Symmetry indicators and anomalous surface states of topological crystalline insulators

TL;DR

This paper develops a unified theoretical framework to analyze topological crystalline insulators (TCIs) using surface Dirac fermions, revealing how symmetry indicators predict gapless surface states across all crystal symmetries.

Contribution

It introduces a comprehensive surface theory for TCIs, connecting symmetry indicators to the presence of gapless surface states for all 230 space groups.

Findings

All 32 point groups and 230 space groups analyzed.

Symmetry indicators reliably predict gapless surface states.

Connection established between bulk topology and surface phenomena.

Abstract

The rich variety of crystalline symmetries in solids leads to a plethora of topological crystalline insulators (TCIs) featuring distinct physical properties, which are conventionally understood in terms of bulk invariants specialized to the symmetries at hand. While isolated examples of TCI have been identified and studied, the same variety demands a unified theoretical framework. In this work, we show how the surfaces of TCIs can be analyzed within a general surface theory with multiple flavors of Dirac fermions, whose mass terms transform in specific ways under crystalline symmetries. We identify global obstructions to achieving a fully gapped surface, which typically lead to gapless domain walls on suitably chosen surface geometries. We perform this analysis for all 32 point groups, and subsequently for all 230 space groups, for spin-orbit-coupled electrons. We recover all previously discussed TCIs in this symmetry class, including those with "hinge" surface states. Finally, we make connections to the bulk band topology as diagnosed through symmetry-based indicators. We show that spin-orbit-coupled band insulators with nontrivial symmetry indicators are always accompanied by surface states that must be gapless somewhere on suitably chosen surfaces. We provide an explicit mapping between symmetry indicators, which can be readily calculated, and the characteristic surface states of the resulting TCIs.