Article,
Crossover finite-size scaling at first-order transitions
Journal of Statistical Physics, 69 (3): 487--537 (Nov 1, 1992)
DOI: 10.1007/bf01050424
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=8), and the crossover regime from hypercubic to cylindrical scaling. For the two-phase case (N=2 we find that the crossover length?L is given by O(Lw)exp(?sLv), where? is the inverse temperature, s 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 ?L=exp?sLv+O(Lv- 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 that?L=O(Lw)exp(?sLv) withw=0 forg=0 andw=1/2 for 0g?1.
