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 a 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 P edge α , where the integer α enumerates the edges. We introduce a particular manner of cutting a crystal, dubbed "Wannier cut," which allows us to compute P edge α. We find that P edge α 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 P edge α , only 2n − 1 are independent of the choice of Wannier cut and correspond to physical observables: corner charges and edge dipoles.

Many-Body Quadrupolar Sum Rule for Higher-Order Topological Insulator

“…Since our formulation works for disordered and interacting systems, one could also test our sum rules for those cases. Finally, finding connections between previous works [20][21][22] and our sum rule would be interesting.…”

“…We see that the edge-localized polarizations P edge where Γ2 = σ 1 ⊗ σ 1 , Γ1 = −σ 1 ⊗ σ 2 , and all the other Γ matrices are the same as the ones in Eq. (20).…”

“…and all Γ and σ matrices are the same as in Eq. (20). Similar to the quadrupole insulator, which has π-flux per plaquette, the bulk quadrupole moment of the halffilled ground state of Eq.…”

“…A natural question is then whether such a correspondence for the bulk quadrupole can be formulated. In recent studies 20,21 , bulk-boundary correspondences for the quadrupole moment were presented using Wannier functions, where the corner charge is expressed as a sum of quantities that depend on the choice of the bulk Wannier functions, although the corner charge itself is not. We also note another recent study 22 , where the corner charge of band insulators is computed using the adiabatic current flowing along the edges.…”

Change of Corner Charge and Adiabatic Current Distribution in Two Dimensional Insulators with Inversion Symmetry

“…The proposal of higher order topological insulator (HOTI) [1][2][3][4][5][6][7][8][9][10] has revived the wide interest for the electric multipole moments in crystalline insulators [11][12][13][14][15][16][17][18][19][20][21][22] . Although electric multipole moments are fundamental concepts in electromagnetism, the search for a self-consistent quantum mechanical theory is surprisingly challenging and brings about a change of paradigm [23][24][25][26][27][28] .…”