Condensed Matter Seminar "Integers in gapped quantum lattice systems"
We propose an index for gapped quantum lattice systems that conserve a U(1)-charge. This index takes integer values and it is therefore stable under perturbations. Our formulation is general, but we show that the index reduces to (i) an index of projections in the non-interacting case, (ii) the filling factor for translational invariant systems, (iii) the quantum Hall conductance in the two-dimensional setting without any additional symmetry. Example (ii) recovers the Lieb-Schultz-Mattis theorem, (iii) provides a new and short proof of quantization of Hall conductance in interacting many-body systems. Additionally, we show Avron-Dana-Zak relations for both integer and fractional quantum Hall effect.