The free energy singularity of the asymmetric six - vertex model and the excitations of the asymmetric XXZ chain

We consider the asymmetric six-vertex model, widely used to describe the equilibrium shape of crystals, and the relevant asymmetric X X Z chain. By means of the Bethe-ansatz solution we determine the free energy singularity, as function of the external field, at two special points on the phase boundary. We confirm the exponent 3 (checked experimentally), as the antiferroelectric ordered phase is reached from the incommensurate phase normally to this boundary, and we determine a new singularity along the tangential direction. Both singularities describe the rounding off of the crystal near a facet. At this point the hole excitations of the spin chain show dispersion relations A E ~ ( A P ) I / 2 at small momenta, leading to a finite-size scaling AE ~ N -~/2 for the |ow-lying excited states, N being the chain size. We discuss the nature of the phase transition and the behavior of arrow-arrow correlation lengths in the ordered phase. @ 1997 Published by Elsevier Science B.V. PACS: 05.70,Jk; 64.60.-; 64.60.Fr; 75.10.Jm