Topological Insulators beyond Energy Band Characterization

Topological phases of matter are generally characterized by topological properties of energy bands of a system. Their transitions under preserved symmetries occur through closing a gap of energy bands, leading to topologically protected edge states in energy spectra in topological phases. Here we predict a new topological phase that emerges through closing a gap of bands constructed by energy bands, instead of through closing an energy gap with preserved symmetries. From this perspective, topological phases may arise from topological properties of the "bands of bands" associated with their gap closure and corresponding edge states. We demonstrate this idea by studying a tight-binding model. We find that the Wannier bands constructed by energy bands exhibit a gap closure associated with a change of a winding number, while the energy bands remain gapped and trivial without any zero energy modes. In addition, the topological Wannier bands give rise to quantized edge polarizations. Since the emergence of this topological phase does not involve any energy gap closure, we expect its appearance under unitary time evolution. Indeed, this phase appears as we perform a quench dynamics. Our study opens a new direction for exploring topological phases beyond conventional energy band characterization.