Antiunitary symmetry protected higher-order topological phases

Higher-order topological (HOT) phases feature boundary (such as corner and hinge) modes of codimension d_c>1 . We here identify an 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 weak anomalous Hall insulator and spin-orbital density-wave orderings, and is characterized by a quantized quadrupolar moment Q_{x y}=0.5 . Similar conclusions can be drawn for the time-reversal symmetry breaking HOT p +i d 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.