Electric polarization and discrete shift from boundary and corner charge in crystalline Chern insulators

Abstract

Recently, it has been shown how topological phases of matter with crystalline symmetry and $U(1)$ charge conservation can be partially characterized by a set of many-body invariants, the discrete shift ${\mathcal{S}}_{\text{o}}$ and electric polarization ${\vec{\mathcal{P}}}_{\text{o}}$, where $\text{o}$ labels a high symmetry point. Crucially, these can be defined even with non-zero Chern number and/or magnetic field. One manifestation of these invariants is through quantized fractional contributions to the charge in the vicinity of a lattice disclination or dislocation. In this paper, we show that these invariants can also be extracted from the length and corner dependence of the total charge (mod 1) on the boundary of the system. We provide a general formula in terms of ${\mathcal{S}}_{\text{o}}$ and ${\vec{\mathcal{P}}}_{\text{o}}$ for the total charge of any subregion of the system which can include full boundaries or bulk lattice defects, unifying boundary, corner, disclination, and dislocation charge responses into a single general theory. These results hold for Chern insulators, despite their gapless chiral edge modes, and for which an unambiguous definition of an intrinsically two-dimensional electric polarization has been unclear until recently. We also discuss how our theory can fully characterize the topological response of quadrupole insulators.

15 pages, 14 figures

References in corpus (14)

• Topological Crystalline Insulators
• Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators
• Classifying fractionalization: symmetry classification of gapped Z2 spin liquids in two dimensions
• Framing Anomaly in the Effective Theory of Fractional Quantum Hall Effect
• Electric polarization in a Chern insulator
• Fractional disclination charge and discrete shift in the Hofstadter butterfly
• Quantized charge polarization as a many-body invariant in (2+1)D crystalline topological states and Hofstadter butterflies
• Non-perturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in (2+1)D
• Interacting Topological Quantum Chemistry in 2D: Many-body Real Space Invariants
• Complete crystalline topological invariants from partial rotations in (2+1)D invertible fermionic states and Hofstadter's butterfly
• Characterization and classification of interacting (2+1)D topological crystalline insulators with orientation-preserving wallpaper groups
• Effective action approach to the filling anomaly in crystalline topological matter
• Crystalline-electromagnetic responses of higher order topological semimetals
• Rotational Symmetry Protected Edge and Corner States in Abelian topological phases

Cited by in corpus (2)

• Disclinations, dislocations, and emanant flux at Dirac criticality
• Fractionally Quantized Electric Polarization and Discrete Shift of Crystalline Fractional Chern Insulators