Chern number in Ising models with spatially modulated real and complex fields

TL;DR

This paper investigates a one-dimensional transverse field Ising model with spatially modulated real and complex fields, using the Chern number to characterize quantum phases and how these fields alter phase boundaries.

Contribution

It introduces a method to compute the Chern number in modulated Ising models, revealing how real and complex fields reshape phase diagrams.

Findings

Spatially modulated fields change phase boundaries.

Chern numbers distinguish different quantum phases.

Exact solutions relate to Dirac and biorthonormal inner products.

Abstract

We study an one-dimensional transverse field Ising model with additional periodically modulated real and complex fields. It is shown that both models can be mapped on a pseudo spin system in the k space in the aid of an extended Bogoliubov transformation. This allows us to introduce the geometric quantity, the Chern number, to identify the nature of quantum phases. Based on the exact solution, we find that the spatially modulated real and complex fields rearrange the phase boundaries from that of the ordinary Ising model, which can be characterized by the Chern numbers defined in the context of Dirac and biorthonormal inner products, respectively.