The study of topological bandstructures is an active area of research in condensed matter physics and beyond. Here, we combine recent progress in this field with developments in machine-learning, another rising topic of interest. Specifically, we introduce an unsupervised machine-learning approach that searches for and retrieves paths of adiabatic deformations between Hamiltonians, thereby clustering them according to their topological properties. The algorithm is general as it does not rely on a specific parameterization of the Hamiltonian and is readily applicable to any symmetry class. We demonstrate the approach using several different models in both one and two spatial dimensions and for different symmetry classes with and without crystalline symmetries. Accordingly, it is also shown how trivial and topological phases can be diagnosed upon comparing with a generally designated set of trivial atomic insulators.
Machine learning Majorana topology using unsupervised and supervised learning
Condensed Matter Theory Center（凝聚态理论中心）；Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742, USA（联合量子研究所、物理系、马里兰大学 College Park 分校，马里兰州，20742, USA）；Department of Physics（物理系）；Astronomy, Center for Materials Theory, Rutgers University, Piscataway, NJ 08854, USA（天文学、材料理论中心、罗格斯大学 Piscataway 分校，新泽西州，08854, USA）
Topology and machine learning are two actively researched topics not only in condensed matter physics, but also in data science. Here, we propose the use of topological data analysis in unsupervised learning of the topological phase diagrams. This is possible because the quantum distance can capture the shape of the space formed by the Bloch wavefunctions as we sweep over the Brillouin zone. Therefore, if we minimize the volume of the space formed by the wavefunction through a continuous deformation, the wavefunctions will end up forming distinct spaces which depend on the topology of the wave…
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 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…
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…
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…
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…
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}\,\, ({…
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 …
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…
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 …
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…
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 …
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,…
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…
We show that the entire continuous model of twisted bilayer graphene (TBG) (and not just the two active bands) with particle-hole symmetry is anomalous and hence incompatible with a lattice model. Previous works, e.g., [Phys. Rev. Lett. 123, 036401], [Phys. Rev. X 9, 021013], [Phys. Rev. B 99, 195455], and others [1-4] found that the two flat bands in TBG possess a fragile topology protected by the $C_{2z}T$ symmetry. [Phys. Rev. Lett. 123, 036401] also pointed out an approximate particle-hole symmetry ($\mathcal{P}$) in the continuous model of TBG. In this work, we numerically confirm that $\…
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 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.
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…
We construct a simple model for electrons in a three-dimensional crystal where a combination of short-range hopping and spin-orbit coupling results in nearly flat bands characterized by a non-trivial Z2 topological index. The flat band is separated from other bands by a bandgap much larger than the bandwidth. We discuss the fate of the many-body ground state of electrons in the flat band in the presence of repulsive interactions at partial filling and conjecture that it may become a three-dimensional fractional topological insulator if conventional magnetic instabilities can be avoided.
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…
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 …
Topological band theory establishes a standardized framework for classifying different types of topological matters. Recent investigations have shown that hyperbolic lattices in non-Euclidean space can also be characterized by hyperbolic Bloch theorem. This theory promotes the investigation of hyperbolic band topology, where hyperbolic topological band insulators protected by first Chern numbers have been proposed. Here, we report a new finding on the construction of hyperbolic topological band insulators with a vanished first Chern number but a non-trivial second Chern number. Our model posse…
We propose Landau levels as a probe for the topological character of electronic bands in two-dimensional moiré superlattices. We consider two configurations of twisted double bilayer graphene (TDBG) that have very similar band structures, but show different valley Chern numbers of the flat bands. These differences between the AB-AB and AB-BA configurations of TDBG clearly manifest as different Landau level sequences in the Hofstadter butterfly spectra calculated using the tight-binding model. The Landau level sequences are explained from the point of view of the distribution of orbital magneti…
Two-dimensional systems with $C_{2}\mathcal{T}$ ($P\mathcal{T}$) symmetry exhibit the Euler class topology $E\in\mathbb{Z}$ in each two-band subspace realizing a fragile topology beyond the symmetry indicators. By systematically studying the energy levels of Euler insulating phases in the presence of an external magnetic field, we reveal the robust gaplessness of the Hofstadter butterfly spectrum in the flat-band limit, while for the dispersive bands the gapping of the Landau levels is controlled by a hidden symmetry. We also find that the Euler class $E$ of a two-band subspace gives a lower b…
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…
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 analyze the modulational instability of nonlinear Bloch waves in topological photonic lattices. In the initial phase of the instability development captured by the linear stability analysis, long wavelength instabilities and bifurcations of the nonlinear Bloch waves are sensitive to topological band inversions. At longer timescales, nonlinear wave mixing induces spreading of energy through the entire band and spontaneous creation of wave polarization singularities determined by the band Chern number. Our analytical and numerical results establish modulational instability as a tool to probe …
We report thickness-tuned electrical transport and Hall resistivity in highly anisotropic three-dimensional Dirac semimetal ZrTe5 nanosheets. We find that when the thickness of the nanosheet is blow about 40 nm, the system takes a clear transition from topological semimetal with two bands carriers to a single band with conventional hole carriers. The resistivity peak temperature T* decreases systematically with decreasing thickness down to about 40 nm, then shifts up with the further decrease of the thickness. Analysis of the data below 40 nm indicates that the hole carriers completely dominat…
Controlling Liouvillian topological phases via Hamiltonian band topology under chiral symmetry
Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area (Guangdong)（粤港澳大湾区量子科学中心）；School of Physics and Astronomy & Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing, Sun Yat-Sen University (Zhuhai Campus)（中山大学珠海校区物理与天文学院及广东省量子计量与传感重点实验室）；Department of Physics, Southern University of Science and Technology（南方科技大学物理系）；Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）
'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…
The non-trivial magnon band topology and its consequent responses have been extensively studied in two-dimensional magnetisms. However, the triangular lattice antiferromagnet (TLAF), the best-known frustrated two-dimensional magnet, has received less attention than the closely related Kagome system, because of the spin-chirality cancellation in the umbrella ground state of the undistorted TLAF. In this work, we study the band topology and the thermal Hall effect (THE) of the TLAF with (anti-)trimerization distortion under the external perpendicular magnetic field using the linearized spin wave…
Elementary band representations are the fundamental building blocks of atomic limit band structures. They have the defining property that at partial filling they cannot be both gapped and trivial. Here, we give two examples -- one each in a symmorphic and a non-symmorphic space group -- of elementary band representations realized with an energy gap. In doing so, we explicitly construct a counterexample to a claim by Michel and Zak that single-valued elementary band representations in non-symmorphic space groups with time-reversal symmetry are connected. For each example, we construct a topolog…
We analyze the effect of weak localization (WL) and weak antilocalization (WAL) in the electronic transport through HgTe/CdTe quantum wells. We show that for increasing Fermi energy the magnetoconductance of a diffusive system with inverted band ordering features a transition from WL to WAL and back, if spin-orbit interactions from bulk and structure inversion asymmetry can be neglected. This, and an additional splitting in the magnetoconductance profile, is a signature of the Berry phase arising for inverted band ordering and not present in heterostructures with conventional ordering. In pres…
We revisit the problem of characterizing band topology in dynamically-stable quadratic bosonic Hamiltonians that do not conserve particle number. We show this problem can be rigorously addressed by a smooth and local adiabatic mapping procedure to a particle number conserving Hamiltonian. In contrast to a generic fermionic pairing Hamiltonian, such a mapping can always be constructed for bosons. Our approach shows that particle non-conserving bosonic Hamiltonians can be classified using known approaches for fermionic models. It also provides a simple means for identifying and calculating appro…
We study the Hofstadter butterfly and Landau levels of the twisted bilayer graphene (TBG). We show that the nontrivial fragile topology of the lowest two bands near the charge neutral point makes their Hofstadter butterfly generically connected with higher bands, closing the gap between the first and second conduction (valence) bands at a certain magnetic flux per unit cell. We also develop a momentum space method for calculating the TBG Hofstadter butterfly, from which we identify three phases where the Hofstadter butterflies of the lowest two bands and the higher bands are connected in diffe…
Crystalline semimetals with certain space group symmetries may possess unusual electronic structure topology, distinct from the conventional Weyl and Dirac semimetals. Characteristic property of these materials is the existence of band-touching points with multiple (higher than two-fold) degeneracy and nonzero topological charge. CoSi is a representative of this group of materials exhibiting the so-called "new fermions". We report on an ab initio calculation of the electronic structure of CoSi using density functional methods, taking into account the spin-orbit interactions. We demonstrate the…
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…
