Nontrivial Topological Invariants Research Articles

Rotational symmetry protected edge and corner states in Abelian topological phases

Spatial symmetries can enrich the topological classification of interacting quantum matter and endow systems with non-trivial strong topological invariants (protected by internal symmetries) with additional "weak" topological indices. In this paper, we study the edge physics of systems with a non-trivial shift invariant, which is protected by either a continuous $\text{U}(1)_r$ or discrete $\text{C}_n$ rotation symmetry, along with internal $\text{U}(1)_c$ charge conservation. Specifically, we construct an interface between two systems which have the same Chern number but are distinguished by their Wen-Zee shift and, through analytic arguments supported by numerics, show that the interface hosts counter-propagating gapless edge modes which cannot be gapped by arbitrary local symmetry-preserving perturbations. Using the Chern-Simons field theory description of two-dimensional Abelian topological orders, we then prove sufficient conditions for continuous rotation symmetry protected gapless edge states using two complementary approaches. One relies on the algebraic Lagrangian sub-algebra framework for gapped boundaries while the other uses a more physical flux insertion argument. For the case of discrete rotation symmetries, we extend the field theory approach to show the presence of fractional corner charges for Abelian topological orders with gappable edges, and compute them in the case where the Abelian topological order is placed on the two-dimensional surface of a Platonic solid. Our work paves the way for studying the edge physics associated with spatial symmetries in symmetry enriched topological phases.

Inner skin effects on non-Hermitian topological fractals

Non-Hermitian (NH) crystals, quasicrystals, and amorphous network display an accumulation of a macroscopic number of states near one of its specific interfaces with vacuum, such as edge, surface, hinge, or corner. This phenomenon is known as the NH skin effect, which can only be observed with open boundary condition. In this regard self-similar fractals, manifesting inner boundaries in the interior of the system, harbor a novel phenomenon, the inner skin effect (ISE). Then the NH skin effect appears at the inner boundaries of the fractal lattice with periodic boundary condition. We showcase this observation by implementing prominent models for NH insulators and superconductors on representative planar Sierpinski carpet fractal lattices. They accommodate both first-order and second-order ISEs at inner edges and corners, respectively, for charged as well as neutral Majorana fermions. Furthermore, over extended parameter regimes ISEs are tied with nontrivial bulk topological invariants, yielding intrinsic ISEs. With the recent success in engineering NH topological phases on highly tunable metamaterial platforms, such as photonic and phononic lattices, as well as topolectric circuits, the proposed ISEs can be observed experimentally at least on fractal metamaterials with periodic boundary condition.

First-principles study on the electronic structures and topological properties of Bi(110)/IV-VI and Bi(110)/V-V van der Waals heterostructures

Based on the first-principles calculations, numerous theoretical research efforts have been made to exploit electronic structures and topological properties in Bi(110) and IV-VI groups (Sn0.5Se0.5, Sn0.5Se0.25Te0.25, Sn0.5Te0.5), V-V groups (Sb0.5As0.5, Bi0.5As0.5, α-Bi0.5Sb0.5, α2-Bi0.5Sb0.5) substrates heterostructures. Molecular dynamics confirmed that the proposed bilayer heterostructures are stable. The electronic analysis results show that the valence and conduction bands near the Fermi level are mainly contributed by Bi(110). More importantly, Bi(110)/Sn0.5Te0.5 and Bi(110)/Sn0.5Se0.25Te0.25 have non-topological properties. However, when applying 20% biaxial tensile strain, both of them have nontrivial Z2 topological invariants and topological edge states. Our findings not only breathe new power into the family of 2D materials but also highlight these 2D semiconductors as attractive materials potential for future electronics and topological devices.

Ta4SiTe4 : A possible one-dimensional topological insulator

Using the first-principles calculations, we present a prediction of the nontrivial topological phase in one-dimensional (1D) ${\mathrm{Ta}}_{4}{\mathrm{SiTe}}_{4}$ with high stability further verified by the molecular dynamics simulation and phonon spectrum. In this material, the band inversion intrinsically occurs between $\mathrm{Ta}\text{\ensuremath{-}}d$ and $\mathrm{Te}\text{\ensuremath{-}}p$ orbitals without the spin-orbit coupling effect. As a novel topological insulator, protected by time-reversal symmetry Tˆ and inversion symmetry Pˆ, we explicitly demonstrate the existence of nontrivial topological invariants and the protected edge states in it with a large bulk band gap of $\ensuremath{\sim}163\phantom{\rule{0.28em}{0ex}}\mathrm{meV}$, which could facilitate the experimental verification at the room temperature. In addition, we find its topological states are robust against the external pressure. Our results uncover a potential 1D topological-insulating ${\mathrm{Ta}}_{4}{\mathrm{SiTe}}_{4}$ and promote it as a concrete material platform for exploring the intriguing physics of low-dimensional topological phases.

Topological dissipation in a time-multiplexed photonic resonator network

Topological phases feature robust edge states that are protected against the effects of defects and disorder. These phases have largely been studied in conservatively coupled systems, in which non-trivial topological invariants arise in the energy or frequency bands of a system. Here we show that, in dissipatively coupled systems, non-trivial topological invariants can emerge purely in a system’s dissipation. Using a highly scalable and easily reconfigurable time-multiplexed photonic resonator network, we experimentally demonstrate one- and two-dimensional lattices that host robust topological edge states with isolated dissipation rates, measure a dissipation spectrum that possesses a non-trivial topological invariant, and demonst rate topological protection of the network’s quality factor. The topologically non-trivial dissipation of our system exposes new opportunities to engineer dissipation in both classical and quantum systems. Moreover, our experimental platform’s straightforward scaling to higher dimensions and its ability to implement inhomogeneous, non-reciprocal and long range couplings may enable future work in the study of synthetic dimensions.

Gapless Topological Phases and Symmetry-Enriched Quantum Criticality

We introduce topological invariants for gapless systems and study the associated boundary phenomena. More generally, the symmetry properties of the low-energy conformal field theory (CFT) provide discrete invariants, establishing the notion of symmetry-enriched quantum criticality. The charges of nonlocal scaling operators, or more generally of symmetry defects, are topological and imply the presence of localized edge modes. We primarily focus on the $1+1d$ case where the edge has a topological degeneracy, whose finite-size splitting can be exponential or algebraic in system size depending on the involvement of additional gapped sectors. An example of the former is given by tuning the spin-1 Heisenberg chain to a symmetry-breaking Ising phase. An example of the latter arises between the gapped Ising and cluster phases: this symmetry-enriched Ising CFT has an edge mode with finite-size splitting $\sim 1/L^{14}$. In addition to such new cases, our formalism unifies various examples previously studied in the literature. Similar to gapped symmetry-protected topological phases, a given CFT can split into several distinct symmetry-enriched CFTs. This raises the question of classification, to which we give a partial answer -- including a complete characterization of symmetry-enriched $1+1d$ Ising CFTs. Non-trivial topological invariants can also be constructed in higher dimensions, which we illustrate for a symmetry-enriched $2+1d$ CFT without gapped sectors.

Second-order topological insulator state in hexagonal lattices and its abundant material candidates

We propose two mechanisms to realize the second order topological insulator (SOTI) state in spinless hexagonal lattices, viz., chemical modification and anti-Kekul\'e/Kekul\'e distortion of hexagonal lattice. Correspondingly, we construct two models and demonstrate the nontrivial band topology of the SOTI state characterized by the second Stiefel-Whitney class $w_2$ in the presence of inversion symmetry ($\textit{P}$) and time-reversal symmetry ($\textit{T}$). Based on the two mechanisms and using first-principles calculations and symmetry analysis, we predict three categories of real light element material candidates, i.e., hydrogenated and halogenated 2D hexagonal group IV materials XY (X=C, Si, Ge, Sn, Y=H, F, Cl), 2D hexagonal group V materials (blue phosphorene, blue arsenene, and black phosphorene, black arsenene), and the recent experimentally synthesized anti-Kekul\'e/Kekul\'e order graphenes and the counterparts of silicene/germanene/stanene. We explicitly demonstrate the nontrivial topological invariants and existence of the protected corner states with fractional charge for these candidates with giant bulk band gap (up to 3.5 eV), which could facilitate the experimental verification by STM. Our approaches and proposed abundant real material candidates will greatly enrich 2D SOTIs and promote their intriguing physics research.

Specialty topological fiber using periodic lattice geometries

Topological photonics provides a new platform to control and manipulate the flow of light in photonic devices that exhibit topological phase transitions. One way to introduce such transition of geometric phases in photonic waveguides is to modulate the structure of the photonic crystals, array waveguides, or the metamaterials in such a way that it eventually changes the bulk topology. Such topological waveguides localize the light at the boundary of the two distinct bulk systems with different topological invariant. Based on this fundamental physics, we demonstrate a robust and controlled propagation of light in a class of specialty optical fiber which exhibits the topological phase transition between a nontrivial one-dimensional (1D) periodic cylinder and its topologically trivial counterpart. The key idea is to explore the possibility of topological phase transition of 1D photonic bands over the momentum space by tuning the unit-cell dimension and its individual thicknesses. We have systematically computed the trivial and the nontrivial topological invariants, the Zak phase, of designed cylindrical 1D lattices with inversion symmetry over the first Brillouin zone. To establish the presence and the robustness of the localized edge mode, we have studied the reflectance of the geometry in the presence of disorder. Also, the simulated field profile and the back-reflectionless propagation of the edge mode through the fiber have been shown. Besides conventional fibers, such topological optical fibers are definitely a waveguide geometry for robust delivery of light and thus would be very significant to open up a new paradigm in guided wave photonics.

Observation of a chiral wave function in the twofold-degenerate quadruple Weyl system BaPtGe

Topological states in quantum materials are defined by non-trivial topological invariants, such as the Chern number, which are properties of their bulk wave functions. A remarkable consequence of topological wave functions is the emergence of edge modes, a phenomenon known as bulk-edge correspondence, that gives rise to quantized or chiral physical properties. While edge modes are widely presented as signatures of non-trivial topology, how bulk wave functions can manifest explicitly topological properties remains unresolved. Here, using high-resolution inelastic x-ray spectroscopy (IXS) combined with first principles calculations, we report experimental signatures of chiral wave functions in the bulk phonon spectrum of BaPtGe, which we show to host a previously undiscovered twofold degenerate quadruple Weyl node. The chirality of the degenerate phononic wave function yields a non-trivial phonon dynamical structure factor, S(Q,$\omega$), along high-symmetry directions, that is in excellent agreement with numerical and model calculations. Our results establish IXS as a powerful tool to uncover topological wave functions, providing a key missing ingredient in the study of topological quantum matter.

Atomic Line Defects and Topological Superconductivity in Unconventional Superconductors

Topological superconductors (TSCs) are correlated quantum states with simultaneous off-diagonal long-range order and nontrivial topological invariants. They produce gapless or zero energy boundary excitations, including Majorana zero modes and chiral Majorana edge states with topologically protected phase coherence essential for fault-tolerant quantum computing. Candidate TSCs are very rare in nature. Here, we propose a novel route toward emergent quasi-one-dimensional (1D) TSCs in naturally embedded quantum structures such as atomic line defects in unconventional spin-singlet $s$-wave and $d$-wave superconductors. We show that inversion symmetry breaking and charge transfer due to the missing atoms lead to the occupation of incipient impurity bands and mixed parity spin singlet and triplet Cooper pairing of neighboring electrons traversing the line defect. Nontrivial topological invariants arise and occupy a large part of the parameter space, including the time reversal symmetry breaking Zeeman coupling due to applied magnetic field or defect-induced magnetism, creating TSCs in different topological classes with robust Majorana zero modes at both ends of the line defect. Beyond providing a novel mechanism for the recent discovery of zero-energy bound states at both ends of an atomic line defect in monolayer Fe(Te,Se) superconductors, the findings pave the way for new material realizations of the simplest and most robust 1D TSCs using embedded quantum structures in unconventional superconductors with large pairing energy gaps and high transition temperatures.

Nodal lines and mapping to mirror Chern numbers in Ca2As family

We study topological properties of materials in the Ca$_2$As family without spin-orbit coupling (SOC) by combining the first-principles calculation and a tight-binding model calculation. As a result of the calculation, we reveal that the Ca$2_$As family consists of one insulator phase and three nodal line phases including an intersecting nodal ring phase, though one of the phases is not found with realistic material parameters. Additionally, we discuss what kind of nontrivial topological invariants will emerge from each nodal line phase when SOC is introduced. We also find a mapping from a nodal line semimetal without SOC to a topological crystalline insulator with SOC. This mapping can be used to specify the realized topological phase from the candidates given by the previous phase classification method.

Nanopolaritonic second-order topological insulator based on graphene plasmons

Ultrastrong confinement, long lifetime, and gate-tunability of graphene plasmon polaritons (GPPs) motivate wide-ranging efforts to develop GPP-based active nanophotonic platforms. Incorporation of topological phenomena into such platforms will ensure their robustness as well as enrich their capabilities as promising test beds of strong light–matter interactions. A recently reported approach suggests an experimentally viable platform for topological graphene plasmonics by introducing nanopatterned gates—metagates. We propose a metagate-tuned GPP platform emulating a second-order topological crystalline insulator. The metagate imprints its crystalline symmetry onto graphene by modulating its chemical potential via field-effect gating. Depending on the gate geometry and applied voltage, the resulting two-dimensional crystal supports either one-dimensional chiral edge states or zero-dimensional midgap corner states. The proposed approach to achieving the hierarchy of nontrivial topological invariants at all dimensions lower than the dimension of the host material paves the way to utilizing higher-order topological effects for on-chip and ultracompact nanophotonic waveguides and cavities.

Four-dimensional topological lattices through connectivity

Thanks to recent advances, the 4D quantum Hall (QH) effect is becoming experimentally accessible in various engineered setups. In this paper, we propose a type of 4D topological system that, unlike other 2D and 4D QH models, does not require complicated (artificial) gauge fields and/or time-reversal symmetry breaking. Instead, we show that there are 4D QH systems that can be engineered for spinless particles by designing the lattice connectivity with real-valued hopping amplitudes, and we explain how this physics can be intuitively understood in analogy with the 2D Haldane model. We illustrate our discussion with a specific 4D lattice proposal, inspired by the widely-studied 2D honeycomb and brickwall lattice geometries. This also provides a minimal model for a topological system in class AI, which supports nontrivial topological band invariants only in four spatial dimensions or higher.

Discovery of topological Weyl fermion lines and drumhead surface states in a room temperature magnet.

Topological matter is known to exhibit unconventional surface states and anomalous transport owing to unusual bulk electronic topology. In this study, we use photoemission spectroscopy and quantum transport to elucidate the topology of the room temperature magnet Co2MnGa. We observe sharp bulk Weyl fermion line dispersions indicative of nontrivial topological invariants present in the magnetic phase. On the surface of the magnet, we observe electronic wave functions that take the form of drumheads, enabling us to directly visualize the crucial components of the bulk-boundary topological correspondence. By considering the Berry curvature field associated with the observed topological Weyl fermion lines, we quantitatively account for the giant anomalous Hall response observed in this magnet. Our experimental results suggest a rich interplay of strongly interacting electrons and topology in quantum matter.

Bulk–Boundary Correspondence for Sturmian Kohmoto-Like Models

We consider one dimensional tight binding models on $\ell^2(\mathbb Z)$ whose spatial structure is encoded by a Sturmian sequence $(\xi_n)_n\in \{a,b\}^\mathbb Z$. An example is the Kohmoto Hamiltonian, which is given by the discrete Laplacian plus an onsite potential $v_n$ taking value $0$ or $1$ according to whether $\xi_n$ is $a$ or $b$. The only non-trivial topological invariants of such a model are its gap-labels. The bulk-boundary correspondence we establish here states that there is a correspondence between the gap label and a winding number associated to the edge states, which arises if the system is augmented and compressed onto half space $\ell^2(\mathbb N)$. This has been experimentally observed with polaritonic waveguides. A correct theoretical explanation requires, however, first a smoothing out of the atomic motion via phason flips. With such an interpretation at hand, the winding number corresponds to the mechanical work through a cycle which the atomic motion exhibits on the edge states.

Robust two-dimensional topological insulators in derivatives of group-VA oxides with large band gap: Tunable quantum spin Hall states

Recently, group-VA 2D materials have attracted intense attention due to excellent properties in various aspects. Particularly, the strong spin–orbital coupling effect is a remarkable virtue of group-VA 2D materials, which is more widely concerned by researchers. Here, based on first-principles calculations, we provide a systematical investigation on the electronic and topological properties of a novel series 2D compound VA2OX (VA=Bi, Sb, As, P; X=S, Se, Te). Combining with the nontrivial Z2-type topological invariants, it is confirmed that VA2OX 2D crystals possess topologically nontrivial characteristics. The 2D TIs VA2OX demonstrate large and tunable nontrivial energy gaps, from 0 to 592meV, indicating that the tunable QSH effect can be realized at room temperature. Moreover, as an example of VA2OX systems, the band gap of Bi2OS 2D crystal almost remains unchanged when tensile strain not larger than 8%. Notably, under a large range of strain (±10%), Bi2OS keeps the inversion-asymmetric topological insulator phases, which emerges the robustness of nontrivial topology against mechanical deformation and makes it competent for realizing new topological phenomena. The above results are expected to promote further experimental investigation for fundamental exploration and practical application, which will significantly broaden the scientific and technological impact of the QSH effect.

Angle resolved photoemission spectroscopy studies on three dimensional strong topological insulators and magnetic topological insulators

<sec> The discovery of topological materials – condensed matter systems that have nontrivial topological invariants – marked the commencement of a new era in condensed matter physics and materials science. Three dimensional topological insulators (3D TIs) are one of the first discovered and the most studied among all topological materials. The bulk material of the TIs have the characteristics of the insulator, having a complete energy gap. Their surface electronic states, on the other hand, have the characteristics of a conductor, with energy band passes continuously through the Fermi surface. The conductivity of this topological surface state (TSS) is protected by the time reversal symmetry of the bulk material. The TSS is highly spin-polarized and form a special spin-helical configuration that allows electrons with specific spin to migrate only in a specific direction on the surface. By this means, surface electrons in TIs can " bypass” the influence of local impurities, achieving a lossless transmission of spin-polarized current. The existence of TIs directly leads to a variety of novel transport, magnetic, electrical, and optical phenomena, such as non-local quantum transport, quantum spin Hall effect, etc., promising wide application prospects. Recently, several research groups have searched all 230 non-magnetic crystal space groups, exhausting all the found or undiscovered strong/weak TIs, topological crystalline insulators (TCI), and topological semimetals. This series of work marks that theoretical understanding of non-magnetic topological materials has gone through a period of one-by-one prediction and verification, and entered the stage of the large-area material screening and optimization.</sec><sec> Parallel to non-magnetic TIs, magnetic topological materials constructed by ferromagnetic or antiferromagnetic long range orders in topological systems have always been an important direction attracting theoretical and experimental efforts. In magnetic TIs, the lack of time reversal symmetry brings about new physical phenomena. For example, when a ferromagnetic order is introduced into a three-dimensional TI, the Dirac TSS that originally intersected at one point will open a magnetic gap. When the Fermi surface is placed just in the gap, the quantum anomalous Hall effect can be implemented. At present, the research on magnetic topology systems is still in the ascendant. It is foreseeable that these systems will be the main focus and breakthrough point of topology material research in the next few years. </sec><sec> Angle-resolved photoemission spectroscopy (ARPES) is one of the most successful experimental methods of solid state physics. Its unique <i>k</i>-space-resolved single-electron detection capability and simple and easy-to-read data format make it a popular choice for both theoretists and experimentalists. In the field of topological materials, ARPES has always been an important experimetnal technique. It is able to directly observe the bulk and surface band structure of crystalline materials, and in a very intuitive way. With ARPES, it is incontrovertible to conclude whether a material is topological, and which type of topological material it belongs to.</sec><sec> This paper reviews the progress of ARPES research on TIs since 2008, focusing on the experimental energy band characteristics of each series of TIs and the general method of using ARPES to study this series of materials. Due to space limitations, this paper only discusses the research progress of ARPES for strong 3D TIs (focusing on the Bi<sub>2</sub>Se<sub>3</sub> series) and magnetic TIs (focusing on the MnBi<sub>2</sub>Te<sub>4</sub> series). Researches involving TCIs, topological Kondo insulators, weak 3D TIs, topological superconductors and heterostructures based on topological insulators will not be discussed. This paper assumes that the reader has the basic knowledge of ARPES, so the basic principles and system components of ARPES are not discussed.</sec>

Topological invariants for holographic semimetals

We study the behavior of fermion spectral functions for the holographic topological Weyl and nodal line semimetals. We calculate the topological invariants from the Green functions of both holographic semimetals using the topological Hamiltonian method, which calculates topological invariants of strongly interacting systems from an effective Hamiltonian system with the same topological structure. Nontrivial topological invariants for both systems have been obtained and the presence of nontrivial topological invariants further supports the topological nature of the holographic semimetals.

Topological edge states for disordered bosonic systems

Quadratic bosonic Hamiltonians over a one-particle Hilbert space can be described by a Bogoliubov-de Gennes (BdG) Hamiltonian on a particle-hole Hilbert space. In general, the BdG Hamiltonian is not self-adjoint, but only J-self-adjoint on the particle-hole space viewed as a Krein space. Nevertheless, its energy bands can have non-trivial topological invariants like Chern numbers or winding numbers. By a thorough analysis for tight-binding models, it is proved that these invariants lead to bosonic edge modes which are robust to a large class of possibly disordered perturbations. Furthermore, general scenarios are presented for these edge states to be dynamically unstable even though the bulk modes are stable.

Topological triplon modes and bound states in a Shastry–Sutherland magnet

The twin discoveries of the quantum Hall effect, in the 1980's, and of topological band insulators, in the 2000's, were landmarks in physics that enriched our view of the electronic properties of solids. In a nutshell, these discoveries have taught us that quantum mechanical wavefunctions in crystalline solids may carry nontrivial topological invariants which have ramifications for the observable physics. One of the side effects of the recent topological insulator revolution has been that such physics is much more widespread than was appreciated ten years ago. For example, while topological insulators were originally studied in the context of electron wavefunctions, recent work has led to proposals of topological insulators in bosonic systems: in photonic crystals, in the vibrational modes of crystals, and in the excitations of ordered magnets. Here we confirm the recent proposal that, in a weak magnetic field, the dimerized quantum magnet SrCu$_{2}$(BO$_{3}$)$_2$ is a bosonic topological insulator with nonzero Chern number in the triplon bands and topologically protected chiral edge excitations.