Phase diagram and topology of the XXZ chain with alternating bonds and staggered magnetic field

The XXZ spin-half chain has Heisenberg exchange interactions ${J}_z$ $({J}_{\ensuremath{\perp}})$ in the $z$ $(x,y)$ direction. The model has a transition from the spin-fluid phase for $\ensuremath{-}{J}_{\ensuremath{\perp}}<{J}_z<{J}_{\ensuremath{\perp}}$ to the N\'eel phase for ${J}_z>{J}_{\ensuremath{\perp}}>0$. When bond alternation $\ensuremath{\delta}$ is included, the N\'eel phase transitions to the dimer phase for a finite value of $\ensuremath{\delta}$. We determine the phase diagram using simple topological indicators related to the polarization of both spins. When a staggered magnetic field $B$ is included, a contour plot of these indicators as a function of $\ensuremath{\delta}$ and $B$ determine the amount of topological quantized spin pumping around closed circuits in the $(\ensuremath{\delta},B)$ plane.