  Topological materials are of great interest for applications in quantum
computing, providing intrinsic robustness against environmental noises. A
popular direction is to look for Majorana modes in integrated systems
interfaced with superconducting materials. However, is superconductivity
necessary for materials to exhibits non-abelian statistics? Here we predict
with strong theoretical and numerical evidences that there exist topologically
phases in a class of one-dimensional single crystals, which contain large
bandgaps and are within experimental reach. Specifically, the nontrivial Zak
phases are associated with gapless boundary states, which provides the
non-Abelian statistics required for constructing topologically-protected
quantum gates, even without superconductivity and magnetic field. Another
anomalous feature of these materials is that the electric polarization is
quantized and a transverse field induces a longitudinal polarization. This
novel material property provides an experimentally-accessible mechanism for
braiding the non-Abelian anyons.
