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

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