Morse theory study on the evolution of nodal lines in $\mathcal{PT}$-symmetric nodal-line semimetals

TL;DR

This study applies Morse theory to classify and understand the possible topological changes of nodal lines in $ ext{PT}$-symmetric nodal-line semimetals, revealing only three types of topology change events.

Contribution

It introduces a systematic classification of nodal line topology changes using Morse theory, extending to systems with additional symmetries.

Findings

Only three types of topology changes are possible: creation, reconnection, and annihilation.

The topology changes are characterized by Morse indices 0, 1, and 2.

The theory is extended to include rotational and mirror symmetries.

Abstract

A nodal-line semimetal is a topological gapless phase containing one-dimensional degeneracies called nodal lines. The nodal lines are deformed by a continuous change of the system such as pressure and they can even change their topology, but it is not systematically understood what kind of changes of topology of nodal lines are possible. In this paper, we classify the events of topology change of nodal lines by the Morse theory and reveal that only three types of topology changes of nodal lines, i.e., creation, reconnection, and annihilation, are possible in the spinless nodal-line semimetal protected by inversion and time-reversal symmetries. They are characterized by an index having the values 0, 1, and 2 for the above three types in the Morse theory. Moreover, we extend our theory to systems with rotational symmetries and mirror symmetry and disclose the possible events of topology change of nodal lines under each symmetry.