# Coexistence of four-band nodal rings and triply-degenerate nodal points in centrosymmetric metal diborides

## TL;DR

This paper predicts novel topological band features in metal diborides, including four-band nodal rings with Dirac-cone surface states and triply-degenerate nodal points that transform into Dirac points with spin-orbit coupling, offering new insights into topological metals.

## Contribution

It introduces the concept of four-band nodal rings and their unique surface states, along with the coexistence of triply-degenerate nodal points in centrosymmetric metal diborides, expanding the understanding of topological band crossings.

## Key findings

- Identification of four-band nodal rings requiring four entangled bands
- Discovery of Dirac-cone-like surface states associated with FNRs
- Transformation of TNPs into Dirac points with spin-orbit coupling

## Abstract

Topological metals with protected band-crossing points have been attracting great interest. Here we report novel topological band features in a family of metal diboride materials. Using first- principles calculations, we show that these materials are metallic, and close to Fermi level, there appears coexistence of one pair of nodal rings and one pair of triply-degenerate nodal points (TNPs). The nodal ring here is distinct from the previously studied ones in that its formation requires four entangled bands, not just two as in previous cases, hence it is termed as a four-band nodal ring (FNR). Remarkably, we show that FNR features Dirac-cone-like surface states, in contrast to the usual drumhead surface states for two-band nodal rings. Due to the presence of inversion symmetry, the TNP here is also different from those discussed previously in inversion-asymmetric systems. Especially, when spin-orbit coupling is included, the TNP here transforms into a novel Dirac point that is close to the borderline between the type-I and type-II Dirac point categories. We discuss their respective symmetry protections, and construct effective models for their characterization. The large linear energy range (> 2 eV) in these materials should facilitate the experimental detection of the signatures of these nontrivial band crossings.