Understand
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.
- Here we develop a theory of symmetry-protected quantized invariants for topological phases defined on a lattice, where quantized invariants with no continuum analog can arise.
- We develop topological field theories using discrete crystalline gauge fields to fully characterize quantized invariants of (2+1)D Abelian topological orders with symmetry group $G = U(1) \times G_{\text{space}}$, where $G_{\text{space}}$ consists of orientation-preserving space group symmetries on the lattice.
- We show how discrete rotational and translational symmetry fractionalization can be 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.