Clean isotropic quantum Hall fluids in the continuum possess a host of symmetry-protected 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) × Gspace, where Gspace consists of orientation-preserving 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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.