A rigorous theory of finite-size scaling at first-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 magnetizationMper(h, L) in cubes of sizeLdunder periodic boundary conditions on the external fieldh is analyzed. For the case where two phases coexist at the infinite-volume transition pointht, we prove that, independent of the details of the model, the finite-volume magnetization per lattice site behaves likeMper(ht)+M tanh[MLd(h−ht)] withMper(h) denoting the infinite-volume magnetization and M=1/2[Mper(ht+0)−Mper(ht−0)]. Introducing the finite-size transition pointhm(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 isht−hm(L)=O(L−2d), 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 pointhtfrom finite-size data with a shift that is exponentially small inL. Finally, the finite-size effects are discussed also in the case where more than two phases coexist.