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

We calculate the quantum phase diagram of an 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 XX chain (Delta (i)=0) and the isotropic model with ferromagnetic phase J(2). For Delta (1) < -1, a ferromagnetic phase and two antiferromagnetic phases exist. For \(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 (1)=Delta (2)=1 suggests that a small gap is present.