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Gauge Field of Bloch Electrons in dual space

First considered in context of QHEKohmoto 1985

Principle of Quantum MechanicsEigenstate does not depend on overall phase factor

Gauge invariant magnetic field in dual space

Superconductivity and Quantized Flux

ratio of wave functions on sublattice A and B

write

U(1) field

Meissner effect

Hall conductanceKohmoto 1985

but if C contain Dirac point

no magnetic fieldtime reversal symmetry is not brokenHall conductance has to be zerosum of fluxes is zero monopole-antimonopole confinement

Dual Space of Honeycomb lattice

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Quantum Hall Effect

magnetic field induces breaking of the timereversal symmetry

Jx = σxyEy

•Laughlin•TKNN

Key advantages:magnetic fieldDissipationless response, sincetopological origin, due to gauge field in momentum spaceKohmoto (1985)also appear in Berry phase expression (k->adiabatic parameter)

Gauge field does not related to external magnetic field explicitly

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Hofstadter Butterfly

nonzero TKNN Hall conductance

nonzero TKNN Hall conductance

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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

CrystalBand Jahn-Teller effectPeirerls instability of one-dimensional half-filled band(dimerization)

Dirac Mode

massive massless

doubly degenerateNon degenerate

Jahn-Teller instability mode

K-K’

period 3 direction

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tA tB 5%

Period 2 modulation

Period 3 modulation

SummaryHoneycomb Lattice: sublattice A and B, non-Bravais latticetopological dual space:

Dirac zero mode: break down of Bloch’s theoremground 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

No magnetic fieldTime reversal symmetry -> zero Hall conductance -> monopole confinement

With a magnetic fieldHofstadter mechanism, 2q dual subspaces, nonzero TKNN Hall conductancemonopole deconfinement