Topological Band Theory
Periodic lattices let us label electronic states by crystal momentum. Explore each principle from mathematical definition to physical evidence and limitations.
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Concept 15Bloch theorem and Brillouin torusPeriodic lattices let us label electronic states by crystal momentum.Concept 16Chern number from Berry curvatureIntegrating Berry curvature over a closed momentum-space surface yields an integer topological invariant.Concept 17TKNN quantization and Hall conductanceHall conductivity can be locked to integers because it is the integral of band geometry.Concept 18Landau levels and integer quantum Hall effectA perpendicular magnetic field organizes two-dimensional electron motion into quantized cyclotron levels.Concept 19Bulk–boundary correspondenceA change of bulk topology across an interface requires gap-crossing boundary states under suitable assumptions.Concept 20Haldane model and Chern insulatorsComplex next-neighbor hopping can generate a quantum Hall-like phase without net magnetic flux per unit cell.Concept 21SSH model and winding numberA one-dimensional dimerized chain has distinct gapped winding sectors.Concept 22Zak phase and polarizationThe Berry phase across a one-dimensional Brillouin zone is related to electric polarization.Concept 23Time-reversal symmetry and Z2 topologyTime-reversal symmetry can protect helical edge modes even when total Chern number vanishes.Concept 24Topological insulators in three dimensionsThree-dimensional time-reversal-invariant insulators can have robust surface Dirac states.Concept 25Weyl semimetals and monopolesWeyl nodes act as sources or sinks of Berry curvature in three-dimensional momentum space.Concept 26Disorder and topological robustnessTopological properties can survive many imperfections because small perturbations cannot change a discrete invariant continuously.