L ×t, and, as a corollary, the finite-size scaling of the shape of the order parameter in a hypercubic box (t=L), the infinite cylinder (t=∞), and the crossover regime from hypercubic to cylindrical scaling. For the two-phase case (N=2 we find that the crossover lengthξ _L is given by O(L^w)exp(ΒσL^v), whereΒ is the inverse temperature, σ is the surface tension, and w=1/2 if v+1=2 whilew=0 if v+1 >2. For the standard Ising model we also consider free boundary conditions, showing that ξ_L=exp[ΒσL^v+O(L^v− 1)] for any dimension v+1⩾2. For v+1=2 we finally discuss a class of boundary conditions which interpolate between free (corresponding to the interpolating parameter g=0) and periodic boundary conditions (corresponding to g=1), finding thatξ _L=O(L^w)exp(ΒσL ^v) withw=0 forg=0 andw=1/2 for 0<g⩽1.