Weyl Orbit Research Articles

Quantum description of Fermi arcs in Weyl semimetals in a magnetic field

For a Weyl semimetal (WSM) in a magnetic field, a semiclassical description of the Fermi-arc surface state dynamics is usually employed for explaining various unconventional magnetotransport phenomena, e.g., Weyl orbits, the three-dimensional quantum Hall effect, and the high transmission through twisted WSM interfaces. For a half-space geometry, we determine the low-energy quantum eigenstates for a four-band model of a WSM in a magnetic field perpendicular to the surface. The eigenstates correspond to in- and out-going chiral Landau level (LL) states, propagating (anti)parallel to the field direction near different Weyl nodes, which are coupled by evanescent surface-state contributions generated by all other LLs. These replace the Fermi arc in a magnetic field. Computing the phase shift accumulated between in- and out-going chiral LL states, we compare our quantum-mechanical results to semiclassical predictions. We find quantitative agreement between both approaches. Published by the American Physical Society 2024

Photon-assisted extrinsic Weyl orbits and three-dimensional quantum Hall effect in surface-doped topological Dirac semimetals

Photon-assisted extrinsic Weyl orbits and three-dimensional quantum Hall effect in surface-doped topological Dirac semimetals

Weyl orbits as probe of chiral separation effect in magnetic Weyl semimetals

We consider magnetic Weyl semimetals. First of all we review relation of intrinsic anomalous Hall conductivity, band contribution to intrinsic magnetic moment, and the conductivity of chiral separation effect (CSE) to the topological invariants written in terms of the Wigner transformed Green functions (with effects of interaction and disorder taken into account). Next, we concentrate on the CSE. The corresponding bulk axial current is accompanied by the flow of the states in momentum space along the Fermi arcs. Together with the bulk CSE current this flow forms closed Weyl orbits. Their detection can be considered as experimental discovery of chiral separation effect. Previously it was proposed to detect Weyl orbits through the observation of quantum oscillations (Potter et al 2014 Nat. Commun. 5 5161). We propose the alternative way to detect existence of Weyl orbits through the observation of their contributions to Hall conductance.

Chiral anomaly and Weyl orbit in three-dimensional Dirac semimetal Cd3As2 grown on Si

Preparing Cd3As2, which is a three-dimensional (3D) Dirac semimetal in certain crystal orientation, on Si is highly desirable as such a sample may well be fully compatible with existing Si CMOS technology. However, there is a dearth of such a study regarding Cd3As2 films grown on Si showing the chiral anomaly. Here, for the first time, we report the novel preparation and fabrication technique of a Cd3As2 (112) film on a Si (111) substrate with a ZnTe (111) buffer layer which explicitly shows the chiral anomaly with a nontrivial Berry’s phase of π. Despite the Hall carrier density ( n3D≈ 9.42×1017 cm−3) of our Cd3As2 film, which is almost beyond the limit for the portents of a 3D Dirac semimetal to emerge, we observe large linear magnetoresistance in a perpendicular magnetic field and negative magnetoresistance in a parallel magnetic field. These results clearly demonstrate the chiral magnetic effect and 3D Dirac semimetallic behavior in our silicon-based Cd3As2 film. Our tailoring growth of Cd3As2 on a conventional substrate such as Si keeps the sample quality, while also achieving a low carrier concentration.

Thin film Weyl semimetals with turning number of Fermi surface greater than unity

The development of Weyl semimetals (WSMs) as a new type of quantum material for a variety of novel phenomena is underway. In this context WSM with broken time reversal symmetry and inversion symmetry confined to a thin film which exhibits topologically distinct Fermi surface in the low energy regime is presented. The equi-energy contours possess a topological invariant called turning number ν. We extend the numerical investigation of the magnetotransport in the WSM to include systems with the turning number of Fermi surface greater than unity, ν=2, which is due to the self intersection of equi-energy contours. This results in exotic Weyl orbits which are a hybrid of bulk and surface states and are crucial for magnetotransport. This is in contrast to the conventional electron gas in which orbits are closed curves without self intersection. The contour of Weyl orbits in real space is connected to the shape of Fermi surface. In the presence of a perpendicular magnetic field on a slab geometry, the probability density of edge states are wide and narrow on each edge rather than symmetric on both edges (the hall mark of conventional quantum Hall effect). We explore the effect of disorder on these asymmetric edge states and find that the conductance remains quantized at 2e2/h up to a critical disorder strength. Our findings show that the transport characteristics of WSM are more robust to disorder in a particular direction for these systems with higher turning number. These systems also show unusual conductance by a slanted magnetic field as the slant can modify the Landau levels at the Fermi level.

Anisotropic resistance with a 90° twist in a ferromagnetic Weyl semimetal, Co2MnGa

Weyl semimetals exhibit exotic magnetotransport phenomena such as the chiral anomaly and surface-to-bulk quantum oscillations (Weyl orbits) due to chiral bulk states and topologically protected surface states. Here we report a unique transport property in crystals of the ferromagnetic nodal-line Weyl semimetal Co2MnGa that have been polished to micron thicknesses using a focused ion beam. These thin crystals exhibit a large planar resistance anisotropy (10 × ) with axes that rotate by 90 degrees between opposite faces of the crystal. We use symmetry arguments and electrostatic simulations to show that the observed anisotropy resembles that of an isotropic conductor with surface states that are impeded from hybridization with bulk states. The origin of these states awaits further experiments that can correlate the surface bands with the observed 90° twist.

Two types of three-dimensional quantum Hall effects in multilayer WTe2

Interplay between topological surface states and bulk states gives rise to diverse exotic transport phenomena in topological materials. The recently proposed Weyl orbit in topological semimetals in the presence of magnetic field is a remarkable example. This novel closed magnetic orbit consists of Fermi arcs on two spatially separated sample surfaces which are connected by the bulk chiral zero mode, which can contribute to transport. Here we report Shubnikov--de Haas oscillation and its evolution into quantum Hall effect (QHE) in multilayered type-II Weyl semimetal $\mathrm{W}{\mathrm{Te}}_{2}$. We observe both the three-dimensional (3D) QHE from bulk states by parallel stacking of confined two-dimensional layers in the low magnetic field region and the 3D QHE in the quantized surface transport due to Weyl orbits in the high magnetic field region. Our study of the two types of QHEs controlled by magnetic field and our demonstration of the crossover between quantized bulk and surface transport provide an essential platform for the future quantized transport studies in topological semimetals.

Quantum Particle on Dual Weight Lattice in Even Weyl Alcove

Even subgroups of affine Weyl groups corresponding to irreducible crystallographic root systems characterize families of single-particle quantum systems. Induced by primary and secondary sign homomorphisms of the Weyl groups, free propagations of the quantum particle on the refined dual weight lattices inside the rescaled even Weyl alcoves are determined by Hamiltonians of tight-binding types. Described by even hopping functions, amplitudes of the particle’s jumps to the lattice neighbours are together with diverse boundary conditions incorporated through even hopping operators into the resulting even dual-weight Hamiltonians. Expressing the eigenenergies via weighted sums of the even Weyl orbit functions, the associated time-independent Schrödinger equations are exactly solved by applying the discrete even Fourier–Weyl transforms. Matrices of the even Hamiltonians together with specifications of the complementary boundary conditions are detailed for the C2 and G2 even dual-weight models.

Weyl orbits of matrix morsifications and a Coxeter spectral classification of positive signed graphs and quasi-Cartan matrices of Dynkin type [formula omitted

By applying an advanced technique of Weyl orbits of matrix morsifications of Dynkin graphs we obtain a complete classification (up to the strong Gram Z-congruence) of the connected positive signed graphs Δ, with n≥2 vertices, of Dynkin type An (see Problem 2.1(a)), by a reduction of the problem to the combinatorial properties of the Weyl group of type An and to the classification of the conjugacy classes of the integer orthogonal matrices C in Mn+1(Z) such that the characteristic polynomial of C is of the form charC(t)=tn+1−1. Following the usual Coxeter spectral analysis technique of edge-bipartite graphs applied in Simson (2018) [31], we study such positive signed graphs Δ, with n≥2 vertices, by means of the non-symmetric Gram matrix GˇΔ∈Mn(Z) defining Δ, its Gram quadratic form qΔ:Zn→Z, v↦v⋅GˇΔ⋅vtr, the complex spectrum speccΔ⊂S1:={z∈C,|z|=1} of the Coxeter matrix CoxΔ:=−GˇΔ⋅GˇΔ−tr∈Mn(Z), called the Coxeter spectrum of Δ, and the Coxeter polynomial coxΔ(t):=det⁡(t⋅E−CoxΔ)∈Z[t]. We define Δ to be positive if the quadratic form qΔ:Zn→Z is positive definite, or equivalently the symmetric Gram matrix GΔ:=12[GˇΔ+GˇΔtr]=E+12AdΔ∈Mn(12Z) of Δ is positive definite, where AdΔ∈Mn(Z) is the adjacency matrix of Δ.Here we classify such positive signed graphs, up to the strong Gram Z-congruence Δ≈ZΔ′, where Δ≈ZΔ′ means that GˇΔ′=Btr⋅GˇΔ⋅B, for some B∈Mn(Z) with det⁡B=±1. The main result of the paper asserts that, given such a connected positive signed graph Δ of Dynkin type An (equivalently, det⁡2GΔ=n+1), we have(i)the Coxeter polynomial coxΔ(t) has the form tn+tn−1+⋯+t+1 (i.e., it coincides with the Coxeter polynomial coxAn(t) of the Dynkin graph An), and(ii)there is a strong Gram Z-congruence Δ≈ZAn.As a consequence we obtain the following solution of the Coxeter spectral hypothesis stated in [26] and [32]:(a)Given a pair Δ,Δ′ of connected positive signed graphs of Dynkin type An, with n≥2 vertices, there is a congruence Δ≈ZΔ′ if and only if speccΔ=speccΔ′, and(b)Given a positive definite connected quasi-Cartan matrix C∈Mn(Z) (in the sense of [4]) of Dynkin type An (equivalently, det⁡C=n+1), the Coxeter polynomial coxC(t)∈Z[t] of C coincides with the Coxeter polynomial coxAn(t)=tn+tn−1+⋯+t+1 of the Dynkin graph An and C is strongly Z-congruent with the canonical symmetrizable Cartan matrix An of the root system of type An.

Anisotropic Three-Dimensional Quantum Hall Effect and Magnetotransport in Mesoscopic Weyl Semimetals.

Weyl semimetals are emerging to become a new class of quantum-material platform for various novel phenomena. Especially, the Weyl orbit made from surface Fermi arcs and bulk relativistic states is expected to play a key role in magnetotransport, leading even to a three-dimensional quantum Hall effect (QHE). It is experimentally and theoretically important although yet unclear whether it bears essentially the same phenomenon as the conventional two-dimensional QHE. We discover an unconventional fully three-dimensional anisotropy in the quantum transport under a magnetic field. Strong suppression and even disappearance of the QHE occur when the Hall-bar current is rotated away from being transverse to parallel with respect to the Weyl point alignment, which is attributed to a peculiar absence of conventional bulk-boundary correspondence. Besides, transport along the magnetic field can exhibit a remarkable reversal from negative to positive magnetoresistance. These results establish the uniqueness of this QHE system as a novel three-dimensional quantum matter.

On electron propagation in triangular graphene quantum dots

Tight-binding models of electron propagation in single-layer triangular graphene quantum dots with armchair and zigzag edges are developed. The electron hoppings to the nearest and next-to-nearest neighbours on the honeycomb lattice as well as interactions with the confining Dirichlet and Neumann walls are incorporated into the resulting tight-binding Hamiltonians. Associated to the irreducible crystallographic root system A 2, the armchair and zigzag honeycomb Weyl orbit functions together with the related discrete Fourier–Weyl transforms provide explicit exact forms of the electron wave functions and energy spectra. The electronic probability densities corresponding to the armchair and zigzag dots are evaluated and their contrasting behaviour exemplified.

Anisotropic Shubnikov-de Haas effect in topological Weyl semimetal MoTe2

Newly emergent type-II Weyl semimetals with topological surface states so-called Fermi arcs have attracted much attention for their novel physical properties and potential application in quantum devices. Here, we investigate the in-plane anisotropic structure and inversion symmetry breaking by angle-resolved polarized Raman and second harmonic generation and observe the anisotropic Shubnikov-de Haas effect in Weyl Semimetal MoTe2, which is only present in the b-axis (armchair chain) direction. First-principles calculation depicts the type-II Weyl points and clear topological Fermi arcs. A nontrivial π Berry's phase from Landau quantization and an extra-quantum oscillation frequency arising by Weyl orbit are obtained, which provide evidence for the existence of an anisotropic type-II Weyl state in MoTe2. This work reveals the nontrivial topological surface state of Weyl semimetal MoTe2 in both theory and experiment, providing a promising platform for unique physical properties and applications in quantum information processing.

Three-dimensional topological plasmons in Weyl semimetals

We systematically investigate the properties of bulk, surface and edge plasmons in Weyl semimetals in presence of a magnetic field. It is found that unidirectional plasmons with different properties exist on different surfaces, which is in consistent with the nontrivial topology of the three-dimensional (3D) bulk plasmons. These novel plasmons possess momentum-location lock and may travel between surfaces. The anomalous Hall conductivity brings about abundant anisotropic plasmon dispersions, from linear to parabolic or even hyperbolic bands. With the help of a semiclassical picture for the formation of the Weyl orbits, we point out the Fermi-arc plasmons at opposite surfaces can make up another unique 3D topological plasmon. Furthermore, there is a gapless unidirectional edge plasmon protected by the topology whose direction and position can be controlled by external field. Our work thus uncovers topological features of Weyl plasmons, which may have important applications in photoelectric devices based on chiral/topological plasmonics.

Conductance oscillation in surface junctions of Weyl semimetals

Fermi arc surface states, the manifestation of the bulk-edge correspondence in Weyl semimetals, have attracted much research interest. In contrast to the conventional Fermi loop, the disconnected Fermi arcs provide an exotic 2D system for exploration of novel physical effects on the surface of Weyl semimetals. Here, we propose that visible conductance oscillation can be achieved in the planar junctions fabricated on the surface of Weyl semimetal with a pair of Fermi arcs. It is shown that Fabry-P\'{e}rot-type interference inside the 2D junction can generate conductance oscillation with its visibility strongly relying on the shape of the Fermi arcs and their orientation relative to the strip electrodes, the latter clearly revealing the anisotropy of the Fermi arcs. Moreover, we show that the visibility of the oscillating pattern can be significantly enhanced by a magnetic field perpendicular to the surface taking advantage of the bulk-surface connected Weyl orbits. Our work offers an effective way for the identification of Fermi arc surface states through transport measurement and predicts the surface of Weyl semimetal as a novel platform for the implementation of 2D conductance oscillation.

Elliptic blowup equations for 6d SCFTs. Part IV. Matters

Given the recent geometrical classification of 6d (1, 0) SCFTs, a major question is how to compute for this large class their elliptic genera. The latter encode the refined BPS spectrum of the SCFTs, which determines geometric invariants of the associated elliptic non-compact Calabi-Yau threefolds. In this paper we establish for all 6d (1, 0) SCFTs in the atomic classification blowup equations that fix these elliptic genera to large extent. The latter fall into two types: the unity and the vanishing blowup equations. For almost all rank one theories, we find unity blowup equations which determine the elliptic genera completely. We develop several techniques to compute elliptic genera and BPS invariants from the blowup equations, including a recursion formula with respect to the number of strings, a Weyl orbit expansion, a refined BPS expansion and an ϵ1, ϵ2 expansion. For higher-rank theories, we propose a gluing rule to obtain all their blowup equations based on those of rank one theories. For example, we explicitly give the elliptic blowup equations for the three higher-rank non-Higgsable clusters, ADE chain of −2 curves and conformal matter theories. We also give the toric construction for many elliptic non-compact Calabi- Yau threefolds which engineer 6d (1, 0) SCFTs with various matter representations.

Author Correction: Cycling Fermi arc electrons with Weyl orbits

Author Correction: Cycling Fermi arc electrons with Weyl orbits

Discrete cosine and sine transforms generalized to honeycomb lattice II. Zigzag boundaries

The discrete cosine and sine transforms are generalized to a triangular fragment of the honeycomb lattice with zigzag boundaries. The zigzag honeycomb point sets are constructed by subtracting the weight lattice from the refined root lattice points of the crystallographic root system A2. The two-variable (anti)symmetric orbit functions of the Weyl group of A2, discretized simultaneously on the triangular fragments of the root and weight lattices, induce a novel parametric family of zigzag extended Weyl and Hartley orbit functions. As specific linear combinations of the original orbit functions, the zigzag extended orbit functions retain the Neumann and Dirichlet boundary conditions. Three types of discrete complex Fourier–Weyl transforms and real-valued Hartley–Weyl transforms are detailed. The corresponding unitary transform matrices and interpolating behavior of the discrete transforms are exemplified.

Cycling Fermi arc electrons with Weyl orbits

Cycling Fermi arc electrons with Weyl orbits

Intrinsic coupling between spatially-separated surface Fermi-arcs in Weyl orbit quantum Hall states

Topological semimetals hosting bulk Weyl points and surface Fermi-arc states are expected to realize unconventional Weyl orbits, which interconnect two surface Fermi-arc states on opposite sample surfaces under magnetic fields. While the presence of Weyl orbits has been proposed to play a vital role in recent observations of the quantum Hall effect even in three-dimensional topological semimetals, actual spatial distribution of the quantized surface transport has been experimentally elusive. Here, we demonstrate intrinsic coupling between two spatially-separated surface states in the Weyl orbits by measuring a dual-gate device of a Dirac semimetal film. Independent scans of top- and back-gate voltages reveal concomitant modulation of doubly-degenerate quantum Hall states, which is not possible in conventional surface orbits as in topological insulators. Our results evidencing the unique spatial distribution of Weyl orbits provide new opportunities for controlling the novel quantized transport by various means such as external fields and interface engineering.

Central Splitting of A2 Discrete Fourier–Weyl Transforms

Two types of bivariate discrete weight lattice Fourier–Weyl transforms are related by the central splitting decomposition. The two-variable symmetric and antisymmetric Weyl orbit functions of the crystallographic reflection group A2 constitute the kernels of the considered transforms. The central splitting of any function carrying the data into a sum of components governed by the number of elements of the center of A2 is employed to reduce the original weight lattice Fourier–Weyl transform into the corresponding weight lattice splitting transforms. The weight lattice elements intersecting with one-third of the fundamental region of the affine Weyl group determine the point set of the splitting transforms. The unitary matrix decompositions of the normalized weight lattice Fourier–Weyl transforms are presented. The interpolating behavior and the unitary transform matrices of the weight lattice splitting Fourier–Weyl transforms are exemplified.