Phase diagram of the non-hermitean asymmetric XXZ spin chain

The low-lying excitations of the asymmetric $XXZ$ spin chain are derived explicitly in the antiferromagnetic regime through the Bethe Ansatz. It is found that a massless and conformal invariant phase with central charge $c=1$ is separated from a massive phase by a line on which the low-lying excitations surprisingly scale with the lattice length as $\Delta E \sim N^{-\frac{1}{2}}$. The mass gap vanishes with an exponent $\frac{1}{2}$ as one approaches the massless phase. The connection with the asymmetric $6$--vertex model and some physical consequences are discussed.