Topological materials are quantum materials with nontrivial ground-state entanglement that are irremovable so long as certain rules, like invariance under symmetries and the existence of an energy gap, are respected. They showcase unconventional properties like robust anomalous surface states and quantized physical responses. The intense research efforts in understanding topological materials result in a modernized perspective on the decades-old theory of symmetry representations in electronic band structures, and inspire the development of general theories that enable the efficient diagnosis of topological materials using only symmetry data. One example is the theory of symmetry indicators of band topology, which is the focus of this topical review. We will aim at providing a pedagogical introduction to the key concepts and constructions in the theory, alongside with a brief summary of the latest development.
The systematic diagnosis of band topology enabled by the method of "symmetry indicators" underlies the recent advances in the search for new materials realizing topological crystalline insulators. Such an efficient method has been missing for superconductors because the band structure in the superconducting phase is not usually available. In this work, we establish symmetry indicators for weak-coupling superconductors that detect nontrivial topology based on the representations of the metallic band structure in the normal phase, assuming a symmetry property of the gap function. We demonstrate …
Stable Real-Space Invariants and Topology Beyond Symmetry Indicators
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Imperial College London（伦敦帝国理工学院）；University of the Basque Country UPV/EHU（巴斯克国立大学）；Peking University（北京大学）；Princeton University（普林斯顿大学）
We present a framework to systematically address topological phases when finer partitionings of bands are taken into account, rather than only considering the two subspaces spanned by valence and conduction bands. Focusing on $C_2\mathcal{T}$-symmetric systems that have gained recent attention, for example in the context of layered van-der-Waals graphene heterostructures, we relate these insights to homotopy groups of Grassmannians and flag varieties, which in turn correspond to cohomology classes and Wilson-flow approaches. We furthermore make use of a geometric construction, the so-called Pl…
Recently twisted bilayer graphene (t-BLG) emerges as a new strongly correlated physical platform near a magic twist angle, which hosts many exciting phenomena such as the Mott-like insulating phases, unconventional superconducting behavior and emergent ferromagnetism. Besides the apparent significance of band flatness, band topology may be another critical element in determining strongly correlated twistronics yet receives much less attention. Here we report compelling evidence for nontrivial noninteracting band topology of t-BLG moiré Dirac bands through a systematic nonlocal transport study,…
We characterize non-Hermitian band structures by symmetry indicator topological invariants. Enabled by crystalline inversion symmetry, these indicators allow us to short-cut the calculation of conventional non-Hermitian topological invariants. In particular, we express the three-dimensional winding number of point-gapped non-Hermitian systems, which is defined as an integral over the whole Brillouin zone, in terms of symmetry eigenvalues at high-symmetry momenta. Furthermore, for time-reversal symmetric non-Hermitian topological insulators, we find that symmetry indicators characterize the ass…
Topological crystalline superconductors are known to have possible higher-order topology, which results in Majorana modes on $d-2$ or lower-dimensional boundaries. Given the rich possibilities of boundary signatures, it is desirable to have topological invariants that can predict the type of Majorana modes from band structures. Although symmetry indicators, a type of invariant that depends only on the band data at high-symmetry points, have been proposed for certain crystalline superconductors, there exist symmetry classes in which symmetry indicators fail to distinguish superconductors with d…
In this work, we revisit the model of PbTe presented in [E. Fradkin, E. Dagotto, and D. Boyanovsky, Phys. Rev. Lett. 57, 2967 (1986)]. We show that the low energy theory of this model corresponds to a (higher-order) topological crystalline insulator in space group $Fm\bar{3}m1'$, diagnosable by symmetry indicators. We show that the gapless fermions found on antiphase domain walls are the topological boundary modes of the system, due to a nonvanishing mirror Chern number. Furthermore, we show that any symmetric completion of the model must be in this same topological phase. Finally, we comment …
In this letter we summarise our study of Su-Schrieffer-Hegger(SSH) model with a one dimensional non-orientable manifold as bulk. For this purpose a SSH model with any bulk (SAB) is introduced. We observe the following: (1) The topology of the SSH model is preserved even with a non-orientable bulk, (2) The appearance of a doubly degenerate flat band exactly at the Fermi level, (3) At half filling the band structure is metallic due to the complete overlap of valence and conduction band but the electrons will be localised because the bands are flat and (4) This system is unique because it has fla…
While the symmetry-based diagnosis of topological insulators and semimetals has enabled large-scale discovery of topological materials candidates, the extension of these approaches to the diagnosis of topological superconductors remains a major open question. One important new ingredient in the analysis of topological superconductivity is the presence of $\mathbb Z_2$-valued Pfaffian invariants associated with certain high-symmetry momenta. Such topological invariants lie beyond the conventional scope of symmetry representation theory for band structures, and as such they are nontrivial to inc…
In flat bands, superconductivity can lead to surprising transport effects. The superfluid "mobility", in the form of the superfluid weight $D_s$, does not draw from the curvature of the band but has a purely band-geometric origin. In a mean-field description, a non-zero Chern number or fragile topology sets a lower bound for $D_s$, which, via the Berezinskii-Kosterlitz-Thouless mechanism, might explain the relatively high superconducting transition temperature measured in magic-angle twisted bilayer graphene (MATBG). For fragile topology, relevant for the bilayer system, the fate of this bound…
The occurrence of superconducting and insulating phases is well-established in twisted graphene bilayers, and they have also been reported in other arrangements of graphene layers. We investigate three such arrangements: untwisted AB bilayer graphene on an hBN substrate, two graphene bilayers twisted with respect to each other, and a single ABC stacked graphene trilayer on an hBN substrate. Narrow bands with different topology occur in all cases, producing a high density of states which enhances the role of interactions. We investigate the effect of the long range Coulomb interaction, treated …
The unambiguous detection of the band topology for topological nodal systems remains an urgent problem in this field. Usually in experiments this relies on the detection for the topological edge modes, which may requires the high demands for the preparation and purifications of the materials and may give ambiguities. In this work, we propose a new approach towards the unambiguous detection of the bulk band crossing nodes of topological nodal systems without evolving the edge mode. The spirit of our proposal is to couple our targeted Hamiltonian to an weak external field $\mathbf {A}$ with a sp…
The properties of electrons in magnetically ordered crystals are of interest both from the viewpoint of realizing novel topological phases, such as magnetic Weyl semimetals, and from the applications perspective of creating energy-efficient memories. A systematic study of symmetry and topology in magnetic materials has been challenging given that there are 1651 magnetic space groups (MSGs). Here, by using an efficient representation of allowed band structures, we obtain a systematic description of several basic properties of free electrons in all MSGs in three dimensions as well as in the 528 …
Topological phases, such as Chern insulators, are defined in terms of additive indices that are stable against the addition of trivial degrees of freedom. Such topology presents an obstruction to any Wannier representation, namely, the representation of the electronic states in terms of symmetric, exponentially localized Wannier functions. Here, we address the converse question: Do obstructions to Wannier representation imply stable band topology? We answer this in the negative, pointing out that some bands can also display a distinct type of "fragile topology." Bands with fragile topology do …
Topological crystalline superconductors have attracted rapidly rising attention due to the possibility of higher-order phases, which support Majorana modes on boundaries in $d-2$ or lower dimensions. However, although the classification and bulk topological invariants in such systems have been well studied, it is generally difficult to faithfully predict the boundary Majoranas from the band-structure information due to the lack of well-established bulk-boundary correspondence. Here we propose a protocol for deriving symmetry indicators that depend on a minimal set of necessary symmetry data of…
Topology is a central notion in the classification of band insulators and characterization of entangled many-body quantum states. In some cases, it manifests as quantized observables such as quantum Hall conductance. However, being inherently a global property depending on the entirety of the system, its direct measurement has remained elusive to local experimental probes in many cases. Here, we demonstrate that various topological band indices can be directly probed by resonant inelastic x-ray scattering. Specifically, we show that the crystalline symmetry eigenvalues at the high-symmetry mom…
Bloch theory describes the electronic states in crystals whose energies are distributed as bands over the Brillouin zone. The electronic states corresponding to a (few) isolated energy band(s) thus constitute a vector bundle. The topological properties of these vector bundles provide new characteristics of the corresponding electronic phases. We review some of these properties in the case of (topological) insulators and semi-metals.
We theoretically demonstrate how competition between band inversion and spin-orbit coupling (SOC) results in nontrivial evolution of band topology, taking antiperovskite Ba$_3$SnO as a prototype material. A key observation is that when the band inversion dominates over SOC, there appear "twin" Dirac cones in the band structure. Due to the twin Dirac cones, the band shows highly peculiar structure in which the upper cone of one of the twin continuously transforms to the lower cone of the other. Interestingly, the relative size of the band inversion and SOC is controlled in this series of antipe…
Although the richness of spatial symmetries has led to a rapidly expanding inventory of possible topological crystalline (TC) phases of electrons, physical realizations have been slow to materialize due to the practical difficulty to ascertaining band topology in realistic calculations. Here, we integrate the recently established theory of symmetry indicators of band topology into first-principle band-structure calculations, and test it on a databases of previously synthesized crystals. The combined algorithm is found to efficiently unearth topological materials and predict topological propert…
In this paper we show that for a range of configurations of inhomogeneous magnetic fields it is possible to create flat bands of Majorana states localized on the edges of 2-d lattices. Majorana bound states have been predicted to exist in both one dimensional and two dimensional systems with Rashba spin-orbit coupling, magnetic fields, and placed in proximity to a superconductor. For the proposed systems we present the bulk topological phase diagrams, and we study the conditions for weak topology which predicts the formation of bands of Majorana states. The Majorana bands are demonstrated to b…
We show that the Wannier obstruction and the fragile topology of the nearly flat bands in twisted bilayer graphene at magic angle are manifestations of the nontrivial topology of two-dimensional real wave functions characterized by the Euler class. To prove this, we examine the generic band topology of two dimensional real fermions in systems with space-time inversion $I_{ST}$ symmetry. The Euler class is an integer topological invariant classifying real two band systems. We show that a two-band system with a nonzero Euler class cannot have an $I_{ST}$-symmetric Wannier representation. Moreove…
We uncover topological features of neutral particle-hole pair excitations of correlated quantum anomalous Hall (QAH) insulators whose approximately flat conduction and valence bands have equal and opposite non-zero Chern number. Using an exactly solvable model we show that the underlying band topology affects both the center-of-mass and relative motion of particle-hole bound states. This leads to the formation of topological exciton bands whose features are robust to nonuniformity of both the dispersion and the Berry curvature. We apply these ideas to recently-reported broken-symmetry spontane…
The rapid development of topological photonics and acoustics calls for accurate understanding of band topology in classical waves, which is not yet achieved in many situations. Here, we present the Wilson-loop approach for exact numerical calculation of the topological invariants for several photonic/sonic crystals. We demonstrate that these topological photonic/sonic crystals are topological crystalline insulators with fragile topology, a feature which has been ignored in previous studies. We further discuss the bulk-edge correspondence in these systems with emphasis on symmetry broken on the…
'Magic'-angle twisted bilayer graphene has received a lot of interest due to its flat bands with potentially non-trivial topology that lead to intricate correlated phases. A spectrum with flat bands, however, does not require a twist between multiple sheets of van der Waals materials, but rather can be realized with the application of an appropriate periodic potential. Here, we propose the imposition of a tailored periodic potential onto a single graphene layer through local perturbations that could be created via lithography or adatom manipulation, which also results in an energy spectrum fea…
Electron energy bands of crystalline solids generically exhibit degeneracies called band-structure nodes. Here, we introduce non-Abelian topological charges that characterize line nodes inside the momentum space of crystalline metals with space-time inversion ($\mathcal{PT}$) symmetry and with weak spinorbit coupling. We show that these are quaternion charges, similar to those describing disclinations in biaxial nematics. Starting from two-band considerations, we develop the complete many-band description of nodes in the presence of $\mathcal{PT}$ and mirror symmetries, which allows us to inve…
Electronic materials can sustain a variety of unusual, but symmetry protected touchings of valence and conduction bands, each of which is identified by a distinct topological invariant. Well-known examples include linearly dispersing pseudo-relativistic fermions in monolayer graphene, Weyl and nodal-loop semimetals, biquadratic (bicubic) band touching in bilayer (trilayer) graphene, as well as mixed dispersions in multi-Weyl systems. Here we show that depending on the underlying band curvature, the shear viscosity in the collisionless regime displays a unique power-law scaling with frequency a…
The symmetry-based indicator [H. C. Po, A. Vishwanath, H. Watanabe, Nat. Commun. 8, 50 (2017)] is a practical tool to diagnose topological materials in the band theory. In this note, we present two directions to generalize the symmetry-based indicator for other classes of topological materials. The one is for superconductors. The careful definition of the atomic insulators and the trivial vacuum Hamiltonian yields the symmetry-based indicators specific to superconductors. The other is for ingap boundary states. The quotient of the group of atomic insulators by a subset of atomic insulators suc…
In this work, we examine the topological phases that can arise in triangular lattices with disconnected elementary band representations. We show that, although these phases may be "fragile" with respect to the addition of extra bands, their topological properties are manifest in certain nontrivial holonomies (Wilson loops) in the space of nontrivial bands. We introduce an eigenvalue index for fragile topology, and we show how a nontrivial value of this index manifests as the winding of a hexagonal Wilson loop; this remains true even in the absence of time-reversal or sixfold rotational symmetr…
We elucidate the general rule governing the response of dislocation lines in three-dimensional topological band insulators. According to this ${\bf K}\text{-}{\bf b}\text{-}{\bf t}$ rule, the lattice topology, represented by dislocation lines oriented in direction ${\bf t}$ with Burgers vector ${\bf b}$, combines with the electronic-band topology, characterized by the band-inversion momentum ${\bf K}_{\rm inv}$, to produce gapless propagating modes when the plane orthogonal to the dislocation line features a band inversion with a nontrivial ensuing flux $Φ={\bf K}_{\rm inv}\cdot {\bf b}\,\, ({…
Based on first-principles calculations and tight-binding model analysis, we propose monolayer graphdiyne as a candidate material for a two-dimensional higher-order topological insulator protected by inversion symmetry. Despite the absence of chiral symmetry, the higher-order topology of monolayer graphdiyne is manifested in the filling anomaly and charge accumulation at two corners. Although its low energy band structure can be properly described by the tight-binding Hamiltonian constructed by using only the $p_z$ orbital of each atom, the corresponding bulk band topology is trivial. The nontr…
In many atomically thin materials their optical absorption is dominated by excitonic transitions. It was recently found that optical selection rules in these materials are influenced by the band topology near the valleys. We propose that gate-controlled band ordering in a single atomic monolayer, through changes in the valley winding number and excitonic transitions, can be probed in helicity-resolved absorption and photoluminescence. This predicted tunable band topology is confirmed by combining an effective Hamiltonian and a Bethe-Salpeter equation for an accurate description of excitons, wi…
We study the electronic structure and band topology of fully relaxed twisted multi-layer graphene (TMLG). Isolated flat bands emerge in TMLG with the number of layers [$M+N$ with $M$ the layer number of the bottom few-layer graphene (FLG)] up to 10 and with various stacking orders, and most of them are on the hole side. The touched bands of FLGs around the Fermi level are split by the moiré coupling through the electron-hole asymmetry in low-energy bands of FLGs and by the vertical hopping between next-nearest layers. The full structural relaxation leads to global gaps that completely isolate …
We classify the topology of bands defined by the energy states of quantum systems with scale separation between slow and fast degrees of freedom, invariant under fermionic time reversal. Classical phase space transforms differently from momentum space under time reversal, and as a consequence the topology of adiabatic bands is different from that of Bloch bands. We show that bands defined over a two-dimensional phase space are classified by the Chern number, whose parity must be equal to the parity of the band rank. Even-rank bands are equivalently classified by the Kane-Mele index, an integer…
Motivated by the recent experimental realizations of hyperbolic lattices in circuit quantum electrodynamics and in classical electric-circuit networks, we study flat bands and band-touching phenomena in such lattices. We analyze noninteracting nearest-neighbor hopping models on hyperbolic analogs of the kagome and dice lattices with heptagonal and octagonal symmetry. We show that two characteristic features of the energy spectrum of those models, namely the fraction of states in the flat band as well as the number of touching points between the flat band and the dispersive bands, can both be c…
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 …
The geometric properties of a lattice can have profound consequences on its band spectrum. For example, symmetry constraints and geometric frustration can give rise to topologicially nontrivial and dispersionless bands, respectively. Line-graph lattices are a perfect example of both of these features: their lowest energy bands are perfectly flat, and here we develop a formalism to connect some of their geometric properties with the presence or absence of fragile topology in their flat bands. This theoretical work will enable experimental studies of fragile topology in several types of line-gra…
Various properties of the energy band structures (electronic, phonon, etc.), including systematic band degeneracy, sticking and extremes, following from the full line group symmetry of the single-wall carbon nanotubes are established. The complete set of quantum numbers consists of quasi momenta (angular and linear or helical) and parities with respect to the z-reversal symmetries and, for achiral tubes, the vertical plane. The assignation of the electronic bands is performed, and the generalized Bloch symmetry adapted eigen functions are derived. The most important physical tensors are charac…
We show that topology of the low-energy band structure in bilayer graphene critically depends on mechanical deformations of the crystal which may easily develop in suspended graphene flakes. We describe the Lifshitz transition that takes place in strained bilayers upon splitting the parabollic bands at intermediate energies into several Dirac cones at the energy scale of few meV. Then, we show how this affects the electron Landau level spectra and the quantum Hall effect.
We show that band topology can dramatically change the photophysics of two-dimensional (2D) semiconductors. For systems in which states near the band extrema are of multiple orbitals character and the spinors describing the orbital components (pseudospins) pick up nonzero winding numbers (topological invariants) around the extremal k-point, the optical strength and nature (i.e., helicity) of the excitonic states are dictated by the optical matrix element winding number, a unique and heretofore unrecognized characteristic. We illustrate these findings in three gapped graphene systems - monolaye…
A critical point of the energy dispersion is the momentum where electron velocity vanishes. At the corresponding energy, the density of states (DOS) exhibits non-analyticity such as divergence. Critical points can be first classified as ordinary and high-order ones, and the ordinary critical points have been studied thoroughly by Léon van Hove. In this work, we describe and classify high-order critical points based on topology, scaling and symmetry, which are beyond Léon van Hove's framework. We show that high-order critical points can have power-law divergent DOS with particle-hole asymmetry,…
Electron motion in crystals is governed by the coupling between crystal momentum and internal degrees of freedom such as spin implicit in the band structure. The description of this coupling in terms of a momentum-dependent effective field and the resultant Berry phase has greatly advanced our understanding of diverse phenomena including various Hall effects and has lead to the discovery of new states of matter exemplified by topological insulators. While experimental studies on topological systems have focused on the gapless states that emerge at the surfaces or edges, the underlying nontrivi…
