A Study of Curvature Theory for Different Symmetry Classes of Hamiltonian

TL;DR

This paper explores curvature properties of Hamiltonians across different symmetry classes, revealing symmetry-dependent curvature behaviors, the role of torsion in topological phases, and applying curvature theory to topological matter models for the first time.

Contribution

It introduces the application of curvature theory to Hamiltonians of various symmetry classes in topological matter, analyzing their geometric properties and transformations.

Findings

Mirror symmetric curvature in BDI class

Absence of mirror symmetry in AIII class

Evidence of torsion in A class Hamiltonian

Abstract

We study and present the results of curvature for different symmetry classes (BDI, AIII and A) model Hamiltonians and also present the transformation of model Hamiltonian from one distinct symmetry class to other based on the curvature property. We observe the mirror symmetric curvature for the Hamiltonian with BDI symmetry class but there is no evidence of such behavior for Hamiltonians of AIII symmetry class. We show the origin of torsion and its consequences on the parameter space of topological phase of the system. We find the evidence of torsion for the Hamiltonian of A symmetry class. We present Serret-Frenet equations for all model Hamiltonians in $\mathbf{R}^3$ space. To the best of our knowledge, this is the first application of curvature theory to the model Hamiltonian of different symmetry classes which belong to the topological state of matter.