# $d$-Orbital Topological Insulator and Semimetal in Antifluorite Cu$_2$S Family: Contrasting Spin Helicities, Nodal Box, and Hybrid Surface States

## TL;DR

This paper identifies a new class of three-dimensional $d$-orbital topological materials in the Cu$_2$S family, exhibiting diverse topological phases driven by spin-orbit coupling and symmetry breaking, including insulators, semimetals, and Weyl states.

## Contribution

It introduces the $d$-orbital topological phases in Cu$_2$S, revealing how SOC sign and symmetry breaking lead to various topological states with unique surface features.

## Key findings

- Negative SOC yields topological insulators with Dirac surface states.
- Positive SOC results in topological semimetals with contrasting helicities.
- Breaking symmetry induces nodal lines and Weyl points, with coexisting Fermi arcs and Dirac cones.

## Abstract

We reveal a class of three-dimensional $d$-orbital topological materials in the antifluorite Cu$_2$S family. Derived from the unique properties of low-energy $t_{2g}$ states, their phases are solely determined by the sign of spin-orbit coupling (SOC): topological insulator for negative SOC, whereas topological semimetal for positive SOC; both having Dirac-cone surface states but with contrasting helicities. With broken inversion symmetry, the semimetal becomes one with a nodal box consisting of butterfly-shaped nodal lines that are robust against SOC. Further breaking the tetrahedral symmetry by strain leads to an ideal Weyl semimetal with four pairs of Weyl points. Interestingly, the Fermi arcs coexist with a surface Dirac cone on the (010) surface, as required by a $Z_2$-invariant.