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.
Recent developments in higher-order topological phases have elucidated on the relationship between nontrivial multipole moments in the bulk and the emergence of fractionally quantized charges at the boundary. Here, we put on the table a proposal of the three-dimensional octupole insulator with fractionally quantized corner charges $\pm e/8$: sodium chloride, commonly known as table salt. The fractional quantization of the corner charge is unaffected by the quantum fluctuation of the electric charge per ion, as we demonstrate explicitly with ab initio calculations. We further show that the elec…
Electric multipole moments are the most fundamental properties of insulating materials. However, the general formulation of bulk multipoles has been a long standing problem. The solution for the electric dipole moment was provided decades ago by King-Smith, Vanderbilt, and Resta. Recently, there have been attempts at generalizing Resta's formula to higher-order multipoles. We point out several issues in the recent proposals.
Nonlinear topological laser based on multipole insulators
North China University of Water Resources and Electric Power（华北水利水电大学）；Lanzhou University（兰州大学）；Chongqing University（重庆大学）；Aerospace Information Research Institute, Chinese Academy of Sciences（中国科学院空天信息创新研究院）；The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））
Higher order topological insulators (HOTI) have emerged as a new class of phases, whose robust in-gap "corner" modes arise from the bulk higher-order multipoles beyond the dipoles in conventional topological insulators. Here, we incorporate Floquet driving into HOTI's, and report for the first time a dynamical polarization theory with anomalous non-equilibrium multipoles. Further, a proposal to detect not only corner states but also their dynamical origin in cold atoms is demonstrated, with the latter one never achieved before. Experimental determination of anomalous Floquet corner modes are a…
The properties of topological systems are inherently tied to their dimensionality. Higher-dimensional physical systems exhibit topological properties not shared by their lower dimensional counterparts and, in general, offer richer physics. One example is a d-dimensional quantized multipole topological insulator, which supports multipoles of order up to 2^d and a hierarchy of gapped boundary modes with topological 0-D corner modes at the top. While multipole topological insulators have been successfully realized in electromagnetic and mechanical 2D systems with quadrupole polarization, and a 3D…
Recently there is a surge of interests in the so-called topologically protected corner states in 2D and 3D systems. Such systems are considered as high order topological insulators. Wannier centers are used as topological invariants to characterize bulk systems. The existence of corner states is considered as a reflection of the topological non-triviality of bulk energy bands. We demonstrate that the Wannier centers are not topological invariants by showing they depend on the choices of unit cells. The same bulk system can be considered as both topological and trivial with two equally possible…
In this article we extend the celebrated Berry-phase formulation of electric polarization in crystals to higher electric multipole moments. We determine the necessary conditions under which, and minimal models in which, the quadrupole and octupole moments are topologically quantized electromagnetic observables. Such systems exhibit gapped boundaries that are themselves lower-dimensional topological phases. Furthermore, they manifest topologically protected corner states carrying fractional charge, i.e., fractionalization at the boundary of the boundary. To characterize these new insulating pha…
The quantum mechanical position operators, and their products, are not well-defined in systems obeying periodic boundary conditions. Here we extend the work of Resta who developed a formalism to calculate the electronic polarization as an expectation value of a many-body operator, to include higher multipole moments, e.g., quadrupole and octupole. We define $n$-th order multipole operators whose expectation values can be used to calculate the $n$-th multipole moment when all of the lower moments are vanishing (modulo a quantum). We show that changes in our operators are tied to flows of $n-1$-…
Majorana bound states often occur at the end of 1D topological superconductor or at the $π$ Josephson junction mediated by a helical edge state. Validated by a new bulk invariant and an intuitive edge argument, we show the emergence of one Majorana Kramers pair at each corner of a square-shaped 2D topological insulator proximitized by an $s_\pm$-wave (e.g., Fe-based) superconductor. We obtain a phase diagram that emphasizes the roles of bulk parameters and edge orientations. We propose several experimental realizations in lattice-matched candidate materials. Our scheme offers a high-temperatur…
The modern theory of electric polarization has recently been extended to higher multipole moments, such as quadrupole and octupole moments. The higher electric multipole insulators are essentially topological crystalline phases protected by underlying crystalline symmetries. Henceforth, it is natural to ask what are the consequences of symmetry breaking in these higher multipole insulators. In this work, we investigate topological phases and the consequences of symmetry breaking in generalized electric quadrupole insulators. Explicitly, we generalize the Benalcazar-Bernevig-Hughes model by add…
In vortex-like spin arrangements, multiple spins can combine into emergent multipole moments. Such multipole moments have broken space-inversion and time-reversal symmetries, and can therefore exhibit linear magnetoelectric (ME) activity. Three types of such multipole moments are known: toroidal, monopole, and quadrupole moments. So far, however, the ME-activity of these multipole moments has only been established experimentally for the toroidal moment. Here, we propose a magnetic square cupola cluster, in which four corner-sharing square-coordinated metal-ligand fragments form a noncoplanar b…
We propose a new correlated topological state which we call a higher-order topological Mott insulator (HOTMI). This state exhibits a striking bulk-boundary correspondence due to electron correlations. Namely, the topological properties in the bulk, characterized by the Z3 spin-Berry phase, result in gapless corner modes emerging only in spin excitations (i.e., the single-particle excitations remain gapped around the corner). We demonstrate the emergence of the HOTMI in a Hubbard model on the kagome lattice, and elucidate how strong correlations change gapless corner modes at the noninteracting…
We submit that the magnetic space-group Cac (#9.41) is consistent with the established magnetic structure of BaFe2Se3, with magnetic dipole moments in a motif that uses two ladders [Caron J M et al 2011 Phys. Rev. B 84 180409(R)]. The corresponding crystal class m1' allows axial and polar dipoles and forbids bulk ferromagnetism. The compound supports magneto-electric multipoles, including a magnetic charge (monopole) and an anapole (magnetic toroidal dipole) visible in the Bragg diffraction of x-rays and neutrons. Our comprehensive simulation of neutron Bragg diffraction by BaFe2Se3 exploits e…
The bulk-boundary correspondence relates quantized edge states to bulk topological invariants in topological phases of matter. In one-dimensional symmetry-protected topological systems (SPTs), quantized topological Thouless pumps directly reveal this principle and provide a sound mathematical foundation. Symmetry-protected higher-order topological phases of matter (HOSPTs) also feature a bulk-boundary correspondence, but its connection to quantized charge transport remains elusive. Here we show that quantized Thouless pumps connecting $C_4$-symmetric HOSPTs can be described by a tuple of four …
Quantum many-body systems have a rich structure in the presence of boundaries. We study the groundstates of conformal field theories (CFTs) and Lifshitz field theories in the presence of a boundary through the lens of the entanglement entropy. For a family of theories in general dimensions, we relate the universal terms in the entanglement entropy of the bulk theory with the corresponding terms for the theory with a boundary. This relation imposes a condition on certain boundary central charges. For example, in $2+1$ dimensions, we show that the corner-induced logarithmic terms of free CFTs an…
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 $C_n$-symmetric crystalline insulators with vanishing polarization, having as consequence the existence of corner-localized charges quantized in multiples of $\frac{e}{n}$. We characterize the existence of this charge systematically and build topological indices that relate the symmetry rep…
We study two-dimensional spinful insulating phases of matter that are protected by time-reversal and crystalline symmetries. To characterize these phases we employ the concept of corner charge fractionalization: Corners can carry charges that are fractions of even multiples of the electric charge. The charges are quantized and topologically stable as long as all symmetries are preserved. We classify the different corner charge configurations for all point groups, and match them with the corresponding bulk topology. For this we employ symmetry indicators and (nested) Wilson loop invariants. We …
