The free energy singularity of the asymmetric 6--vertex model and the excitations of the asymmetric XXZ chain

We consider the asymmetric six--vertex model, {\it i.e.} the symmetric six--vertex model in an external field with both horizontal and vertical components, and the relevant asymmetric $XXZ$ chain. The model is widely used to describe the equilibrium shape of a crystal. By means of the Bethe Ansatz solution we determine the exact free energy singularity, as function of both components of the field, at two special points on the phase boundary. We confirm the exponent $\frac{3}{2}$ (already checked experimentally), as the antiferroelectric ordered phase is reached from the incommensurate phase normally to the phase boundary, and we determine a new singularity along the tangential direction. Both singularities describe the rounding off of the crystal near a facet. The hole excitations of the spin chain at this point on the phase boundary show dispersion relations with the striking form $\Delta E\sim (\Delta P)^{\half}$ at small momenta, leading to a finite size scaling $\Delta E \sim N^{-\half}$ for the low--lying excited states, where $N$ is the size of the chain. We conjecture that a Pokrovskii--Talapov phase transition is replaced at this point by a transition with diverging correlation length, but not classified in terms of conformal field theory.