# Antiunitary symmetry protected higher order topological phases

## TL;DR

This paper uncovers an antiunitary symmetry that protects higher-order topological phases, enabling robust corner and hinge modes even with certain symmetry-breaking perturbations.

## Contribution

It introduces an antiunitary symmetry framework that stabilizes higher-order topological phases against specific weak symmetry-breaking effects.

## Key findings

- Corner modes are protected by antiunitary symmetry at zero energy.
- HOT phases persist with weak anomalous Hall and density wave orders.
- Majorana zero modes survive in HOT superconductors with perturbations.

## Abstract

Higher-order topological (HOT) phases feature boundary (such as corner and hinge) modes of codimension $d_c>1$. We here identify an \emph{antiunitary} operator that ensures the spectral symmetry of a two-dimensional HOT insulator and the existence of cornered localized states ($d_c=2$) at precise zero energy. Such an antiunitary symmetry allows us to construct a generalized HOT insulator that continues to host corner modes even in the presence of a \emph{weak} anomalous Hall insulator and a spin-orbital density wave orderings, and is characterized by a quantized quadrupolar moment $Q_{xy}=0.5$. Similar conclusions can be drawn for the time-reversal symmetry breaking HOT $p+id$ superconductor and the corner localized Majorana zero modes survive even in the presence of weak Zeeman coupling and $s$-wave pairing. Such HOT insulators also serve as the building blocks of three-dimensional second-order Weyl semimetals, supporting one-dimensional hinge modes.