# Chiral topological insulating phases from three-dimensional nodal loop semimetals

## TL;DR

This paper introduces a topological Z index for 3D chiral insulators with P*T symmetry, linking nodal loops to topological invariants and edge states, verified through a Bi2Te3 family material.

## Contribution

It defines a new topological invariant for 3D nodal loop semimetals with chiral symmetry and P*T symmetry, connecting geometric properties to topological phases.

## Key findings

- Defined a Z invariant as a winding number for nodal loops
- Established correspondence between winding numbers and Dirac cone edge states
- Validated the approach with a Bi2Te3 family topological insulator model

## Abstract

We identify a topological Z index for three dimensional chiral insulators with P*T symmetry where two Hamiltonian terms define a nodal loop. Such systems may belong in the AIII or DIII symmetry class. The Z invariant is a winding number assigned to the nodal loop and has a correspondence to the geometric relation between the nodal loop and the zeroes of the gap terms. Dirac cone edge states under open boundary conditions are in correspondence with the winding numbers assigned to the nodal loops. We verify our method with the low-energy effective Hamiltonian of a three-dimensional material of topological insulators in the Bi$_2$Te$_3$ family.