Bulk-and-edge to corner correspondence

Abstract

We show that two-dimensional band insulators, with vanishing bulk polarization, obey bulk-and-edge to corner charge correspondence stating that the knowledge of the bulk and the two corresponding ribbon band structures uniquely determines the fractional part of the corner charge irrespective of the corner termination. Moreover, physical observables related to macroscopic charge density of a terminated crystal can be obtained by representing the crystal as collection of polarized edge regions with polarizations ${\vec{P}}_{\alpha }^{\text{edge}}$, where the integer $\alpha$ enumerates the edges. We introduce a particular manner of cutting a crystal, dubbed "Wannier cut", which allows us to compute ${\vec{P}}_{\alpha }^{\text{edge}}$. We find that ${\vec{P}}_{\alpha }^{\text{edge}}$ consists of two pieces: the bulk piece expressed via quadrupole tensor of the bulk Wannier functions' charge density, and the edge piece corresponding to the Wannier edge polarization---the polarization of the edge subsystem obtained by Wannier cut. For a crystal with $n$ edges, out of $2n$ independent components of ${\vec{P}}_{\alpha }^{\text{edge}}$, only $2n-1$ are independent of the choice of Wannier cut and correspond to physical observables: corner charges and edge dipoles.

arXiv commentpublished version. v2: application to Benalcazar-Bernevig-Hughes model

References in corpus (27)

• Quantum Spin Hall Effect in Graphene
• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Unpaired Majorana fermions in quantum wires
• Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices
• Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures
• Topological Field Theory of Time-Reversal Invariant Insulators
• Maximally localized Wannier functions: Theory and applications
• Periodic table for topological insulators and superconductors
• Non-Abelian statistics and topological quantum information processing in 1D wire networks
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• Magnetoelectric polarizability and axion electrodynamics in crystalline insulators
• Quantization of fractional corner charge in $C_n$-symmetric higher-order topological crystalline insulators
• Electron Localization in the Insulating State
• Orbital magnetization in periodic insulators
• Polarization and localization in insulators: generating function approach
• Classification of Topological Insulators and Superconductors
• Higher-order bulk-boundary correspondence for topological crystalline phases
• Quantum Theory of Orbital Magnetization and its Generalization to Interacting Systems
• Bulk-boundary correspondence from the inter-cellular Zak phase
• Many-Body Electric Multipole Operators in Extended Systems
• Many-Body Order Parameters for Multipoles in Solids
• Difficulties in operator-based formulation of the bulk quadrupole moment
• Higher-order topological band structures
• Surface polarization and edge charges
• Second-Order Bulk-Boundary Correspondence in Rotationally Symmetric Topological Superconductors from Stacked Dirac Hamiltonians
• Geometric orbital magnetization in adiabatic processes

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• Thermodynamic approach to electric quadrupole moments
• ${\mathbb{Z}}_2$ Spin Hopf Insulator: Helical Hinge States and Returning Thouless Pump
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