Crystalline gauge fields and quantized discrete geometric response for Abelian topological phases with lattice symmetry

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. The discrete torsion vector implies a type of crystal momentum fractionalization that is only non-trivial 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, a fractionally quantized angular momentum of disclinations, rotationally symmetric fractional charge polarization and its angular momentum counterpart, constraints on charge and angular momentum per unit cell, and quantized momentum bound to dislocations and units of area. The fractionally quantized charge polarization, which is non-trivial only on a lattice with $2$, $3$, and $4$-fold rotation symmetry, implies a fractional charge bound to lattice dislocations and a fractional charge per unit length along the boundary. An important role is played by a finite group grading on Burgers vectors, which depends on the point group symmetry of the lattice.