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

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 topological phase diagrams in the case J < 0 and the case J > 0.