Topological near fields generated by topological structures NatureComms ScienceAdvances
Thanks to the periodicity of the quasimomentum, the two-dimensional Brillouin zone has a toruslike structure, in which top-bottom and left-right edges should be identified. Energy spectrum of the Harper-Hofstadter model, which is called the Hofstadter butterfly.
. The colored Hofstadter butterfly in which the color of each band gap indicates the topological invariant of the gap, given by the sum of the Chern numbers of all bands below. Warm colors indicate a positive topological invariant, whereas the cool colors indicate a negative topological invariant. The horizontal axes are the energies and the vertical axes are the flux A plaquette of the Haldane model. In addition to the usual nearest-neighbor hoppings, there are also complex next-nearest-neighbor hoppings. For the latter, hopping along the arrows carries a phase of. The phase diagram of the Haldane model as a function of the next-nearest-neighbor hopping phase. Bulk band structure of the honeycomb lattice with nearest-neighbor hopping only. Conical touchings of the bands are Dirac points. Typical bulk band structure of the Haldane model in the presence of a band gap. Typical band structure for a gapless honeycomb lattice with zigzag edges. Flat-band localized edge states connect between the Dirac cones, with one per edge. Typical band structure with edges on both sides of the system when a topological gap opens. Red and blue lines indicate edge states on left and right edges, respectively.A schematic illustration of how the interface of two regions with different topological numbers can host a localized state. The spatial dependence of the mass term
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