Quantized charge polarization as a many-body invariant in (2+1)D crystalline topological states and Hofstadter butterflies

We show how topological phases of matter in (2+1)D can possess a non-trivial intrinsic quantized charge polarization P, even in the presence of non-zero Chern number and magnetic field. For invertible topological states, P is a Z₂ × Z₂, Z₃, Z₂, or Z₁ topological invariant in the presence of M = 2, 3, 4, or 6-fold rotational symmetry, lattice (magnetic) translational symmetry, and charge conservation. P manifests in the bulk of the system as (i) a fractional quantized contribution of P · b mod 1 to the charge bound to lattice disclinations and dislocations with Burgers vector b, (ii) a notion of linear momentum for additionally inserted magnetic flux, and (iii) an oscillatory system size dependent contribution to the effective 1d polarization on a cylinder. We study P in a variety of tight-binding models of spinless free fermions in a magnetic field. We derive predictions from topological field theory, which we match to numerical calculations for the effects (i)-(iii), demonstrating that these can be used to extract P from microscopic models in an intrinsically many-body way. We show how, given a high symmetry point O, there is a corresponding discrete shift S_O, such that P specifies the dependence of S_O on O. We derive colored Hofstadter butterflies, corresponding to the quantized value of P, which further refine the colored butterflies from the Chern number and discrete shift.