Quantized Charge Polarization as a Many-Body Invariant in (2+1)D Crystalline Topological States and Hofstadter Butterflies - Yuxuan Zhang; Naren Manjunath; Gautam Nambiar; Maissam Barkeshli (Physical Review X, 2023)
We show how to define a quantized many-body charge polarization 𝒫[over →] for (2+1)D topological phases of matter, even in the presence of nonzero Chern number and magnetic field. For invertible topological states, 𝒫[over →] is a Z_{2}×Z_{2}, Z_{3}, Z_{2}, or Z_{1} topological invariant in the presence of (M=2, 3, 4, or 6)-fold rotational symmetry, lattice (magnetic) translational symmetry, and charge conservation. 𝒫[over →] manifests in the bulk of the system as (i) a fractional quantized contribution of 𝒫[over →]·b[over →]  mod  1 to the charge bound to lattice disclinations and dislocations with Burgers vector b[over →], (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 𝒫[over →] 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 𝒫[over →] from microscopic models in an intrinsically many-body way. We show how, given a high symmetry point o, there is a topological invariant, the discrete shift 𝒮_{o}, such that 𝒫[over →] specifies the dependence of 𝒮_{o} on o. We derive colored Hofstadter butterflies, corresponding to the quantized value of 𝒫[over →], which further refine the colored butterflies from the Chern number and discrete shift.