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

Abstract

The XXZ spin-half chain has Heisenberg exchange interactions $J_z$ ($J_\perp$) in the $z$ ($x,y$) direction. The model has a transition from the spin-fluid phase for $-J_\perp < J_z < J_\perp$ to the N\'{e}el phase for $J_z > J_\perp >0$. When bond alternation $\delta$ is included, the N\'{e}el phase transitions to the dimer phase for a finite value of $\delta$. 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 $\delta$ and $B$ determine the amount of topological quantized spin pumping around closed circuits in the $(\delta,B)$ plane.