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Gauge Field of Bloch Electrons in dual space

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. Superconductivity and Quantized Flux.

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Gauge Field of Bloch Electrons in dual space

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  1. 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

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

  3. Meissner effect but if C contain Dirac point Hall conductance Kohmoto 1985 no magnetic field time reversal symmetry isnotbroken Hall conductance has to be zero sum of fluxes is zero monopole-antimonopole confinement

  4. Dual Space of Honeycomb lattice

  5. Hofstadter Butterfly nonzero TKNN Hall conductance

  6. nonzero TKNN Hall conductance

  7. 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 massive massless Non degenerate doubly degenerate

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

  9. tAtB 5% Period 3 modulation Period 2 modulation

  10. Summary Honeycomb Lattice: sublattice A and B, non-Bravaislattice topological dual space: Dirac zero mode: break down of Bloch’s theorem ground state degeneracy-> Jahn-Tellerperiod 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

  11. 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

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