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We calculate the quantum phase diagram of the {\it XXZ} chain withnearest-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 isdetermined by the method of crossing of excitation spectra. For large$J_{2}/J_{1}$, this method seems to indicate the existence of a second spinfluid critical phase. However, an analysis of the spin stiffness and magneticsusceptibility for $\Delta_{1}=\Delta_{2}=1$ suggest that a small gap ispresent.Comment: 10 pages, 4 psfigure

Phase diagram of theXXZchain with next-nearest-neighbor interactions