# Universal phase transition and band structures for spinless nodal-line and Weyl semimetals

## TL;DR

This paper explores the phase transition mechanisms between spinless topological nodal-line semimetals and Weyl semimetals, classifying nodal lines and analyzing symmetry effects on their band structures.

## Contribution

It introduces a classification of nodal lines and details how symmetry-breaking influences the transition to Weyl semimetals in spinless systems.

## Key findings

- Nodal lines are classified into two types based on position and shape.
- Transition to Weyl semimetals occurs when time-reversal symmetry is broken for type-A nodal lines.
- Crystallographic symmetries can protect nodal lines even after inversion symmetry is broken.

## Abstract

We study a general phase transition between spinless topological nodal-line semimetal and Weyl semimetal phases. We classify topological nodal lines into two types based on their positions and shapes, and their phase transitions depends on their types. We show that the topological nodal-line semimetal becomes the Weyl semimetal by breaking time-reversal symmetry when the nodal lines enclose time-reversal invariant momenta (type-A nodal lines). We also discuss an effect of crystallographic symmetries determining the band structure of the topological nodal-line semimetals. Thanks to protection by the symmetries, the topological nodal-line semimetals can transit into the spinless Weyl semimetals or maintain the nodal lines in many crystals after inversion symmetry is broken.