Ground-state phase diagrams of frustrated spin-SXXZchains: Chiral ordered phases

The ground-state phase diagram of the frustrated spin-S XXZ chain with the competing nearest- and next-nearest-neighbor antiferromagnetic couplings is studied numerically by using the density-matrix renormalization-group method for the cases of $S=1/2,$ $3/2,$ and $2.$ We are particularly interested in the possible gapless and gapped chiral phases, in which the chirality ${\kappa }_l{=S}_l^x{S}_{l+1\mathrm{}}^y-S_l^y{S}_{l+1\mathrm{}}^x$ exhibits a finite long-range order whereas the spin correlation decays either algebraically or exponentially. We show that the gapless chiral phase appears in a broad region of the phase diagram for general S. By contrast, the gapped chiral phase is found for integer S in a narrow region of the phase diagram, while it has not been identified for half-odd integer S within our numerical accuracy. By combining the results with our previous result for $S=1,$ we discuss the S dependence of the phase diagram. The prediction from a bosonization analysis on the decay exponent of the spin correlation is verified.