Recent discoveries on topological characterization of gapless systems have attracted interest in both theoretical studies and experimental realizations. Examples of such gapless topological phases are Weyl semimetals, which exhibit three-dimensional (3D) Dirac cones (Weyl points), and nodal line semimetals, which are characterized by line nodes (two bands touching along a line). Inspired by our...

The dynamic engineering of band structures for ultracold atoms in optical lattices represents an innovative approach to understanding and exploring the fundamental principles topological matter. In particular, folded Floquet spectrum determines associated topology via inversion. We experimentally theoretically study two-frequency phase modulation asymmetrically hybridize lowest two bands a one-...

We prove that the spectrum of a Schrödinger operator with a potential which is periodic in certain directions and superexponentially decaying in the others is purely absolutely continuous. Therefore, we reduce the operator using the Bloch-Floquet-Gelfand transform in the periodic variables, and show that, except for at most a set of quasi-momenta of measure zero, the reduced operators satisfies...