8. Gauge Field of Bloch Electrons in dual space First considered in context of QHE Kohmoto 1985 Principle of Quantum Mechanics Eigenstate does not depend on overall phase factor Gauge invariant magnetic field in dual space

9. Superconductivity and Quantized Flux ratio of wave functions on sublattice A and B write U(1) field

10. Meissner effect Hall conductance Kohmoto 1985 but if C contain Dirac point no magnetic field time reversal symmetry is not broken Hall conductance has to be zero sum of fluxes is zero monopole-antimonopole confinement

11. Dual Space of Honeycomb lattice

13. Hofstadter Butterfly nonzero TKNN Hall conductance

19. Jahn-Teller Theorem Localized Object(molecule, impurity) If a molecule has degeneracy in electronic energy, there is at least one instability mode of symmetry breaking Crystal Band Jahn-Teller effect Peirerls instability of one-dimensional half-filled band(dimerization) Dirac Mode massivemassless doubly degenerate Non degenerate

20. Jahn-Teller instability mode K-K’ period 3 direction

21. t A t B 5% Period 2 modulation Period 3 modulation

22. Summary Honeycomb Lattice: sublattice A and B, non-Bravais lattice topological dual space: Dirac zero mode: break down of Bloch’s theorem ground state degeneracy-> Jahn-Teller period 3 lattice modulation in U(1) gauge field of Bloch electrons in dual space Type II superconductor, Abrikosov quantized vortex, +1, - 1 Magnetic monopole: non-Abelian gauge theory broken to U(1), cf ‘t Hooft

23. No magnetic field Time reversal symmetry -> zero Hall conductance -> monopole confinement With a magnetic field Hofstadter mechanism, 2q dual subspaces, nonzero TKNN Hall conductance monopole deconfinement