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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
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Superconductivity and Quantized Flux
ratio of wave functions on sublattice A and B
write
U(1) field
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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
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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
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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
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Jahn-Teller instability mode
K-K’
period 3 direction
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tA tB 5%
Period 2 modulation
Period 3 modulation
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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
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No magnetic fieldTime reversal symmetry -> zero Hall conductance -> monopole confinement
With a magnetic fieldHofstadter mechanism, 2q dual subspaces, nonzero TKNN Hall conductancemonopole deconfinement