Flat band in twisted bilayer Bravais lattices

Band engineering in twisted bilayers of the five generic two-dimensional Bravais networks is demonstrated. We first derive symmetry-based constraints on the interlayer coupling, which helps us to predict and understand the shape of the potential barrier for the electrons under the influence of the moir\'{e} structure without reference to microscopic details. It is also pointed out that the generic constraints becomes best relevant when the typical length scale of the microscopic interlayer coupling is moderate. The concepts are numerically demonstrated in simple tight-binding models to show the band flattening due to the confinement into the potential profile fixed by the generic constraints. On the basis of the generic theory, we propose the possibility of anisotropic band flattening, in which quasi one-dimensional band dispersion is generated from relatively isotropic original band dispersion. In the strongly correlated regime, anisotropic band flattening leads to a spin-orbital model where intertwined magnetic and orbital ordering can give rise to rich physics.Comment: 10 page

Section Chern number for a 3D photonic crystal and the bulk-edge correspondence

We have characterized the robust propagation modes of electromagnetic waves in helical structures by the section Chern number that is defined for a two-dimensional (2D) section of the three- dimensional (3D) Brillouin zone. The Weyl point in the photonic bands is associated with a dis- continuous jump of the section Chern number. A spatially localized Gaussian basis set is used to calculate the section Chern numbers where we have implemented the divergence-free condition on each basis function in 3D. The validity of the bulk-edge correspondence in a 3D photonic crystal is discussed in relation to the broken inversion symmetry

ZN Berry Phases in Symmetry Protected Topological Phases

We show that the ZN Berry phase (Berry phase quantized into 2π/N) provides a useful tool to characterize symmetry protected topological phases with correlation that can be directly computed through numerics of a relatively small system size. The ZN Berry phase is defined in a N−1-dimensional parameter space of local gauge twists, which we call the “synthetic Brillouin zone,” and an appropriate choice of an integration path consistent with the symmetry of the system ensures exact quantization of the Berry phase. We demonstrate the usefulness of the ZN Berry phase by studying two 1D models of bosons, SU(3) and SU(4) Affleck-Kennedy-Lieb-Tasaki models, where topological phase transitions are captured by Z3 and Z4 Berry phases, respectively. We find that the exact quantization of the ZN Berry phase at the topological transitions arises from a gapless band structure (e.g., Dirac cones or nodal lines) in the synthetic Brillouin zone