Implications of direct-product ground states in the one-dimensional quantumXYZandXYspin chains

We state the conditions under which the general spin-s quantum XYZ ferromagnet ($H_{\mathrm{-}}$) and antiferromagnet ($H_{+}$) with an external magnetic field along one axis, specified by the Hamiltonian $H_{\pm }$=± ${\mathcal{J}}_{l=1\mathrm{}}^N$ ($J_x$ $S_l^x$ $S_{l+1\mathrm{}}^x$+$J_y$ $S_l^y$S ${}_{l+1\mathrm{}}{}^y{}_{}{}^{}$+$J_z$ $S_l^z$ $S_{l+1\mathrm{}}^z$)-h ${\mathcal{J}}_{l=1\mathrm{}}^N$ $S_l^z$ exhibits a fully ordered ground state described by a wave function which is a direct product of single-site wave functions. We present a detailed analysis of the implications for the zero-temperature dynamical properties of this model. In particular, we derive a rigorous relation between the three dynamic structure factors $S_{\mu \mu }$(q,ω), μ=x,y,z at T=0. For the special case of the s=(1/2) anisotropic XY model ($J_z$=0), these relations are used to determine the dynamic structure factors $S_{\mathrm{xx}}$(q,ω) and $S_{\mathrm{yy}}$(q,ω) at T=0 and h=($J_x$ $J_y$ ${)}^{1/2}$ in terms of the known dynamic structure factor $S_{\mathrm{zz}}$(q,ω).