# Geometric phase and topological phase diagram of the one-dimensional $XXZ$ Heisenberg spin chain in a longitudinal field

## TL;DR

This paper investigates the geometric phase and topological phase diagram of the one-dimensional $XXZ$ Heisenberg spin chain in a longitudinal field, combining semi-analytical mean-field theory with numerical matrix-product-state methods.

## Contribution

It develops a semi-analytical mean-field approach to compute the geometric phase and maps the topological phase diagram of the $XXZ$ chain in a longitudinal field, validated by numerical simulations.

## Key findings

- Topological phase diagrams are mapped for $J<0$ and $J>0$ cases.
- Mean-field theory results agree with numerical MPS calculations.
- The geometric phase reveals topological properties of the $XXZ$ chain.

## Abstract

In this paper, we determine the geometric phase for the one-dimensional $XXZ$ Heisenberg chain with spin-$1/2$, the exchange couple $J$ and the spin anisotropy parameter $\Delta$ in a longitudinal field(LF) with the reduced field strength $h$. Using the Jordan-Wigner transformation and the mean-field theory based on the Wick's theorem, a semi-analytical theory has been developed in terms of order parameters which satisfy the self-consistent equations. The values of the order parameters are numerically computed using the matrix-product-state(MPS) method. The validity of the mean-filed theory could be checked through the comparison between the self-consistent solutions and the numerical results. Finally, we draw the the topological phase diagrams in the case $J<0$ and the case $J>0$.