Quadrupole moments, edge polarizations, and corner charges in the Wannier representation

Abstract

The modern theory of polarization allows for the determination of the macroscopic end charge of a truncated one-dimensional insulator, modulo the charge quantum $e$, from a knowledge of bulk properties alone. A more subtle problem is the determination of the corner charge of a two-dimensional insulator, modulo $e$, from a knowledge of bulk and edge properties alone. While previous works have tended to focus on the quantization of corner charge in the presence of symmetries, here we focus on the case that the only bulk symmetry is inversion, so that the corner charge can take arbitrary values. We develop a Wannier-based formalism that allows the corner charge to be predicted, modulo $e$, only from calculations on ribbon geometries of two different orientations. We elucidate the dependence of the interior quadrupole and edge dipole contributions upon the gauge used to construct the Wannier functions, finding that while these are individually gauge-dependent, their sum is gauge-independent. From this we conclude that the edge polarization is not by itself a physical observable, and that any Wannier-based method for computing the corner charge requires the use of a common gauge throughout the calculation. We satisfy this constraint using two Wannier construction procedures, one based on projection and another based on a gauge-consistent nested Wannier construction. We validate our theory by demonstrating the correct prediction of corner charge for several tight-binding models. We comment on the relations between our approach and previous ones that have appeared in the literature.

21 pages, 10 figures, typo fixed, references updated, published version

References in corpus (10)

• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• $(d-2)$-dimensional edge states of rotation symmetry protected topological states
• Orbital magnetization in periodic insulators
• Towards a bulk theory of flexoelectricity
• Bulk-corner correspondence of time-reversal symmetric insulators: deduplicating real-space invariants
• Corner charge and bulk multipole moment in periodic systems
• Fractional Corner Charge of Sodium Chloride
• Bulk-and-edge to corner correspondence
• Spin-orbit spillage as a measure of band inversion in insulators
• Thermodynamic approach to electric quadrupole moments

Cited by in corpus (16)

• General corner charge formula in two-dimensional C_n-symmetric higher-order topological insulators
• Bound states at partial dislocation defects in multipole higher-order topological insulators
• Filling anomaly for general 2D and 3D $C_4$ symmetric lattices
• Higher-order topological phases in crystalline and non-crystalline systems: a review
• Thouless Pumps and Bulk-Boundary Correspondence in Higher-Order Symmetry-Protected Topological Phases
• $N$-band Hopf insulator
• Fractional Corner Charge of Sodium Chloride
• Topological edge and corner states and fractional corner charges in blue phosphorene
• Universal higher-order bulk-boundary correspondence of triple nodal points
• Towards a theory of surface orbital magnetization
• Homotopic classification of band structures: Stable, fragile, delicate, and stable representation-protected topology
• Wannier Topology and Quadrupole Moments for a generalized Benalcazar-Bernevig-Hughes Model
• Fractional corner magnetization of collinear antiferromagnets
• Many-Body Quadrupolar Sum Rule for Higher-Order Topological Insulator
• Topological insulators and geometry of vector bundles
• Theory of thermopolarization effect