Majorana charges, winding numbers and Chern numbers in quantum Ising models

TL;DR

This paper explores the topological properties of quantum Ising models, establishing the equivalence of Chern numbers, winding numbers, and a newly introduced Majorana charge to characterize different phases.

Contribution

It introduces the concept of Majorana charge and demonstrates its equivalence to Chern and winding numbers in quantum Ising models with various interactions.

Findings

Majorana charge equals Chern and winding numbers.

Topological invariants characterize phase diagrams.

Analytical and numerical verification of Majorana charge.

Abstract

Mapping a many-body state on a loop in parameter space is a simple way to characterize a quantum state. The connections of such a geometrical representation to the concepts of Chern number and Majorana zero mode are investigated based on a generalized quantum spin system with short and long-range interactions. We show that the topological invariants, the Chern numbers of corresponding Bloch band is equivalent to the winding number in the auxiliary plane, which can be utilized to characterize the phase diagram. We introduce the concept of Majorana charge, the magnitude of which is defined by the distribution of Majorana fermion probability in zero-mode states, and the sign is defined by the type of Majorana fermion. By direct calculations of the Majorana modes we analytically and numerically verify that the Majorana charge is equal to Chern numbers and winding numbers.