Higher-order topological insulators (HOTIs) or multipole insulators, hosting peculiar corner states, were discovered [arXiv:1611.07987, arXiv:1708.03636]. It was independently discovered [arXiv:1702.00624] that continuum 5-dimensional (5D) Weyl semimetals generically host the corner states, and so do 4D class A and 3D class AIII topological insulators. In this paper we further confirm that the 5D Weyl semimetals, upon dimensional reduction, lead to universal higher-order topology. First we explain a discrete symmetry protecting the 5D Weyl semimetals, and describe dimensional reductions of the 5D Weyl semimetals to the popular HOTIs in the continuum limit. We calculate the topological charge carried by edge states of the 5D Weyl semimetal, for the most generic boundary condition. The topological charge is a Dirac monopole, which can also be seen from that edge Hamiltonians are always of the form of a 3D Weyl semimetal. This edge topology leads to the edge-of-edge states, or the corner states, generically, suggesting that the 5D Weyl semimetal is thought of as a physical structural origin of corner states in HOTIs. In addition, we explicitly calculate a nested Wilson loop of the 5D Weyl semimetal and find that the topological structure is identical to that of a Wilson loop of a Dirac monopole.
Nested Feature Spectrum Topology: Tripartite Topological Equivalence of Feature, Entanglement, and Wilson Loop Spectrum
Institute of Physics, Academia Sinica（中央研究院物理研究所）；Department of Physics, Northeastern University（东北大学物理系）；Quantum Materials and Sensing Institute, Northeastern University（东北大学量子材料与传感研究所）
The topological states of matter and topological materials have been attracting extensive interests as one of the frontier topics in condensed matter physics and materials science since the discovery of quantum Hall effect in 1980s. So far all the topological phases such as quantum Hall effect, quantum spin Hall effect and topological insulators and superconductors are characterized by a nonzero integer or Z and Z2 topological invariant. None is a half-integer or fractional. Here we propose a novel type of semimetals which hosts a single cone of Wilson fermions instead of Dirac fermions. The W…
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 …
We study interaction effects, including both long-ranged Coulomb and short-range interactions, in three-dimensional topological triple-Weyl semimetals whose triple-Weyl points are protected by crystal symmetries. By performing Wilsonian renormalization group analysis of the low-energy effective field theory of the minimal model with triple-Weyl nodes, we find that the fixed point of noninteracting triple-Weyl fermions is unstable in the presence of Coulomb interactions and flows to a nontrivial stable fixed point representing marginal Fermi liquids with anisotropic screening effects. We furthe…
We study the multi-gap topology of the periodic spectra of Wilson loop operators (WLOs) in mirror symmetric insulators. We develop two topological invariants each associated with a mirror-invariant gap in the Wilson loop spectrum. We propose that both topological invariants in combination determine the general higher-order bulk-boundary correspondence in 2D mirror symmetric, boundary-obstructed topological insulators. Finally, we demonstrate that these new multi-gap topological invariants apply to anomalous cases beyond those captured by the nested Wilson loop, and we subsequently develop an u…
The magnetoelectric coupling of electrons in a three-dimensional solid can be effectively described by axion electrodynamics. Here we report the discovery of the fractional magnetoelectric effect in chiral anomalous semimetals of the three-dimensional massless Wilson fermions, which have linear dispersions at the energy crossing point, and break the chiral symmetry at generic momenta. In the presence of electric and magnetic fields, the time-reversal and parity symmetry breaking give rise to the quarter-quantized topological magnetoelectric effect, which is directly related to the winding numb…
Weyl semimetals harbor unusual surface states known as Fermi arcs, which are essentially disjoint segments of a two dimensional Fermi surface. We describe a prescription for obtaining Fermi arcs of arbitrary shape and connectivity by stacking alternate two dimensional electron and hole Fermi surfaces and adding suitable interlayer coupling. Using this prescription, we compute the local density of states -- a quantity directly relevant to scanning tunneling microscopy -- on a Weyl semimetal surface in the presence of a point scatterer and present results for a particular model that is expected …
We study the electronic structure of the nodal line semimetal ZrSiTe both experimentally and theoretically. We find two different surface states in ZrSiTe - topological drumhead surface states and trivial floating band surface states. Using the spectra of Wilson loops, we show that a non-trivial Berry phase that exists in a confined region within the Brillouin Zone gives rise to the topological drumhead-type surface states. The $\mathbb{Z}_2$ structure of the Berry phase induces a $\mathbb{Z}_2$ 'modular arithmetic' of the surface states, allowing surface states deriving from different nodal l…
Electron-phonon interactions and instabilities in Weyl semimetals under magnetic fields and torsional strain
Pontificia Universidad Católica de Chile（智利天主教 Pontificia 大学）；Universidad Nacional Autónoma de México（墨西哥国立自治大学）；Center for Nanotechnology and Advanced Materials CIEN-UC（纳米技术与先进材料中心 CIEN-UC）
Based on irreducible representations (or symmetry eigenvalues) and compatibility relations, a material can be predicted to be a topological/trivial insulator [satisfying compatibility relations] or a topological semimetal [violating compatibility relations]. However, Weyl semimetals usually go beyond this symmetry-based strategy. In other words, Weyl nodes could emerge in a material, no matter if its occupied bands satisfy compatibility relations, or if the symmetry indicators are zero. In this work, we propose a new topological invariant $χ$ for the systems with S$_4$ symmetry [i.e., the impr…
We proposed that BaHgSn is a Dirac semimetal (DSM) which can host hourglass-like surface states (HSSs) as protected by nonsymmorphic glide symmetry. Compared to KHgSb, an isostructural topological crystalline insulator with the same HSSs, BaHgSn has an additional band inversion at $Γ$ point. This band inversion is induced by the stronger interlayer coupling among Hg-Sn honeycomb layers than that among Hg-Sb-layers in KHgSb, which leads to bulk Dirac nodes in BaHgSn along the layer stacking direction $Γ$-$A$. In addition, the mirror Chern number $C_{i}$ protected by the mirror plane $\overline{…
We study the band topology of the superconducting nodal-line semimetal (SC-NLSM) protected by the inversion symmetry with and without the spin-orbital coupling. Without the spin-orbital coupling, both the $s$-wave SC-NLSM and the chiral $p$-wave SC-NLSM are topologically nontrivial and can be described by the nonzero winding number. Based on the Wilson loop method, we verify that they are both the fragile topological superconductors, namely, their nontrivial band topologies can be moved off by coupling to additional topologically trivial bands. The fragile topological phase persists in spinful…
Following a general protocol of periodically driving static first-order topological phases (supporting surface states) with suitable discrete symmetry breaking Wilson-Dirac masses, here we construct a hierarchy of higher-order Floquet topological phases in three dimensions. In particular, we demonstrate realizations of both second-order and third-order Floquet topological states, respectively supporting dynamic hinge and corner modes at zero quasienergy, by periodically driving their static first-order parent states with one and two discrete symmetry breaking Wilson-Dirac mass(es). While the s…
We propose a general principle for constructing higher-order topological (HOT) phases. We argue that if a $D$-dimensional first-order or regular topological phase involves $m$ Hermitian matrices that anti-commute with additional $p-1$ mutually anti-commuting matrices, it is conceivable to realize an $n$th-order HOT phase, where $n=1, \cdots, p$, with appropriate combinations of discrete symmetry-breaking Wilsonian masses. An $n$th-order HOT phase accommodates zero modes on a surface with codimension $n$. We exemplify these scenarios for prototypical three-dimensional gapless systems, such as a…
We study the band topology and the associated linking structure of topological semimetals with nodal lines carrying $Z_{2}$ monopole charges, which can be realized in three-dimensional systems invariant under the combination of inversion $P$ and time reversal $T$ when spin-orbit coupling is negligible. In contrast to the well-known $PT$-symmetric nodal lines protected only by $π$ Berry phase in which a single nodal line can exist, the nodal lines with $Z_{2}$ monopole charges should always exist in pairs. We show that a pair of nodal lines with $Z_{2}$ monopole charges is created by a {\it dou…
Hidden topology and strong quantum metric bounds in trivial systems
Hefei National Laboratory（合肥国家实验室）；University of Science and Technology of China（中国科学技术大学）；Fudan University（复旦大学）；Westlake University（西湖大学）；Great Bay University（大湾区大学）
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…
We theoretically investigate a periodically driven semimetal based on a square lattice. The possibility of engineering both Floquet Topological Insulator featuring Floquet edge states and Floquet higher order topological insulating phase, accommodating topological corner modes has been demonstrated starting from the semimetal phase, based on Floquet Hamiltonian picture. Topological phase transition takes place in the bulk quasi-energy spectrum with the variation of the drive amplitude where Chern number changes sign from $+1$ to $-1$. This can be attributed to broken time-reversal invariance (…
The axion insulator (AXI) has long been recognized as the simplest example of a 3D magnetic topological insulator (TI). The most familiar AXI results from magnetically gapping the surface states of a 3D $\mathbb{Z}_{2}$ TI while preserving the bulk gap. Like the 3D TI, it exhibits a quantized magnetoelectric polarizability of $θ=π$, and can be diagnosed from bulk symmetry eigenvalues when inversion symmetric. However, whereas a 3D TI is characterized by bulk Wilson loop winding, 2D surface states, and the pumping of the 2D $\mathbb{Z}_{2}$ TI index, we show that an AXI with a large number of b…
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…
We study the bulk and boundary properties of fragile topological insulators (TIs) protected by inversion symmetry, mostly focusing on the class A of the Altland-Zirnbauer classification. First, we propose an efficient method for diagnosing fragile band topology by using the symmetry data in momentum space. Using this method, we show that among all the possible parity configurations of inversion-symmetric insulators, at least 17 percent of them have fragile topology in 2D while fragile TIs are less than 3 percent in 3D. Second, we study the bulk-boundary correspondence of fragile TIs protected …
In recent years, transition metal dichalcogenides (TMDs) have garnered great interest as topological materials -- monolayers of centrosymmetric $β$-phase TMDs have been identified as 2D topological insulators (TIs), and bulk crystals of noncentrosymmetric $γ$-phase MoTe$_2$ and WTe$_2$ have been identified as type-II Weyl semimetals. However, ARPES and STM probes of these TMDs have revealed huge, "arc-like" surface states that overwhelm, and are sometimes mistaken for, the much smaller topological surface Fermi arcs of bulk type-II Weyl points. In this letter, we use first-principles calculati…
We analyze the topological properties of a chiral ${p}+i{p}$ superconductor for a two-dimensional metal/semimetal with four Dirac points. Such a system has been proposed to realize second-order topological superconductivity and host corner Majorana modes. We show that with an additional $\mathsf{C}_4$ rotational symmetry, the system is in an intrinsic higher-order topological superconductor phase, and with a lower and more natural $\mathsf{C}_2$ symmetry, is in a boundary-obstructed topological superconductor phase. The boundary topological obstruction is protected by a bulk Wannier gap. Howev…
Band topology of anomalous quantum Hall insulators can be precisely addressed by computing Chern numbers of constituent non-degenerate bands that describe quantized, Abelian Berry flux through two-dimensional Brillouin zone. Can Chern numbers be defined for $SU(2)$ Berry connection of two-fold degenerate bands of materials preserving space-inversion ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetries or combined $\mathcal{PT}$ symmetry, without detailed knowledge of underlying basis? We affirmatively answer this question by employing a non-Abelian generalization of Stokes' theorem and…
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 concept of topological phases has been generalized to higher-order topological insulators and superconductors with novel boundary states on corners or hinges. Meanwhile, recent experimental advances in controlling dissipation (such as gain and loss) open new possibilities in studying non-Hermitian topological phases. Here, we show that higher-order topological corner states can emerge by simply introducing staggered on-site gain/loss to a Hermitian system in trivial phases. For such a non-Hermitian system, we establish a general bulk-corner correspondence by developing a biorthogonal neste…
Higher-order topological phases and real topological phases are two emerging topics in topological states of matter, which have been attracting considerable research interest. However, it remains a challenge to find realistic materials that can realize these exotic phases. Here, based on first-principles calculations and theoretical analysis, we identify graphyne, the representative of the graphyne-family carbon allotropes, as a two-dimensional (2D) second-order topological insulator and a real Chern insulator. We show that graphyne has a direct bulk band gap at the three $M$ points, forming t…
We extend the theory of dipole moments in crystalline insulators to higher multipole moments. In this paper, we expand in great detail the theory presented in Ref. 1, and extend it to cover associated topological pumping phenomena, and a novel class of 3D insulator with chiral hinge states. In quantum-mechanical crystalline insulators, higher multipole bulk moments manifest themselves by the presence of boundary-localized moments of lower dimension, in exact correspondence with the electromagnetic theory of classical continuous dielectrics. In the presence of certain symmetries, these moments …
Recently extended from the modern theory of electric polarization, quantized multipole topological insulators (QMTIs) describe higher-order multipole moments, lying in nested Wilson loops, which are inherently quantized by lattice symmetries. Overlooked in the past, QMTIs reveal new types of gapped boundaries, which themselves represent lower-dimensional topological phases and host topologically protected zero-dimensional (0D) corner states. Inspired by these pioneering theoretical predictions, tremendous efforts have been devoted to the experimental observation of topological quantized quadru…
Over a long period of exploration, the successful observation of quantized version of anomalous Hall effect (AHE) in thin film of magnetically-doped topological insulator completed a quantum Hall trio---quantum Hall effect (QHE), quantum spin Hall effect (QSHE), and quantum anomalous Hall effect (QAHE). On the theoretical front, it was understood that intrinsic AHE is related to Berry curvature and U(1) gauge field in momentum space. This understanding established connection between the QAHE and the topological properties of electronic structures characterized by the Chern number. With the tim…
Exact Chiral Symmetries of 3+1D Hamiltonian Lattice Fermions
Walter Burke Institute for Theoretical Physics, Caltech（加州理工学院沃尔特·伯克理论物理研究所）；Department of Physics, Caltech（加州理工学院物理系）；Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy, University of California, Los Angeles（加州大学洛杉矶分校物理学与天文学系马尼·L·巴乌米克理论物理研究所）
