Phase diagram of the XXZ chain with next-nearest-neighbor interactions

We calculate the quantum phase diagram of the {it XXZ} chain with nearest-neighbor (NN) $J_{1}$ and next-NN exchange $J_{2}$ with anisotropies $Delta_{1}$ and $Delta_{2}$ respectively. In particular we consider the case $Delta_{1}=-Delta_{2}$ to interpolate between the {it XX} chain ($% Delta_{i}=0$) and the isotropic model with ferromagnetic $J_{2}$. For $% Delta_{1}<-1$, a ferromagnetic and two antiferromagnetic phases exist. For $| Delta_{i}| <1$, the boundary between the dimer and spin fluid phases is determined by the method of crossing of excitation spectra. For large $J_{2}/J_{1}$, this method seems to indicate the existence of a second spin fluid critical phase. However, an analysis of the spin stiffness and magnetic susceptibility for $Delta_{1}=Delta_{2}=1$ suggest that a small gap is present.