Universal properties in the free energy of the asymmetricN-state vertex model

Abstract

Electric-field-dependent free energy of the exactly solvable asymmetric N-state vertex model (i.e. in an arbitrary vertical and an arbitrary horizontal electric field v and h) in the low-temperature antiferroelectric phase is obtained in the form of the expansion in terms of small but nonzero polarization p. Exact mapping onto a microscopic surface model (solid-on-solid (SOS) model) is employed to study the vicinal-surface free energy below the roughening temperature TR which determines the equilibrium crystal shape (ECS) near the facet edge of a crystal. The obtained expansion of the free energy is of the well-established Gruber-Mullins-Pokrovsky-Talapov (GMPT) type: f(p,h) = f(0,h) + a(h)p + b(h)p3 + O(p4). It is found that the coefficients a(h) and b(h) are identical with those of the asymmetric six-vertex model. Based on this expansion, in cooperation with the Andreev construction of the ECS, universal properties are verified along the whole facet contour. First, directly from the GMPT-type expansion, the critical exponents governing the rounding of the crystal facet are obtained; the exponent is (3/2) in all direction except the tangential one in which case it is 3. Second, the universal relation between a(h) and b(h) is verified, leading to the universal Gaussian curvature jump at the facet edge and a universal relation between the critical amplitudes of the ECS profiles of the vicinal surface.