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 twodimensional periodic systems. This is an analog of the formula for the surface charge density in terms of the bulk polarization. In the presence of a 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.

The bulk-corner correspondence of time-reversal symmetric insulators

“…For instance, the electric polarization of an inversion-symmetric one-dimensional atomic chain is either integer or semi-integer, with a quantized value that does not depend upon microscopic details, but is rather encoded in a gauge-invariant topological index 28 . More recently, it has been shown that excess electronic charges localized at various topological defects, such as dislocations, can be (fractionally) quantized, thus representing yet other incarnations of bulk quantities encoded in topological invariants [29][30][31][32][33][34] . Quantized charges appearing at the corners and disclinations of two-dimensional crystals have been very recently measured in metamaterials [35][36][37] and proposed to appear in recently synthesized materials structures 38 .…”

Topological edge and corner states and fractional corner charges in blue phosphorene

“…These multiple corner states can also be well described by a simple effective Hamiltonian for uncoupled edge and corner orbitals. In the armchair corner, in particular, we demonstrate that corner states appear right at the Fermi energy, which leads to the emergence of fractional corner charge due to filling anomaly [37][38][39][40][41][42][43][44][45].…”

Competition of First-Order and Second-Order Topology on the Honeycomb Lattice

“…While those TIs in d dimensions exhibit d − 1 dimensional boundary states, these multipole insulators feature d − n dimensional corner or hinge states. These modes correspond to quantized higher electric multipole moments, and because of this unusual bulk-boundary correspondence they are often re-ferred to as higher-order TIs (HOTIs) [26][27][28][29][30][31][32][33][34], in contrast to the previously mentioned first-order TIs. Their theoretical prediction has been quickly matched with the first experimental realizations, with HOTIs realized in bismuth [35,36], topolectrical circuits [37][38][39][40][41][42], photonic crystals [43][44][45], acoustic [46][47][48] and elastic systems [42].…”

“…In a similar fashion, the simplest HOTI model, the square lattice BBH model [24,25] , still requires insertion of π fluxes, which are not necessary for the honeycomb lattice. The honeycomb lattice offers second-order topology solely through anisotropic hopping amplitudes, referred to as Kekulé and anti-Kekulé distortions [49][50][51][52] which was recently realized in graphene [33]. Several works have explored HOTI phases on the honeycomb lattice [53][54][55][56][57][58][59][60][61][62] including experimental work [39].…”