Corner charge and bulk multipole moment in periodic systems

Abstract

A formula for the corner charge in terms of the bulk quadrupole moment is derived for two-dimensional periodic systems. This is an analog of the formula for the surface charge density in terms of the bulk polarization. In the presence of an $n$-fold rotation symmetry with $n=3$, $4$, and $6$, the quadrupole moment is quantized and is independent of the spread or shape of Wannier orbitals, depending only on the location of Wannier centers of filled bands. In this case, our formula predicts the fractional part of the quadrupole moment purely from the bulk property. The system can contain many-body interactions as long as the ground state is gapped and topologically trivial in the sense it is smoothly connected to a product state limit. An extension of these results to three-dimensional systems is also discussed. In three dimensions, in general, even the fractional part of the corner charge is not fully predictable from the bulk perspective even in the presence of point group symmetry.

arXiv comment15 pages, 6 figures, 3 tables. v3: version to appear in PRB

References in corpus (36)

• Maximally-localized generalized Wannier functions for composite energy bands
• Maximally localized Wannier functions: Theory and applications
• Quantized Electric Multipole Insulators
• 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
• Topolectrical circuit realization of topological corner modes
• Observation of a phononic quadrupole topological insulator
• Higher-order topological insulators and semimetals on the breathing Kagome and pyrochlore lattices
• Demonstration of a quantized microwave quadrupole insulator with topologically protected corner states
• Symmetry-based Indicators of Band Topology in the 230 Space Groups
• Quantization of fractional corner charge in $C_n$-symmetric higher-order topological crystalline insulators
• Symmetry indicators and anomalous surface states of topological crystalline insulators
• Higher-Order Topological Insulators in Quasicrystals
• A fractional corner anomaly reveals higher-order topology
• Helical Topological Edge States in a Quadrupole Phase
• Higher-order topological insulators protected by inversion and rotoinversion symmetries
• An anomalous higher-order topological insulator
• Minimal model for higher-order topological insulators and phosphorene
• Fractional disclination charge in two-dimensional $C_n-$symmetric topological crystalline insulators
• Many-Body Electric Multipole Operators in Extended Systems
• Higher-order topological insulators in amorphous solids
• Quadrupole Topological Photonic Crystals
• Many-Body Order Parameters for Multipoles in Solids
• Fractional corner charges in spin-orbit coupled crystals
• Lattice Homotopy Constraints on Phases of Quantum Magnets
• Shift insulators: rotation-protected two-dimensional topological crystalline insulators
• Difficulties in operator-based formulation of the bulk quadrupole moment
• ${\mathbb{Z}}_Q$ Berry Phase for Higher-Order Symmetry-Protected Topological Phases
• Fragile topological phases in interacting systems
• Type-II quadrupole topological insulators
• Geometric orbital magnetization in adiabatic processes
• Higher-Order Entanglement and Many-Body Invariants for Higher-Order Topological Phases
• Bulk-and-edge to corner correspondence
• Identifying Higher Order Topology and Fractional Corner Charge Using Entanglement Spectra
• Variants of the symmetry-based indicator
• Entanglement Signatures of Multipolar Higher Order Topological Phases

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• Thermodynamic approach to electric quadrupole moments
• Topological edge and corner states and fractional corner charges in blue phosphorene
• Universal higher-order bulk-boundary correspondence of triple nodal points
• Higher-Order Topological Insulator on a Martini Lattice and Its Square Root Descendant
• Detecting Bulk Topology of Quadrupolar Phase from Quench Dynamics
• Competition of First-Order and Second-Order Topology on the Honeycomb Lattice
• Symmetry indicators in commensurate magnetic flux
• Fractional hinge and corner charges in various crystal shapes with cubic symmetry
• Homotopic classification of band structures: Stable, fragile, delicate, and stable representation-protected topology
• Characterization of topological insulators based on the electronic polarization with spiral boundary conditions
• Thermodynamic electric quadrupole moments of nematic phases from first-principles calculation
• Wannier Topology and Quadrupole Moments for a generalized Benalcazar-Bernevig-Hughes Model
• Protected Gapless Edge States In Trivial Topology
• Fractional boundary charges with quantized slopes in interacting one- and two-dimensional systems
• Fractional corner magnetization of collinear antiferromagnets
• Recipe for higher-order topology on the triangular lattice
• General corner charge formulas in various tetrahedral and cubic space groups
• Theory of fractional corner charges in cylindrical crystal shapes
• Many-Body Quadrupolar Sum Rule for Higher-Order Topological Insulator
• Global and Local Topological Crystalline Markers for Rotation-Symmetric Insulators
• Local expression of fractional corner charges in obstructed atomic insulators and relationship with the fractional disclination charges
• Topological defects in a double-mirror quadrupole insulator displace diverging charge