Special Condensed Matter Seminar: Masaki Oshikawa, University of Tokyo - “Polarization, Large Gauge Invariance, and Quantum Hall Effect on Lattice”

Quantum systems on a non-simply connected space possess a
“large” gauge invariance. Laughlin utilized this to explain quantum
Hall effect. Later, it was applied to elucidate a universal relation
between filling factor and energy spectrum in quantum many-body
systems on periodic lattices (Lieb-Schultz-Mattis-M.O.-Hastings).
Somewhat surprisingly, the large gauge invariance is also deeply
related to modern theory of electric polarization developed by
Resta et al. Combining these ideas together, we can derive a constraint
on the Hall conductivity of a many-particle system on a periodic
lattice, with a given magnetic flux per plaquette and particle density