  It has been known that an anti-unitary symmetry such as time-reversal or
charge conjugation is needed to realize Z2 topological phases in
non-interacting systems. Topological insulators and superconducting nanowires
are representative examples of such Z2 topological matters. Here we report the
first-known Z2 topological phase protected by only unitary symmetries. We show
that the presence of a nonsymmorphic space group symmetry opens a possibility
to realize Z2 topological phases without assuming any anti-unitary symmetry.
The Z2 topological phases are constructed in various dimensions, which are
closely related to each other by Hamiltonian mapping. In two and three
dimensions, the Z2 phases have a surface consistent with the nonsymmorphic
space group symmetry, and thus they support topological gapless surface states.
Remarkably, the surface states have a unique energy dispersion with the Mobius
twist, which identifies the Z2 phases experimentally. We also provide the
relevant structure in the K-theory.
