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

We show how to define a quantized many-body charge polarization \(\vec{\mathscr{P}}\) for (2+1)D topological phases of matter, even in the presence of non-zero Chern number and magnetic field. For invertible topological states, \(\vec{\mathscr{P}}\) is a \(\mathbb{Z}_2 \times \mathbb{Z}_2\), \(\mathbb{Z}_3\), \(\mathbb{Z}_2\), or \(\mathbb{Z}_1\) topological invariant in the presence of \(M = 2\), \(3\), \(4\), or \(6\)-fold rotational symmetry, lattice (magnetic) translational symmetry, and charge conservation. \(\vec{\mathscr{P}}\) manifests in the bulk of the system as (i) a fractional quantized contribution of \(\vec{\mathscr{P}} \cdot \vec{b} \text{ mod 1}\) to the charge bound to lattice disclinations and dislocations with Burgers vector \(\vec{b}\), (ii) a linear momentum for magnetic flux, and (iii) an oscillatory system size dependent contribution to the effective 1d polarization on a cylinder. We study \(\vec{\mathscr{P}}\) in lattice 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 \(\vec{\mathscr{P}}\) from microscopic models in an intrinsically many-body way. We show how, given a high symmetry point \(\text{o}\), there is a topological invariant, the discrete shift \(\mathscr{S}_{\text{o}}\), such that \(\vec{\mathscr{P}}\) specifies the dependence of \(\mathscr{S}_{\text{o}}\) on \(\text{o}\). We derive colored Hofstadter butterflies, corresponding to the quantized value of \(\vec{\mathscr{P}}\), which further refine the colored butterflies from the Chern number and discrete shift.