Three-dimensional Weyl Semimetals Research Articles

Magnetic control of Weyl nodes and wave packets in three-dimensional warped semimetals

We investigate the topological phase transitions driven by band warping, λ, and a transverse magnetic field, B, for three-dimensional Weyl semimetals. First, we use the Chern number as a mathematical tool to derive the topological λ×B phase diagram. Next, we associate each of the topological sectors to a given angular momentum state of a rotating wave packet. Then we show how the position of the Weyl nodes can be manipulated by a transverse external magnetic field that ultimately quenches the wave packet rotation, first partially and then completely, thus resulting in a sequence of field-induced topological phase transitions. Finally, we calculate the current-induced magnetization and the anomalous Hall conductivity of a prototypical warped Weyl material. Both observables reflect the topological transitions associated with the wave packet rotation and can help to identify the elusive 3D quantum anomalous Hall effect in three-dimensional, warped Weyl materials. Published by the American Physical Society 2024

Evidence for Two-Dimensional Weyl Fermions in Air-Stable Monolayer PtTe1.75.

The Weyl semimetals represent a distinct category of topological materials wherein the low-energy excitations appear as the long-sought Weyl Fermions. Exotic transport and optical properties are expected because of the chiral anomaly and linear energy-momentum dispersion. While three-dimensional Weyl semimetals have been successfully realized, the quest for their two-dimensional (2D) counterparts is ongoing. Here, we report the realization of 2D Weyl Fermions in monolayer PtTe1.75, which has strong spin-orbit coupling and lacks inversion symmetry, by combined angle-resolved photoemission spectroscopy, scanning tunneling microscopy, second harmonic generation, X-ray photoelectron spectroscopy measurements, and first-principles calculations. The giant Rashba splitting and band inversion lead to the emergence of three pairs of critical Weyl cones. Moreover, monolayer PtTe1.75 exhibits excellent chemical stability in ambient conditions, which is critical for future device applications. The discovery of 2D Weyl Fermions in monolayer PtTe1.75 opens up new possibilities for designing and fabricating novel spintronic devices.

Connecting the avoided quantum critical point to the magic-angle transition in three-dimensional Weyl semimetals

Connecting the avoided quantum critical point to the magic-angle transition in three-dimensional Weyl semimetals

Higher rank chiral fermions in three-dimensional Weyl semimetals

Higher rank chiral fermions in three-dimensional Weyl semimetals

Distinct quasiparticle interference patterns for surface impurity scattering on various Weyl semimetals

We examine the response of the Fermi arc in the context of quasiparticle interference (QPI) with regard to a localized surface impurity on various three-dimensional Weyl semimetals (WSMs). Our study also reveals the variation of the local density of states (LDOS), obtained by Fourier transforming the QPI profile, on the two-dimensional surface. We use the T-matrix formalism to numerically (analytically and numerically) capture the details of the momentum space scattering in QPI (real-space decay in LDOS), considering relevant tight-binding lattice and/or low-energy continuum models modeling a range of different WSMs. In particular, we consider multi-WSM (mWSM), hosting multiple Fermi arcs between two opposite chirality Weyl nodes (WNs), where we find a universal 1/r decay (r measuring the radial distance from the impurity core) of the impurity-induced LDOS, irrespective of the topological charge. Interestingly, the inter-Fermi arc scattering is only present for triple WSMs, where we find an additional 1/r3 decay as compared to double and single WSMs. The untilted single (double) [triple] WSM shows a straight line (leaf-like) [oval-shaped] QPI profile. The above QPI profiles are canted for hybrid WSMs where type-I and type-II Weyl nodes coexist; however, hybrid single WSM demonstrates strong nonuniformity, unlike the hybrid double and triple WSMs. We also show that the chirality and the positions of the Weyl nodes imprint marked signatures in the QPI profile. This allows us to distinguish between different WSMs, including the time-reversal-broken WSMs from the time-reversal-invariant WSM, even though both of the WSMs can host two pairs of Weyl nodes. Our study can thus shed light on experimentally obtainable complex QPI profiles and help differentiate different WSMs and their surface band structures. Published by the American Physical Society 2024

Nodal vacancy bound states and resonances in three-dimensional Weyl semimetals

The electronic structure of a cubic $\mathcal{T}$-symmetric Weyl semimetal is analysed in the presence of atomic-sized vacancy defects. Isolated vacancies are shown to generate nodal bound states with $r^{{\scriptscriptstyle -2}}$ asymptotic tails, even when immersed in a weakly disordered environment. These states show up as a significantly enhanced nodal density of states which, as the concentration of defects is increased, reshapes into a nodal peak that is broadened by inter-vacancy hybridisation into a comb of satellite resonances at finite energies. Our results establish point defects as a crucial source of elastic scattering that leads to nontrivial modifications in the electronic structure of Weyl semimetals.

Topological electronic bands in crystalline solids

Topology is now securely established as a means to explore and classify electronic states in crystalline solids. This review provides a gentle but firm introduction to topological electronic band structure suitable for new researchers in the field. I begin by outlining the relevant concepts from topology, then give a summary of the theory of non-interacting electrons in periodic potentials. Next, I explain the concepts of the Berry phase and Berry curvature, and derive key formulae. The remainder of the article deals with how these ideas are applied to classify crystalline solids according to the topology of the electronic states, and the implications for observable properties. Among the topics covered are the role of symmetry in determining band degeneracies in momentum space, the Chern number and topological invariants, surface electronic states, two- and three-dimensional topological insulators, and Weyl and Dirac semimetals

Magneto-transport signatures in periodically-driven Weyl and multi-Weyl semimetals

We investigate the influence of a time-periodic driving (for example, by shining circularly polarized light) on three-dimensional Weyl and multi-Weyl semimetals, in the planar Hall and planar thermal Hall set-ups. We incorporate the effects of the drive by using the Floquet formalism in the large frequency limit. We evaluate the longitudinal magneto-conductivity, planar Hall conductivity, longitudinal thermo-electric coefficient, and transverse thermo-electric coefficient, using the semi-classical Boltzmann transport equations. We demonstrate the explicit expressions of these transport coefficients in certain limits of the system parameters, where it is possible to perform the integrals analytically. We cross-check our analytical approximations by comparing the physical values with the numerical results, obtained directly from the numerical integration of the integrals. The answers obtained show that the topological charges of the corresponding semimetals have profound signatures in these transport properties, which can be observed in experiments.

Projected topological branes

Nature harbors crystals of dimensionality (d) only up to three. Here we introduce the notion of projected topological branes (PTBs): Lower-dimensional branes embedded in higher-dimensional parent topological crystals, constructed via a geometric cut-and-project procedure on the Hilbert space of the parent lattice Hamiltonian. When such a brane is inclined at a rational or an irrational slope, either a new lattice periodicity or a quasicrystal emerges. The latter gives birth to topoquasicrystals within the landscape of PTBs. As such PTBs are shown to inherit the hallmarks, such as the bulk-boundary and bulk-dislocation correspondences, and topological invariant, of the parent topological crystals. We exemplify these outcomes by focusing on two-dimensional parent Chern insulators, leaving its signatures on projected one-dimensional (1D) topological branes in terms of localized endpoint modes, dislocation modes and the local Chern number. Finally, by stacking 1D projected Chern insulators, we showcase the imprints of three-dimensional Weyl semimetals in d = 2, namely the Fermi arc surface states and bulk chiral zeroth Landau level, responsible for the chiral anomaly. Altogether, the proposed PTBs open a realistic avenue to harness higher-dimensional (d > 3) topological phases in laboratory.

High-order harmonic generation in three-dimensional Weyl semimetals with broken time-reversal symmetry

In this paper the nonlinear interaction of a Weyl semimetal (WSM) with a strong driving electromagnetic wave field is investigated. In the scope of the structure-gauge-invariant low-energy nonlinear electrodynamic theory, the polarization-resolved high-order harmonic generation spectra in the WSM with broken time-reversal symmetry are analyzed. The results obtained show that the spectra in the WSM are completely different compared to the two-dimensional graphene case. In particular, at the noncollinear arrangement of the electric and Weyl node momentum separation vectors, anomalous harmonics are generated which are polarized perpendicular to the pump wave electric field. The intensities of anomalous harmonics are quadratically dependent on the momentum space separation of the Weyl nodes. If the right and the left Weyl fermions are merged, we have a four-component trivial massless Dirac fermion and, as a consequence, the anomalous harmonics vanish. In contrast to the anomalous harmonics, the intensities of normal harmonics do not depend on the Weyl nodes' momentum separation vector and the harmonics spectra resemble the picture for a massless three-dimensional Dirac fermion.

Magneto-Seebeck coefficient of the Fermi liquid in three-dimensional Dirac and Weyl semimetals

We investigate dissipationless magneto-Seebeck effect in three-dimensional Dirac/Weyl semimetal. The Hall resistivity $\rho_{yx}$ and thermoelectric Hall coefficient $\alpha_{xy}$ exhibit plateaus at the quantum limit, where electrons occupy only the zeroth Landau level. In this condition, quantum oscillation in the Seebeck coefficient $S_{xx}\approx \rho_{yx}\alpha_{xy} $ is suppressed, and the massless fermions are transformed into a Fermi liquid system. We show that the Seebeck coefficient at the quantum limit is expressed by the harmonic sum of Fermi wavelength and thermal de Broglie wavelength scaled by magnetic length.

Inherited topological superconductivity in two-dimensional Dirac semimetals

Under what conditions does a superconductor inherit topologically protected nodes from its parent normal state? In the context of inter-Fermi-surface pairing in three-dimensional Weyl semimetals with broken time-reversal symmetry, the pairing order parameter is classified by monopole harmonics and is necessarily nodal [Li and Haldane, Phys. Rev. Lett. 120, 067003 (2018)]. Here, we show that a similar conclusion could also be drawn for 2D Dirac semimetals, although the conditions for the existence of nodes are more complex, depending on the pairing matrix structure in the valley and sublattice space. We analytically and numerically analyze the Bogoliubov-de Gennes quasiparticle spectra for Dirac systems based on the monolayer as well as twisted bilayer graphene. We find that in the cases of intravalley intrasublattice and intervalley intersublattice pairings, the point nodes in the BdG spectra (which are inherited from the Dirac cone in the normal state) are protected by a 1D winding number. The nodal structure of the superconductivity is confirmed numerically using tight-binding models of monolayer and twisted bilayer graphene. Notably, the BdG spectrum is nodal even with a momentum-independent ``bare'' pairing, which, however, acquires a momentum dependence and point nodes upon projection to the Bloch states on the topologically nontrivial Fermi surface, similar in spirit to the Li-Haldane monopole superconductor and the Fu-Kane proximity-induced superconductor on the surface of a topological insulator.

Thermal-field electron emission from three-dimensional Dirac and Weyl semimetals

A model is constructed to describe the thermal-field emission of electrons from a three-dimensional (3D) Dirac and Weyl semimetal hosting Dirac/Weyl node(s). The traditional thermal-field electron emission model is generalized to accommodate the 3D nonparabolic energy band structures in the Dirac/Weyl semimetals, such as cadmium arsenide $({\mathrm{Cd}}_{3}{\mathrm{As}}_{2})$, sodium bismuthide $({\mathrm{Na}}_{3}\mathrm{Bi})$, tantalum arsenide (TaAs), and tantalum phosphide (TaP). Due to the nontrivial energy decomposition of the energy dispersion and the vanishing transverse density of states, an unusual dual-peak feature is observed in the total energy distribution spectrum. This nontrivial dual-peak feature, absent from traditional materials, plays a critical role in manipulating the magnitude of the emission current through the variation of an applied field, temperature, and Fermi level. This feature suggests that a higher Fermi level will achieve a larger current density (apart from low work function). At zero temperature limit, a ${F}^{3}$ scaling law for pure field emission is derived and it is different from the well-known Fowler-Nordheim ${F}^{2}$ scaling law. Furthermore, these new behaviors have shown to exist beyond the Dirac cone approximated model. This model expands the recent understandings of electron emission studied for the Dirac two-dimensional (2D) materials into the 3D regime, and thus offers a theoretical foundation for the exploration in using Dirac semimetallic materials as novel electrodes.

Non-Hermitian Weyl semimetals: Non-Hermitian skin effect and non-Bloch bulk–boundary correspondence

We investigate novel features of three-dimensional non-Hermitian Weyl semimetals, paying special attention to the unconventional bulk–boundary correspondence. We use the non-Bloch Chern numbers as the tool to obtain the topological phase diagram, which is also confirmed by the energy spectra from our numerical results. It is shown that, in sharp contrast to Hermitian systems, the conventional (Bloch) bulk–boundary correspondence breaks down in non-Hermitian topological semimetals, which is caused by the non-Hermitian skin effect. We establish the non-Bloch bulk–boundary correspondence for non-Hermitian Weyl semimetals: the topological edge modes are determined by the non-Bloch Chern number of the bulk bands. Moreover, these topological edge modes can manifest as the unidirectional edge motion, and their signatures are consistent with the non-Bloch bulk–boundary correspondence. Our work establishes the non-Bloch bulk–boundary correspondence for non-Hermitian topological semimetals.

Band topology of pseudo-Hermitian phases through tensor Berry connections and quantum metric

Among non-Hermitian systems, pseudo-Hermitian phases represent a special class of physical models characterized by real energy spectra and the absence of non-Hermitian skin effects. Here we show that several pseudo-Hermitian phases in two and three dimensions can be built by employing $q$-deformed matrices, which are related to the representation of deformed algebras. Through this algebraic approach, we present and study the pseudo-Hermitian version of well-known Hermitian topological phases, ranging from two-dimensional Chern insulators and time-reversal-invariant topological insulators to three-dimensional Weyl semimetals and chiral topological insulators. We analyze their topological bulk states through non-Hermitian generalizations of Abelian and non-Abelian tensor Berry connections and quantum metrics. Although our pseudo-Hermitian models and their Hermitian counterparts share the same topological invariants, their band geometries are different. We indeed show that some of our pseudo-Hermitian phases naturally support nearly flat topological bands, opening the route to the study of pseudo-Hermitian strongly interacting systems. Finally, we provide an experimental protocol to realize our models and measure the full non-Hermitian quantum geometric tensor in synthetic matter.

Fermi arc reconstruction at the interface of twisted Weyl semimetals

Three-dimensional Weyl semimetals have pairs of topologically protected Weyl nodes, whose projections onto the surface Brillouin zone are the end points of zero energy surface states called Fermi arcs. At the endpoints of the Fermi arcs, surface states extend into and are hybridized with the bulk. Here, we consider a two-dimensional junction of two identical Weyl semimetals whose surfaces are twisted with respect to each other and tunnel-coupled. Confining ourselves to commensurate angles (such that a larger unit cell preserves a reduced translation symmetry at the interface) enables us to analyze arbitrary strengths of the tunnel-coupling. We study the evolution of the Fermi arcs at the interface, in detail, as a function of the twisting angle and the strength of the tunnel-coupling. We show unambiguously that in certain parameter regimes, all surface states decay exponentially into the bulk, and the Fermi arcs become Fermi loops without endpoints. We study the evolution of the `Fermi surfaces' of these surface states as the tunnel-coupling strengths vary. We show that changes in the connectivity of the Fermi arcs/loops have interesting signatures in the optical conductivity in the presence of a magnetic field perpendicular to the surface.

Three-dimensional quantum Hall effect in the excitonic phase of a Weyl semimetal

The quantum Hall effect (QHE) is usually observed in two-dimensional systems. Recently, the QHE has been proposed to exist in three-dimensional (3D) Weyl semimetals when the Fermi energy is precisely at the Weyl nodes at zero temperature, which is hardly achievable experimentally. However, the QHE has been experimentally realized in the 3D topological semimetal ${\text{Cd}}_{3}{\text{As}}_{2}$ at finite temperatures. It has been found that a Weyl semimetal may translate to a chiral excitonic insulator due to the interplay between the interaction and band topology. In a Weyl semimetal slab, we show that the Fermi arc surface states can exist not only in the Weyl semimetal but also in the chiral excitonic insulator. As long as the Fermi energy crosses the Fermi arc states, the QHE can be generated under the combined action of bulk states and Fermi arc surface states. Therefore, the integer quantization of Hall conductivity can be observed for a finite energy range, or equivalently at finite temperatures.

Topolectric circuits: Theory and construction

We highlight a general theory to engineer arbitrary Hermitian tight-binding lattice models in electrical LC circuits, where the lattice sites are replaced by the electrical nodes, connected to its neighbors and to the ground by capacitors and inductors. In particular, by supplementing each node with $n$ subnodes, where the phases of the current and voltage are the $n$ distinct roots of \emph{unity}, one can in principle realize arbitrary hopping amplitude between the sites or nodes via the \emph{shift capacitor coupling} between them. This general principle is then implemented to construct a plethora of topological models in electrical circuits, \emph{topolectric circuits}, where the robust zero-energy topological boundary modes manifest through a large boundary impedance, when the circuit is tuned to the resonance frequency. The simplicity of our circuit constructions is based on the fact that the existence of the boundary modes relies only on the Clifford algebra of the corresponding Hermitian matrices entering the Hamiltonian and not on their particular representation. This in turn enables us to implement a wide class of topological models through rather simple topolectric circuits with nodes consisting of only two subnodes. We anchor these outcomes from the numerical computation of the on-resonance impedance in circuit realizations of first-order ($m=1$), such as Chern and quantum spin Hall insulators, and second- ($m=2$) and third- ($m=3$) order topological insulators in different dimensions, featuring sharp localization on boundaries of codimensionality $d_c=m$. Finally, we subscribe to the \emph{stacked topolectric circuit} construction to engineer three-dimensional Weyl, nodal-loop, quadrupolar Dirac and Weyl semimetals, respectively displaying surface and hinge localized impedance.

Ultrafast investigation and control of Dirac and Weyl semimetals

Ultrafast experiments using sub-picosecond pulses of light are poised to play an important role in the study and use of topological materials and, particularly, of the three-dimensional Dirac and Weyl semimetals. Many of these materials’ characteristic properties—their linear band dispersion, Berry curvature, near-vanishing density of states at the Fermi energy, and sensitivity to crystalline and time-reversal symmetries—are closely related to their sub- and few-picosecond response to light. Ultrafast measurements offer the opportunity to explore excitonic instabilities and transient photocurrents, the latter depending on the Berry curvature and possibly quantized by fundamental constants. Optical pulses may, through Floquet effects, controllably and reversibly move, split, merge, or gap the materials’ Dirac and Weyl nodes; coherent phonons launched by an ultrafast pulse offer alternate mechanisms for similar control of the nodal structure. This Perspective will briefly summarize the state of research on the ultrafast properties of Dirac and Weyl semimetals, emphasizing important open questions. It will describe the challenges confronting each of these experimental opportunities and suggest what research is needed for ultrafast pulses to achieve their potential of controlling and illuminating the physics of Dirac and Weyl semimetals.

Topological resonance in Weyl semimetals in a circularly polarized optical pulse

We study theoretically the ultrafast electron dynamics of three-dimensional Weyl semimetals in the field of a laser pulse. For a circularly-polarized pulse, such dynamics is governed by topological resonance, which manifests itself as a specific conduction band population distribution in the vicinity of the Weyl points. The topological resonance is determined by the competition between the topological phase and the dynamic phase and depends on the handedness of a circularly polarized pulse. Also, we show that the conduction band population induced by a circularly-polarized pulse that consists of two oscillations with opposite handedness is highly chiral, which represents the intrinsic chirality of the Weyl points.