# Diagnosis of topological nodal lines with nontrivial monopole charge in the presence of rotation symmetries

## TL;DR

This paper presents a comprehensive method to diagnose topological nodal lines with monopole charges protected by rotation symmetries, based on band inversion and eigenvalue analysis, extending previous frameworks.

## Contribution

It introduces a new diagnosis scheme for topological nodal lines with monopole charges, applicable to systems with rotation symmetries and band crossings, surpassing prior methods.

## Key findings

- Nodal lines are generated during shallow band inversion at the $Gamma$-point.
- The types and numbers of nodal lines relate to rotation and inversion eigenvalues.
- A $$-type monopole charge is established for certain nodal lines with fourfold or sixfold rotation axes.

## Abstract

We discuss a general diagnosis scheme for topological nodal lines protected by time-reversal symmetry and space-inversion symmetry, in the absence of spin-orbit coupling. It is shown that when a shallow band inversion (with a small inverted gap) appears at the $\Gamma$-point in the Brillouin zone, nodal lines are generated along with the inversion. Much information on the types and numbers of these nodal lines can be related to the rotation eigenvalues and inversion eigenvalues of the inverted bands. In addition, we establish a $\mathbb{Z}$-type monopole charge for nodal lines with fourfold or sixfold rotation axis. We point out that the new monopole charge is only well-defined in a fragile sense, that is, when the orbitals near the Fermi energy are restricted to those having the same sign under a twofold rotation. The diagnosis scheme goes beyond the one discussed in Phys. Rev. X 8, 031069 (2018), particularly when there is a band crossing along some high-symmetry line, or when the nodal line has non-unity monopole charge.