Crystalline gauge fields and discrete geometric response for Abelian topological phases with lattice symmetry
Abstract
Clean isotropic quantum Hall fluids in the continuum possess a host of symmetryprotected quantized invariants, such as the Hall conductivity, shift and Hall viscosity, and fractional quantum numbers of quasiparticles. Here we develop topological field theories using discrete crystalline gauge fields to fully characterize quantized invariants of (2+1)D Abelian topological orders in the presence of symmetry group $G = U(1) \times G_{\text{space}}$, where $G_{\text{space}}$ consists of orientationpreserving space group symmetries on the lattice. Discrete rotational and translational symmetry fractionalization is characterized by a discrete spin vector, a discrete torsion vector which has no analog in the continuum or in the absence of lattice rotation symmetry, and an area vector, which also has no analog in the continuum. In particular, we find a type of crystal momentum fractionalization that is only nontrivial for $2$, $3$, and $4$fold rotation symmetry. The quantized topological response theory includes a discrete version of the shift which binds fractional charge to disclinations and corners, rotationally symmetric fractional charge polarization, constraints on charge filling and their discrete angular momentum counterparts, momentum bound to dislocations and units of area, and all of their duals.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 arXiv:
 arXiv:2005.10265
 Bibcode:
 2020arXiv200510265M
 Keywords:

 Condensed Matter  Strongly Correlated Electrons;
 High Energy Physics  Theory;
 Quantum Physics
 EPrint:
 5 pages + 14 pages Appendix