Square-root topological phase with time-reversal and particle-hole symmetry

Abstract

Square-root topological phases have been discussed mainly for systems with chiral symmetry. In this paper, we analyze the topology of the squared Hamiltonian for systems preserving time-reversal and particle-hole symmetry. Our analysis elucidates that two-dimensional systems of class CII host helical edge states due to the nontrivial topology of the squared Hamiltonian in contrast to the absence of ordinary topological phases. The emergence of helical edge modes is demonstrated by analyzing a toy model. We also show the emergence of surface states induced by the nontrivial topology of the squared Hamiltonian in three dimensions.