SURFACE-INDUCED FINITE-SINE EFFECTS FOR FIRST-ORDER PHASE-TRANSITIONS

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 dusters, 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 like m(free)(L, h) = m(+) + m(-)/2 + m(+) - m(-)/2 tanh [m(+) - m(-)/2 L(d)(h - h(chi)(L))] + 0 (1/L) where h(chi)(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(chi)(L) is shifted by an amount proportional to 1/L with respect to the infinite-volume transition 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 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 position h(U)(L) of the maximum of the so-called Binder cumulant U-free(L, h). While h(U)(L) 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(chi)(L) is of the much smaller order 1/L(2d). W, 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.