Two-atom-thin topological crystalline insulators lacking out of plane inversion symmetry

TL;DR

This paper demonstrates a two-atom-thick topological crystalline insulator that maintains its topological properties at both bulk and monolayer thickness, challenging the necessity of certain symmetries for topological protection.

Contribution

It shows that zero-energy surface states in TCIs do not require in-plane rotational symmetry and that topological properties persist at monolayer thickness, using Pfaffian calculations.

Findings

Zero-energy states persist without in-plane rotational symmetry.

Topological properties are invariant at monolayer and bulk scales.

Topological characterization is confirmed by Pfaffian calculations.

Abstract

A two-dimensional topological crystalline insulator (TCI) with a single unit cell (u.c.) thickness is demonstrated here. To that end, one first shows that tetragonal ($C_4$ in-plane) symmetry is not a necessary condition for the creation of zero-energy metallic surface states on TCI slabs of finite-thicknesses, because zero-energy states persist even as all the in-plane rotational symmetries--furnishing topological protection--are completely removed. In other words, zero-energy levels on the model are not due to (nor are they protected by) topology. Furthermore, effective twofold energy degeneracies taking place at few discrete $k-$points away from zero energy in the bulk Hamiltonian--that are topologically protected--persist at the u.c.~thickness limit. The chiral nature of the bulk TCI Hamiltonian permits creating a $2\times 2$ square Hamiltonian, whose topological properties remarkably hold invariant at both the bulk and at the single u.c.~thickness limits. The identical topological characterization for bulk and u.c.-thick phases is further guaranteed by a calculation involving Pfaffians. This way, a two-atom-thick TCI is deployed hereby, in a demonstration of a topological phase that holds both in the bulk, and in two dimensions.