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 such as small magnetic clusters, we consider models with free boundary conditions. For a field-driven transition with two coexisting phases at the infinite-volume transition pointh=h _t , we prove that the low-temperature, finite-volume magnetizationm _free(L, h) per site in a cubic volume of sizeL ^d behaves like

$$m_{free} (L,h) = \frac{{m_ + + m_ - }}{2} + \frac{{m_ + - m_ - }}{2}tanh\left[ {\frac{{m_ + - m_ - }}{2}L^d (h - h_\chi (L))} \right] + O\left( {\frac{1}{L}} \right)$$

whereh _x (L) is the position of the maximum of the (finite-volume) susceptibility andm _± are the infinite-volume magnetizations ath=h _t +0 andh=h _t −0, respectively. We show thath _x (L) is shifted by an amount proportional to 1/L with respect to the infinite-volume transition pointh _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 conditions, 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, for example, the positionh _U (L) of the maximum of the so-called Binder cumulantU _free(L,h). Whileh _U (L) is again shifted by an amount proportional to 1/L with respect to the infinite-volume transition pointh _t , its shift with respect toh _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.