Pfaffian formalism for higher-order topological insulators

TL;DR

This paper extends the Pfaffian formalism from topological insulators to higher-order topological insulators, providing new tools for their classification and analysis, especially in three dimensions with specific symmetries.

Contribution

The authors generalize the Pfaffian formalism to 3D chiral HOTIs protected by $C_4$ and $T$ symmetries, linking HOTIs to TIs and enabling efficient $Z_2$ index computation.

Findings

Generalized Fu-Kane parity criterion for HOTIs

Developed a method to compute $Z_2$ index without a global gauge

Revealed a fundamental link between TIs and HOTIs

Abstract

We generalize the Pfaffian formalism, which has been playing an important role in the study of time-reversal invariant topological insulators (TIs), to 3D chiral higher-order topological insulators (HOTIs) protected by the product of four-fold rotational symmetry $C_4$ and the time-reversal symmetry $T$. This Pfaffian description reveals a deep and fundamental link between TIs and HOTIs, and allows important conclusions about TIs to be generalized to HOTIs. As examples, we demonstrate in the Letter how to generalize Fu-Kane's parity criterion for TIs to HOTIs, and also present a general method to efficiently compute the $Z_2$ index of 3D chiral HOTIs without a global gauge.