Spectral measurements of boundary localized in-gap modes are commonly used to identify topological insulators via the bulk-boundary correspondence. This can be extended to high-order topological insulators for which the most striking feature is in-gap modes at boundaries of higher co-dimension, e.g. the corners of a 2D material. Unfortunately, this spectroscopic approach is not always viable since the energies of the topological modes are not protected and they can often overlap the bulk bands, leading to potential misidentification. Since the topology of a material is a collective product of all its eigenmodes, any conclusive indicator of topology must instead be a feature of its bulk band structure, and should not rely on specific eigen-energies. For many topological crystalline insulators the key topological feature is fractional charge density arising from the filled bulk bands, but measurements of charge distributions have not been accessible to date. In this work, we experimentally measure boundary-localized fractional charge density of two distinct 2D rotationally-symmetric metamaterials, finding 1/4 and 1/3 fractionalization. We then introduce a new topological indicator based on collective phenomenology that allows unambiguous identification of higher-order topology, even in the absence of in-gap states. Finally, we demonstrate the higher-order bulk-boundary correspondence associated with this fractional feature by using boundary deformations to spectrally isolate localized corner modes where they were previously unobservable.
We theoretically study emergent edge and corner states in monolayer blue phosphorus (blue phosphorene) using the first-principles calculation and tight-binding model. We show that the existence of the Wannier orbitals at every bond center yields edge states both in zigzag and armchair nanoribbons. The properties of the edge states can be well described by a simple effective Hamiltonian for uncoupled edge orbitals, where the structural relaxation near the boundary significantly affects the edge band structure. For corner states, we examine two types of corner structures consisting of zigzag and…
We demonstrate that a spin degree of freedom can introduce additional texture to higher order topological insulators (HOTIs), manifesting itself in novel topological invariants, phases, and phase transitions. Spin-polarized mid-gap corner states of various multiplicities are predicted for different HOTI phases, and novel bulk-boundary correspondence principles are defined based on quantized bulk invariants such as total and spin corner charge. Those are shown to be robust with respect to symmetry-preserving spin-flipping perturbations. Photonic realizations of spin-linked topological phases ar…
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…
Higher-order topological insulators host gapless states on hinges or corners of three-dimensional crystals. Recent studies suggested that even topologically trivial insulators may exhibit fractionally quantized charges localized at hinges or corners. Although most of the previous studies focused on two-dimensional systems, in this work, we take the initial step toward the systematic understanding of hinge and corner charges in three-dimensional insulators. We consider five crystal shapes of vertex-transitive polyhedra with the cubic symmetry such as a cube, an octahedron and a cuboctahedron. W…
Recent studies revealed that the electric multipole moments of insulators result in fractional electric charges localized to the hinges and corners of the sample. We here explore the magnetic analog of this relation. We show that a collinear antiferromagnet with spin $S$ defined on a $d$-dimensional cubic lattice features fractionally quantized magnetization $M_{\text{c}}^z=S/2^d$ at the corners. We find that the quantization is robust even in the presence of gapless excitations originating from the spontaneous formation of the Néel order, although the localization length diverges, suggesting …
We consider a setup consisting of two coupled sheets of bilayer graphene in the regime of strong spin-orbit interaction, where electrostatic confinement is used to create an array of effective quantum wires. We show that for suitable interwire couplings the system supports a topological insulator phase exhibiting Kramers partners of gapless helical edge states, while the additional presence of a small in-plane magnetic field and weak proximity-induced superconductivity leads to the emergence of zero-energy Majorana corner states at all four corners of a rectangular sample, indicating the trans…
We consider a system of weakly coupled Rashba nanowires in the strong spin-orbit interaction (SOI) regime. The nanowires are arranged into two tunnel-coupled layers proximitized by a top and bottom superconductor such that the superconducting phase difference between them is $π$. We show that in such a system strong electron-electron interactions can stabilize a helical topological superconducting phase hosting Kramers partners of $\mathbb{Z}_{2m}$ parafermion edge modes, where $m$ is an odd integer determined by the position of the chemical potential. Furthermore, upon turning on a weak in-pl…
We study the entanglement spectrum (ES) of two-dimensional $C_{n}$-symmetric second-order topological insulators (TIs). We show that some characteristic higher order topological observables, e.g., the filling anomaly and its associated fractional corner charge, can be determined from the ES of atomic and fragile TIs. By constructing the relationship between the configuration of Wannier orbitals and the number of protected in-gap states in the ES for different symmetric cuts in real space, we express the fractional corner charge in terms of the number of protected in-gap states of the ES. We sh…
Generally, the topological corner state in two-dimensional second-order topological insulator (2D SOTI) is equivalent to the well-known domain wall state, originated from the mass-inversion between two adjacent edges with phase shift of pi. In this work, go beyond this conventional physical picture, we report a fractional mass-kink induced 2D SOTI in monolayer FeSe with canted checkerboard antiferromagnetic (AFM) order by analytic model and first-principles calculations. The canted spin associated in-plane Zeeman field can gap out the quantum spin Hall edge state of FeSe, forming a fractional …
Higher-order topological insulators, which support lower-dimensional topological boundary states than the first-order topological insulators, have been intensely investigated in the integer dimensional systems. Here, we provide a new paradigm by presenting experimentally a higher-order topological phase in a fractal-dimensional system. Through applying the Benalcazar, Bernevig, and Hughes model into a Sierpinski carpet fractal lattice, we uncover a squeezed higher-order phase diagram featuring the abundant corner states, which consist of zero-dimensional outer corner states and 1.89-dimensiona…
We demonstrate that the presence of a localized state at the corner of an insulating domain is not always a predictor of a certain non-trivial higher-order topological invariant, even though they appear to co-exist in the same Hamiltonian parameter space. Our analysis of $C^n$-symmetric crystalline insulators and their multi-layer stacks reveals that topological corner states are not necessarily correlated with other well-established higher-order boundary observables, such as fractional corner charge or filling anomaly. In a $C^3$-symmetric breathing Kagome lattice, for example, we show that t…
We show that unconventional boundary phenomena occur in the $3Q$ charge bond orders on the hexagonal lattices. At the Van Hove singularity with three nesting momenta, $3Q$ orders can trigger a $\text{C}_6$-symmetric insulator under bond modulations. On the kagome lattice, in-gap corner states appear in the energy spectrum and carry fractional corner charge $-2e/3$. Such corner phenomena originate from the corner filling anomaly and indicate a higher-order topological insulator. The in-gap corner states are also observed on the triangular lattice. The honeycomb lattice does not support fraction…
We consider the geometric part of the effective action for Fractional Quantum Hall Effect (FQHE). It is shown that accounting for the framing anomaly of the quantum Chern-Simons theory is essential to the obtain correct gravitational linear response functions. In the lowest order in gradients the linear response generating functional includes Chern-Simons, Wen-Zee and gravitational Chern- Simons terms. The latter term has a contribution from the framing anomaly which fixes the value of thermal Hall conductivity and contributes to the Hall viscosity of the FQH states on a sphere. We also discus…
We establish an analytic low-energy theory describing higher-order topological insulator (HOTI) phases in quasicrystalline systems. We apply this to a model consisting of two stacked Haldane models with oppositely propagating edge modes, analogous to the Kane-Mele model, and with a $30^\circ$ twist. We show that the resulting localized modes at corners, characteristic of a HOTI, are not associated with conventional mass inversions but are instead associated with what we dub "fractional mass kinks". By generalizing the low-energy theory, we establish a classification for arbitrary $ n $-fold ro…
The topology of insulators is usually revealed through the presence of gapless boundary modes: this is the so-called bulk-boundary correspondence. However, the many-body wavefunction of a crystalline insulator is endowed with additional topological properties that do not yield surface spectral features, but manifest themselves as (fractional) quantized electronic charges localized at the crystal boundaries. Here, we formulate such bulk-corner correspondence for the physical relevant case of materials with time-reversal symmetry and spin-orbit coupling. To so do we develop "partial" real-space …
It is shown that the deviation of fractional quantum Hall edge fluid from power law correlation functions with universal exponent $α=1/ν$ as observed in recent experiment may be explained when analyzed from the viewpoint of chiral anomaly and Berry phase. It is observed that at the edge anomaly vanishes and this induces a nonlocal effect in the construction of the electron creation operator in terms of the edge boson fields. This nonlocality is responsible for the deviation of the power law exponent from $α=1/ν$ of the edge fluid. There are gapless edge excitations described by chiral boson fi…
We develop a general method to compute correlation functions of fractional quantum Hall (FQH) states on a curved space. In a curved space, local transformation properties of FQH states are examined through local geometric variations, which are essentially governed by the gravitational anomaly. Furthermore, we show that the electromagnetic response of FQH states is related to the gravitational response (a response to curvature). Thus, the gravitational anomaly is also seen in the structure factor and the Hall conductance in flat space. The method is based on iteration of a Ward identity obtaine…
Outcomes of measurements are characterized by an infinite family of generalized uncertainties, or cumulants, which provide information beyond the mean and variance of the observable. Here, we investigate the cumulants of a conserved charge in a subregion with corners. We derive nonperturbative relations for the area law, and more interestingly, the angle dependence, showing how it is determined by geometric moments of the correlation function. These hold for translation invariant systems under great generality, including strongly interacting ones. We test our findings by using two-dimensional …
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-bo…
When the crystalline symmetries that protect a higher-order topological phase are not preserved at the boundaries of the sample, gapless hinge modes or in-gap corner states cannot be stabilized. Therefore, careful engineering of the sample termination is required. Similarly, magnetic textures, whose quantum fluctuations determine the supported magnonic excitations, tend to relax to new configurations that may also break crystalline symmetries when boundaries are introduced. Here we uncover that antiskyrmion crystals provide an experimentally accessible platform to realize a magnonic topologica…
The recently established paradigm of higher-order topological states of matter has shown that not only, as previously thought, edge and surface states but also states localised to corners can have robust and exotic properties. Here we report on the experimental realisation of novel corner states made out of classical light in three-dimensional photonic structures inscribed in glass samples using femtosecond (fs) laser technology. By creating and analysing waveguide arrays forming two-dimensional breathing kagome lattices in various sample geometries, we establish this as a platform for corner …
Topological multipole insulators are a class of higher order topological insulators (HOTI) in which robust fractional corner charges appear due to a quantized electric multipole moment of the bulk. This bulk-corner correspondence has been expressed in terms of a topological invariant computed using the eigenstates of the Wilson loop operator, a so called "nested Wilson loop" procedure. We show that, similar to the unitary Floquet operator describing periodically driven systems, the unitary Wilson loop operator can realize "anomalous" phases, that are topologically non-trivial despite having a …
A second-order topological insulator in $d$ dimensions is an insulator which has no $d-1$ dimensional topological boundary states but has $d-2$ dimensional topological boundary states. It is an extended notion of the conventional topological insulator. Higher-order topological insulators have been investigated in square and cubic lattices. In this paper, we generalize them to breathing Kagome and pyrochlore lattices. First, we construct a second-order topological insulator on the breathing Kagome lattice. Three topological boundary states emerge at the corner of the triangle, realizing a 1/3 f…
We show that the Chern-Simons-Landau-Ginsburg theory that describes the quantum Hall effect on a bounded sample is anomaly free and thus does not require the addition of extra chiral fermions on the boundary to restore local gauge invariance.
In this paper, we derive a general formula for the quantized fractional corner charge in two-dimensional C_n-symmetric higher-order topological insulators. We assume that the electronic states can be described by the Wannier functions and that the edges are charge neutral, but we do not assume vanishing bulk electric polarization. We expand the scope of the corner charge formula obtained in previous works by considering more general surface conditions, such as surfaces with higher Miller index and surfaces with surface reconstruction. Our theory is applicable even when the electronic states ar…
Floquet phases of matter have attracted great attention due to their dynamical and topological nature that are unique to nonequilibrium settings. In this work, we introduce a generic way of taking any integer $q$th-root of the evolution operator $U$ that describes Floquet topological matter. We further apply our $q$th-rooting procedure to obtain $2^n$th- and $3^n$th-root first- and second-order non-Hermitian Floquet topological insulators (FTIs). There, we explicitly demonstrate the presence of multiple edge and corner modes at fractional quasienergies $\pm(0,1,...2^{n})π/2^{n}$ and $\pm(0,1,.…
Protected edge modes are the cornerstone of topological states of matter. The simplest example is provided by the integer quantum Hall state at Landau level filling unity, which should feature a single chiral mode carrying electronic excitations. In the presence of a smooth confining potential it was hitherto believed that this picture may only be partially modified by the appearance of additional counterpropagating integer-charge modes. Here, we demonstrate the breakdown of this paradigm: The system favors the formation of edge modes supporting fractional excitations. This accounts for previo…
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 $\vec P^\text{edge}_α$, where the integer $α$ enumerates the edges. We introduce a p…
We report the theoretical discovery and characterization of higher-order Floquet topological phases dynamically generated in a periodically driven system with mirror symmetries. We demonstrate numerically and analytically that these phases support lower-dimensional Floquet bound states, such as corner Floquet bound states at the intersection of edges of a two-dimensional system, protected by the nonequilibrium higher-order topology induced by the periodic drive. We characterize higher-order Floquet topologies of the bulk Floquet Hamiltonian using mirror-graded Floquet topological invariants. T…
The bulk-boundary correspondence is a generic feature of topological states of matter, reflecting the intrinsic relation between topological bulk and boundary states. For example, robust edge states propagate along the edges and corner states gather at corners in the two-dimensional first-order and second-order topological insulators, respectively. Here, we report two kinds of topological states hosting anomalous bulk-boundary correspondence in the extended two-dimensional dimerized lattice with staggered flux threading. At $\frac{1}{2}$-filling, we observe isolated corner states with no fract…
Ordering the topological order in the fractional quantum Hall effect
Yale University（耶鲁大学）；University of Maryland（马里兰大学）；University of Texas, Austin（德克萨斯大学奥斯汀分校）；Institute for Advanced Study（高等研究院）；Massachusetts Institute of Technology（麻省理工学院）
Based on first-principles calculations and symmetry analysis, we predict atomically thin ($1-N$ layers) 2H group-VIB TMDs $MX_2$ ($M$ = Mo, W; $X$ = S, Se, Te) are large-gap higher-order topological crystalline insulators protected by $C_3$ rotation symmetry. We explicitly demonstrate the nontrivial topological indices and existence of the hallmark corner states with quantized fractional charge for these familiar TMDs with large bulk optical band gaps ($1.64-1.95$ eV for the monolayers), which would facilitate the experimental detection by STM. We find that the well-defined corner states exist…
We propose the concept of a device based on a square-shaped sample of a two-dimensional second-order topological helical superconductor which hosts two zero-dimensional Majorana quasiparticles at the corners. The two zero-energy modes rely on particle-hole symmetry (PHS) and their spacial position can be shifted by rotating an in-plane magnetic field and tuning proximity-induced spin-singlet pairing. We consider an adiabatic cycle performed on the degenerate ground-state manifold and show that it realizes the braiding of the two modes whereby they accumulate a non-trivial statistical phase $π$…
Seeking the criterion for diagnosing topological phases in real materials has been one of the major tasks in topological physics. Currently, bulk-boundary correspondence based on spectral measurements of in gap topological boundary states and the fractional corner anomaly derived from the measurement of the fractional spectral charge are two main approaches to characterize topologically insulating phases. However, these two methods require a complete band-gap with either in-gap states or strict spatial symmetry of the overall sample which significantly limits their applications to more general…
Recently, a new class of second-order topological insulators (SOTIs) characterized by an electronic dipole has been theoretically introduced and proposed to host topological corner states. As a novel topological state, it has been attracting great interest and experimentally realized in artificial systems of various fields of physics based on multi-sublattice models, e.g., breathing kagome lattice. In order to realize such kind of SOTI in natural materials, we proposed a symmetry-faithful multi-orbital model. Then, we reveal several familiar transition metal dichalcogenide (TMD) monolayers as …
We have studied the $ν=5/2$ fractional quantum Hall state in a density-tunable sample at extremely low electron densities. For the densities accessed in our experiment, the Landau level mixing parameter $κ$ spans the $2.52<κ<2.82$ range. In the vicinity of $5.8 \times 10^{10}$~cm$^{-2}$ or $κ= 2.6$ an anomalously large change in the density dependence of the energy gap is observed. Possible origins of such an anomaly are discussed, including a topological phase transition in the $ν=5/2$ fractional quantum Hall state.
Electronic materials harbor a plethora of exotic quantum phases, ranging from unconventional superconductors to non-Fermi liquids, and, more recently, topological phases of matter. While these quantum phases in integer dimensions are well characterized by now, their presence in fractional dimensions remains vastly unexplored. Here, we theoretically show that a special class of crystalline phases, namely, higher-order topological phases that via an extended bulk-boundary correspondence feature robust gapless modes on lower dimensional boundaries, such as corners and hinges, can be found on a re…
We give field theory descriptions of the time-reversal invariant quantum spin Hall insulator in 2+1 dimensions and the particle-hole symmetric insulator in 1+1 dimensions in terms of massive Dirac fermions. Integrating out the massive fermions we obtain a low-energy description in terms of a topological field theory, which is entirely determined by anomaly considerations. This description allows us to easily construct low-energy effective actions for the corresponding `fractional' topological insulators, potentially corresponding to new states of matter. We give a holographic realization of th…
Second-order topological insulators (SOTIs) are the topological phases of matter in d dimensions that manifest (d-2)-dimensional localized modes at the intersection of the edges. We show that SOTIs can be designed via stacked Chern insulators with opposite chiralities connected by interlayer coupling. To characterize the bulk-corner correspondence, we establish a Jacobian-transformed nested Wilson loop method and an edge theory that are applicable to a wider class of higher-order topological systems. The corresponding topological invariant admits a filling anomaly of the corner modes with frac…
The Moore-Read state is one the most well known non-Abelian fractional quantum Hall states. It supports non-Abelian Ising anyons in the bulk and a chiral bosonic and chiral Majorana modes on the boundary. It has been recently conjectured that these modes are superpartners of each other and described by a supersymmetric conformal field theory [1]. We propose a non-relativistic supergeometric theory that is compatible with this picture and gives rise to an effective spin-3/2 field in the bulk. After breaking supersymmetry through a Goldstino, the spin-3/2 field becomes massive and can be seen as…
Fractional quantum Hall (FQH) states are examples of symmetry-enriched topological states (SETs): in addition to the intrinsic topological order, which is robust to symmetry breaking, they possess symmetry-protected topological invariants, such as fractional charge of anyons and fractional Hall conductivity. In this paper we develop a comprehensive theory of symmetry-protected topological invariants for FQH states with spatial symmetries, which applies to Abelian and non-Abelian topological states, by using a recently developed framework of $G$-crossed braided tensor categories ($G\times$BTCs)…
