An Interplay of Topology and Quantized Geometric Phase for two Different Symmetry-Class Hamiltonians

TL;DR

This paper explores how symmetry, topology, and geometric phase interact in two different symmetry-class Hamiltonians, revealing that similar symmetries can lead to different topological behaviors and emphasizing the nuanced role of auxiliary space in topological states.

Contribution

It provides a novel analysis of the relationship between symmetry, topology, and geometric phase, highlighting that auxiliary space origin alone does not determine topological states across different Hamiltonian classes.

Findings

Auxiliary space origin is necessary but not sufficient for topological states.

Same symmetry-class Hamiltonians can exhibit different topological behaviors.

Distinct topological and geometric phase behaviors are observed in different symmetry classes.

Abstract

Study of symmetry, topology and geometric phase can reveal many new and interesting results on the topological states of matter. Here we present a completely new and interesting result of symmetry, topology and quantization of geometric phase along with the physical explanation for two different symmetry classes. We present a detailed study of the auxiliary space for two different symmetry classes of Hamiltonians. We show explicitly that the origin of the auxiliary space inside the curve is only a necessary condition but it is not a sufficient condition for the topological state. One of the most interesting results is that same symmetry-class Hamiltonians show different behaviour in topology and quantized geometric phase.