Surface-induced finite-size effects for first-order phase transitions

Abstract

We consider classical lattice models describing first-order phase transitions, and study the finite-size scaling of the magnetization and susceptibility. In order to model the effects of an actual surface in systems like small magnetic clusters, we consider models with free boundary conditions. For a field driven transition with two coexisting phases at the infinite volume transition point h = h t , we prove that the low temperature finite volume magnetization m free (L, h) per site in a cubic volume of size L d behaves likewhere h χ (L) is the position of the maximum of the (finite volume) susceptibility and m ± are the infinite volume magnetizations at h = h t + 0 and h = h t − 0, respectively. We show that h χ (L) is shifted by an amound proportional to 1/L with respect to the infinite volume transitions point h t provided the surface free energies of the two phases at the transition point are different. This should be compared with the shift for periodic boundary conditons, which for an asymmetric transition with two coexisting phases is proportional only to 1/L 2d . One can consider also other definitions of finite volume transition points, as, for example, the position h U (L) of the maximum of the so called Binder cummulant U free (L, h). While it is again shifted by an amount proportional to 1/L with respect to the infinite volume transition point h t , its shift with respect to h χ (L) is of the much smaller order 1/L 2d . We give explicit formulas for the proportionality factors, and show that, in the leading 1/L 2d term, the relative shift is the same as that for periodic boundary conditions. † Heisenberg Fellow ‡ Partly supported by the grants GAČR 202/93/0499 and GAUK 376

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General Theory of Lee-Yang Zeros in Models with First-Order Phase Transitions

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