Charge and spin dynamics in the one-dimensional $t-J_z$ and $t-J$ models

The impact of the spin-flip terms on the (static and dynamic) charge and spin correlations in the Luttinger-liquid ground state of the 1D $t-J$ model is assessed by comparison with the same quantities in the 1D $t-J_z$ model, where spin-flip terms are absent. We employ the recursion method combined with a weak-coupling or a strong-coupling continued-fraction analysis. At $J_z/t=0^+$ we use the Pfaffian representation of dynamic spin correlations. The changing nature of the dynamically relevant charge and spin excitations on approach of the transition to phase separation is investigated in detail. The $t-J_z$ charge excitations (but not the spin excitations) at the transition have a single-mode nature, whereas charge and spin excitations have a complicated structure in the $t-J$ model. In the $t-J_z$ model, phase separation is accompanied by N\'eel long-range order, caused by the condensation of electron clusters with an already existing alternating up-down spin configuration (topological long-range order). In the $t-J$ model, by contrast, the spin-flip processes in the exchange coupling are responsible for continued strong spin fluctuations (dominated by 2-spinon excitations) in the phase-separated state.