Critical and off-critical properties of the $XXZ$ chain in external homogeneous and staggered magnetic fields

The phase diagram of the $XXZ$ chain under the influence of homogeneous and staggered magnetic fields is calculated. The model has a rich phase structure with three types of phases. A fully ferromagnetic phase, an antiferromagnetic phase and a massless phase with partial ferromagnetic and antiferromagnetic order. This massless phase has its critical fluctuations governed by a conformal field theory with central charge $c=1$. When the $\sigma^{z}$-anisotropy $\Delta$ is zero the model is exactly integrable through a Jordan-Wigner fermionization and our results are analytic. For $\Delta\neq 0 $ our analysis are done numerically using lattice sizes up to $M=20$. Our results shows that for $-1 \geq \Delta \geq \frac{\sqrt{2}}{2}$ the staggered magnetic field perturbation is relevant and for $1 \leq \Delta \leq \frac{\sqrt{2}}{2}$ it is irrelevant. In the region of relevant perturbations we show that the massive continuum field theory associated to the model has the same mass spectrum as the sine-Gordon model.