Quantization of fractional corner charge in Cn -symmetric higher-order topological crystalline insulators

Abstract

In the presence of crystalline symmetries, certain topological insulators present a filling anomaly: a mismatch between the number of electrons in an energy band and the number of electrons required for charge neutrality. In this paper, we show that a filling anomaly can arise when corners are introduced in Cn-symmetric crystalline insulators with vanishing polarization, having as consequence the existence of corner-localized charges quantized in multiples of e n . We characterize the existence of this charge systematically and build topological indices that relate the symmetry representations of the occupied energy bands of a crystal to the quanta of fractional charge robustly localized at its corners. When an additional chiral symmetry is present, e 2 corner charges are accompanied by zero-energy corner-localized states. We show the application of our indices in a number of atomic and fragile topological insulators and discuss the role of fractional charges bound to disclinations as bulk probes for these crystalline phases.Topological crystalline insulators (TCIs) [1][2][3][4][5][6][7] are known to exhibit a variety of quantized electromagnetic phenomena. They host bulk dipole moments that lead to surface charge densities quantized in fractions of the electronic charge e [8][9][10][11][12][13]. Recently, it was found that TCIs can also host higher bulk multipole moments that manifest lower-order moments bound to their boundaries [14,15]. For example, a quadrupole insulator in two dimensions has edge-bound dipole moments and cornerbound charges, while an octupole insulator in three dimensions has surface-bound quadrupole moments, hingebound dipole moments, and corner-bound charges. Just as in the case of insulators with symmetry-protected dipole moments, crystalline symmetries quantize the boundary signatures in quadrupole or octupole TCIs. Indeed, TCIs with quantized multipole moments are symmetry protected topological phases of matter; their quantization is robust and can change only in discrete jumps at phase transitions [14,15], unless the protecting symmetries are broken.A salient property of TCIs with quantized higher multipole moments is that some of their protected observables at the boundary are at least two dimensions less than the protecting bulk. This property has now been extended to a broader family of TCIs, broadly referred to as higher-order topological insulators . In this paper, we focus on two-dimensional (2D) higher-order TCIs having zero-dimensional topological signatures. A number of studies have recently shown examples of such TCIs which exhibit in-gap corner-localized states [15,[21][22][23][24][25][26][27], some of which have been related to fractionally quantized corner charges [15,22,23,27]. Interestingly, many such TCIs have these corner signatures in spite of vanishing quadrupole moments, and their mechanisms of protection and associated topological invariants are still not completely elucidated.In this article we systematically study 2D second-order TCIs in class AI (spinless and ti...

Bound states in the continuum of higher-order topological insulators

“…A paradigmatic example is the family of topological insulators which manifest quantized dipole moments in their bulk and charge fractionalization at their boundaries [1][2][3], epitomized by the inversion-symmetric onedimensional Su-Schieffer-Hegger model [4]. This property of boundary charge fractionalization has recently been generalized through the discovery of higher-order topological insulators (HOTIs) whose topology is solely protected by crystalline symmetries and which can host corner fractional charges in 2D and 3D [5][6][7][8][9][10][11][12][13][14].…”

“…This model has been recently studied in the context of charge fractionalization in higher-order topological crystalline insulators [11]. In the topological phase, the Wannier centers in all the bands localize at the maximal Wyckoff position 1b (corner of the unit cell), while in the trivial phase the Wannier centers in all the bands are localized at the maximal Wyckoff position 1a (center of the unit cell).…”

“…In the topological phase, the Wannier centers in all the bands localize at the maximal Wyckoff position 1b (corner of the unit cell), while in the trivial phase the Wannier centers in all the bands are localized at the maximal Wyckoff position 1a (center of the unit cell). The displacement of the Wannier centers relative to the center of their unit cells generates dipole moments per unit length quantized by C 2 symmetry to P = ( e 2 , e 2 ) in the first and fourth bands of the lattice [11]. These quantized dipole moments are accompanied by two edge energy bands (i.e., bands with edge-localized states) spectrally isolated from the bulk energy bands [ Fig.…”

Wannier-type photonic higher-order topological corner states induced solely by gain and loss

“…With increasing strength of gain and loss, the effective anisotropy of the photonic graphene grows. Correspondingly, the two Dirac cones (with opposite Berry phases) approach each other and annihilate at high-symmetry point, leaving a gapped insulator phase, which tends out to be a photonic Wannier-type HOTI phase [43,51].…”

“…In Sec. III, we study how the band structure changes with respect to increasing gain/loss strength and show there is a topological phase transition from a photonic semimetal to a photonic Wannier-type SOTI, characterized by nontrivially quantized Wannier center [43,51]. In Sec.…”