Fig. 4: 3D CI surface states.
From: Cubic 3D Chern photonic insulators with orientable large Chern vectors
Projected band-structure on the surface BZ with Miller index (100), for the interface between Ntriv = 8 cells of trivial insulator and NTI = 8 cells of TI, under a \({{{{{{{\bf{B}}}}}}}}={B}_{z}\hat{{{{{{{{\bf{z}}}}}}}}}\) field and with unit Chern number. Edge states cross the topological gap, highlighted in the shaded region. The extended BZ is fully displayed from −Z to Z, since it lacks of TRS due to application of a \(\hat{{{{{{{{\bf{z}}}}}}}}}\) directed bias field. b 2D dispersion of topological surface states on the projected BZ, in the energy range of the topological band-gap. c Equifrequency loops for a cut taken at midgap: the arrows on the plot indicate the direction of the group velocity vg = ∇kf∣a∣/c = (vy, vz). Blue and red colors correspond to chiral partners with opposite vy and same optical chirality \(\bar{c}\) which are located on opposite left/right sides of the interface (L, R). d Edge states dispersion along a direction normal to both the interface and the external magnetic field; highlighted in circles is a pair of counterpropagating m − th edge channels with ±ky: the spatial profile of their total electric field is shown in two following panels. e Counterpropagating chiral partners located on the two opposite left/right surfaces of the sample (L, R): wavefront propagation in the k direction indicated by White arrows on a xy cross section. f Spatial profile of the total electric field on a yz planar cut: green and blue arrows indicate respectively the group velocity vg and the total Poynting vector S, the last being obtained from a spatial integral of the quantity Re(E × H*), calculated directly from the EM field profiles. Differently from what expected from a conventional QHE edge channel, the Poynting vector displays a component parallel to the external magnetic field (Sz ≠ 0), suggesting both QHE and CME features and the possibility of energy flow along the magnetization axis: we excluded such a possibility in the SN 9, by summing up the contributions of all the edge channels constituting the surface state. Source data are provided as a Source Data file.
