A RIGOROUS THEORY OF FINITE-SIZE SCALING AT 1ST-ORDER PHASE-TRANSITIONS

A large class of classical lattice models describing the coexistence of a finite number of stable states at low temperatures is considered. The dependence of the finite-volume magnetization Mper(h, L) in cubes of size Ldunder periodic boundary conditions on the external field h is analyzed. For the case where two phases coexist at the infinite-volume transition point ht, we prove that, independent of the details of the model, the finite-volume magnetization per lattice site behaves like Mper(ht)+M tanh[MLd(h-ht)] with Mper(h) denoting the infinite-volume magnetization and M=1/2[Mper(ht+0)-Mper(ht-0)]. Introducing the finite-size transition point hm(L) as the point where the finite-volume susceptibility attains the maximum, we show that, in the case of asymmetric field-driven transitions, its shift is ht-hm(L)=O(L-2 d), in contrast to claims in the literature. Starting from the obvious observation that the number of stable phases has a local maximum at the transition point, we propose a new way of determining the point htfrom finite-size data with a shift that is exponentially small in L. Finally, the finite-size effects are discussed also in the case where more than two phases coexist. © 1990 Plenum Publishing Corporation.