Glide symmetry protected higher-order topological insulators from semimetals with butterfly-like nodal lines

Xiaoting Zhou,Hsin Lin,Nicholas Kioussis,Cheng-Yi Huang,Mikel Iraola,Juan L Mañes,Chuang-Han Hsu,Maia G Vergniory

doi:10.1038/s41524-021-00672-9

Abstract

Most topological insulators (TIs) discovered today in spinful systems can be transformed from topological semimetals (TSMs) with vanishing bulk gap via introducing the spin-orbit coupling (SOC), which manifests the intrinsic links between the gapped topological insulator phases and the gapless TSMs. Recently, we have discovered a family of TSMs in time-reversal invariant spinless systems, which host butterfly-like nodal-lines (NLs) consisting of a pair of identical concentric intersecting coplanar ellipses (CICE). In this Communication, we unveil the intrinsic link between this exotic class of nodal-line semimetals (NLSMs) and a {{mathbb{Z}}}_{4} = 2 topological crystalline insulator (TCI), by including substantial SOC. We demonstrate that in three space groups (i.e., Pbam (No.55), P4/mbm (No.127), and P42/mbc (No.135)), the TCI supports a fourfold Dirac fermion on the (001) surface protected by two glide symmetries, which originates from the intertwined drumhead surface states of the CICE NLs. The higher order topology is further demonstrated by the emergence of one-dimensional helical hinge states, indicating the discovery of a higher order topological insulator protected by a glide symmetry.

Highlights

The discovery of the quantum spin Hall effect (QSHE)[1–3] and topological insulators (TIs)[4–6] which are protected by time-reversal symmetry (TRS), has indicated that symmetry plays a crucial role in classifying the topology of free fermion states[7–9]
The concentric intersecting coplanar ellipses (CICE) can be sustained by two glide mirror symmetries and only nine space groups (SGs) are feasible to host it[37]
If M can be prohibited by any spatial symmetry Q, the anomalous gapless surface states will persist, Fig. 3c, we find that the hinge band connectivity is the analog of the quantum spin Hall effect

Summary

INTRODUCTION

The discovery of the quantum spin Hall effect (QSHE)[1–3] and topological insulators (TIs)[4–6] which are protected by time-reversal symmetry (TRS), has indicated that symmetry plays a crucial role in classifying the topology of free fermion states[7–9]. Topological crystalline insulators (TCIs)[10] are protected by other space-group symmetries (Q)[11,12], such as mirror[13] and rotational symmetries[14–16] Such type of systems are known to harbor symmetry-protected topological (SPT) phase[17]. An nth order topological insulator describes a TCI in d-dimensions having symmetry protected (d − n)-dimensional gapless boundary states, but gapped otherwise, when the sample geometry is properly selected, being compatible with Q. NLSMs with NLs integrated in various configurations have been studied under the assumption that the spin-orbit coupling (SOC) is negligible or absent, e.g., a chain link[30–32], a Hopf link[31], and a knot[33] From another perspective, it is intuitive to raise the question whether additional topology could be unearthed when these intricate degenerate links are gapped out by substantial.

Zhou et al 2

RESULTS

DISCUSSION