Weyl Spectrum Research Articles

Quantum oscillations in acoustic phonons in Weyl semimetals

We theoretically study the modification of the energy spectrum of long-wavelength acoustic phonons due to the electron-phonon interaction in a three-dimensional topological Weyl semimetal under the influence of quantizing magnetic fields. We find that the dispersion and attenuation of phonons show striking oscillatory behaviors as varying the magnetic field at low temperatures. These oscillations are distinct when the Fermi energy is in different energy regimes. Moreover, the van Hove singularity of the Weyl spectrum can manifest as a transition between different oscillation patterns when increasing the magnetic field. These phonon behaviors could provide testable fingerprints of the Fermi surface morphology and relativistic feature of the Weyl semimetal.

Spectral properties for some extensions of isometric operators

In this paper, we show that the spectrum, Weyl spectrum, and Browder spectrum are continuous on the set of all 2-isometric operators. We also prove that if T, \(S^{*}\) are 2-isometric and invertible isometric operators respectively and X is a Hilbert–Schmidt operator such that \(TX=XS\), then \(T^{*}X=XS^{*}\). Moreover, we show that every Riesz projection E with respect to a non-zero isolated spectral point \(\lambda \) of a 2-isometric operator T is self-adjoint and satisfies \({\mathcal {R}}(E) = {\mathcal {N}}(T-\lambda ) = {\mathcal {N}}(T- \lambda )^{*}\). Further, we show that quasi-2-isometric operator satisfies Bishop’s property (\(\beta \)). Finally, we prove Weyl type theorems for \(f(d_{TS})\), where \( d_{TS}\) denote the generalized derivation or the elementary operator with quasi-2-isometric operator entries T and \(S^{*}\) and \(f \in H(\sigma (d_{TS}))\), the set of analytic functions which are defined on an open neighborhood of \(\sigma (d_{TS})\).

Weyl’s theorem for paranormal closed operators

In this article, we discuss a few spectral properties of paranormal closed operator (not necessarily bounded) defined in a Hilbert space. This class contains closed symmetric operators. First, we show that the spectrum of such an operator is non-empty and give a characterization of closed range operators in terms of the spectrum. Using these results, we prove the Weyl’s theorem: if T is a densely defined closed paranormal operator, then \(\sigma (T)\setminus \omega (T)=\pi _{00}(T)\), where \(\sigma (T),\, \omega (T)\) and \(\pi _{00}(T)\) denote the spectrum, the Weyl spectrum and the set of all isolated eigenvalues with finite multiplicities, respectively. Finally, we prove that the Riesz projection \(E_\lambda \) with respect to any non-zero isolated spectral value \(\lambda \) of T is self-adjoint and satisfies \(R(E_\lambda )=N(T-\lambda I)=N(T-\lambda I)^*\).

Exploiting the violation of Lorentz symmetry for the planar Hall effect

The low energy Dirac and Weyl spectra are allowed to violate the Lorentz symmetry and thereby have a tilted energy dispersion. The tilt in the energy dispersion induces a Hall voltage in the plane spanned by the electric field and the tilt velocity. In the presence of a magnetic field the planar Hall conductivity and resistivity show Shubnikov de Haas oscillations. The oscillations in the planar Hall effect can become a fingerprint to spot the anomalous transport in Dirac and Weyl semimetals.

On Weyl’s Theorem for Functions of Operators

Let H be a complex separable infinite dimensional Hilbert space. In this paper, a variant of the Weyl spectrum is discussed. Using the new spectrum, we characterize the necessary and sufficient conditions for both T and f(T) satisfying Weyl’s theorem, where f ∊ Hol(σ(T)) and Hol(σ(T)) is defined by the set of all functions f which are analytic on a neighbourhood of σ(T) and are not constant on any component of σ(T). Also we consider the perturbations of Weyl’s theorem for f(T).

Spectral mapping theorems for Weyl spectrum and isolated spectral points

Spectral mapping theorems for Weyl spectrum and isolated spectral points

Giant tunable nonreciprocity of light in Weyl semimetals

The propagation of light in Weyl semimetal films is analyzed. The magnetic family of these materials is known by anomalous Hall effect, which, being enhanced by the large Berry curvature, allows one to create strong gyrotropic and nonreciprocity effects without external magnetic field. The existence of nonreciprocal waveguide electromagnetic modes in ferromagnetic Weyl semimetal films in the Voigt configuration is predicted. Thanks to the strong dielectric response caused by the gapless Weyl spectrum and the large Berry curvature, ferromagnetic Weyl semimetals combine the best waveguide properties of magnetic dielectrics or semiconductors with strong anomalous Hall effect in ferromagnets. The magnitude of the nonreciprocity depends both on the internal Weyl semimetal properties, the separation of Weyl nodes, and the external factor, the optical contrast between the media surrounding the film. By tuning the Fermi level in Weyl semimetals, one can vary the operation frequencies of the waveguide modes in THz and mid-IR ranges. Our findings pave the way to the design of compact, tunable, and effective nonreciprocal optical elements.

Spectral mapping theorems for the upper Weyl and upper Browder spectra

We discuss the spectral mapping properties of the upper Weyl and upper Browder spectra of an arbitrary ordered Banach algebra element.

Universal scattering response across the type-II Weyl semimetal phase diagram

The discovery of Weyl semimetals represents a significant advance in topological band theory. They paradigmatically enlarged the classification of topological materials to gapless systems while simultaneously providing experimental evidence for the long-sought Weyl fermions. Beyond fundamental relevance, their high mobility, strong magnetoresistance, and the possible existence of even more exotic effects, such as the chiral anomaly, make Weyl semimetals a promising platform to develop radically new technology. Fully exploiting their potential requires going beyond the mere identification of materials and calls for a detailed characterization of their functional response, which is severely complicated by the coexistence of surface- and bulk-derived topologically protected quasiparticles, i.e., Fermi arcs and Weyl points, respectively. Here, we focus on the type-II Weyl semimetal class where we find a stoichiometry-dependent phase transition from a trivial to a non-trivial regime. By exploring the two extreme cases of the phase diagram, we demonstrate the existence of a universal response of both surface and bulk states to perturbations. We show that quasi-particle interference patterns originate from scattering events among surface arcs. Analysis reveals that topologically non-trivial contributions are strongly suppressed by spin texture. We also show that scattering at localized impurities generate defect-induced quasiparticles sitting close to the Weyl point energy. These give rise to strong peaks in the local density of states, which lift the Weyl node significantly altering the pristine low-energy Weyl spectrum. Visualizing the microscopic response to scattering has important consequences for understanding the unusual transport properties of this class of materials. Overall, our observations provide a unifying picture of the Weyl phase diagram.

A note on the lower Weyl and Lozanovsky spectra of a positive element

One of the central problems investigated in Alekhno (Positivity 11(3):375–386, 2007, Positivity 13(1):3–20, 2009) is that of providing conditions under which the spectral radius of a positive operator T on a complex Banach lattice lies outside the lower Weyl spectrum of T given that it is not an element of its essential spectrum. In this paper the lower Weyl spectrum of an arbitrary positive ordered Banach algebra element is introduced and studied, and work done in the aforementioned papers are extended to general ordered Banach algebras.

Weyl’s theorem for upper triangular operator matrix and perturbations

Let and be infinite dimensional separable complex Hilbert spaces and T a bounded linear operator on . ‘Weyl’s theorem’ holds for operator T when the complement in the spectrum of the Weyl spectrum coincides with the isolated points of the spectrum which are eigenvalues of finite multiplicity. Let . In this paper, we characterize the invariant subset and the diagonal operator matrix satisfying upper triangular operator matrix , where .

Self-adjoint perturbations of spectra for upper triangular operator matrices

This paper is concerned with the self-adjoint perturbations of the spectra for the upper triangular partial operator matrix with given diagonal entries. A necessary and sufficient condition is given under which such operator matrix admits a Weyl (Fredholm) operator completion by choosing some bounded self-adjoint operator. It is shown that the self-adjoint perturbation of the Weyl (essential) spectrum can be the proper set of the general perturbation. Combining the spectral properties, we further characterize the perturbation of the Weyl (essential) spectrum for Hamiltonian operators.

Generalized left and right Weyl spectra of upper triangular operator matrices

In this paper, for given operators $A\in\B(\H)$ and $B\in\B(\K)$, the sets of all $C\in \B(\K,\H)$ such that $M_C=\bmatrix{cc} A&C\\0&B\endbmatrix$ is generalized Weyl and generalized left (right) Weyl, are completely described. Furthermore, the following intersections and unions of the generalized left Weyl spectra $$ \bigcup_{C\in\B(\K,\H)}\sigma^g_{lw}(M_C) \;\;\; \mbox{and} \;\;\; \bigcap_{C\in\B(\K,\H)}\sigma^g_{lw}(M_C) $$ are also described, and necessary and sufficient conditions which two operators $A\in\B(\H)$ and $B\in\B(\K)$ have to satisfy in order for $M_C$ to be a generalized left Weyl operator for each $C\in\B(\K,\H)$, are presented.

Weyl type theorems for unbounded class- $$\mathcal {A}$$ A operators

The study of Weyl’s theorem and properties has been extended from the class of hermitian operators to the class of hyponormal and Toeplitz operator and several other non-normal classes. However, so far, this study has been limited to the bounded operators. In this paper, we introduce the class of unbounded class-\(\mathcal {A}\) operators and study Weyl-type theorems and properties for this class under certain condition. Certain relations have been established between the isolated spectral points with the Weyl spectrum and the B-Weyl spectrum. Also, for bounded operators, some variants of the Weyl’s Theorem imply the others. In the case of unbounded class-\(\mathcal {A}\) operators, several of those variants of the Weyl’s theorem have been shown to be equivalent.

Photocurrents in Weyl semimetals

The generation of photocurrent in an ideal two-dimensional Dirac spectrum is symmetry-forbidden. In sharp contrast, we show that three-dimensional Weyl semimetals can generically support significant photocurrent generations due to the combination of inversion symmetry breaking and finite tilts of the Weyl spectra. Symmetry properties, chirality relations and various dependences on the system and light sources are explored in details. Our results suggest that noncentrosymmetric Weyl materials can be advantageously applied to room temperature detections of mid- and far-infrared radiations.

Type-II Weyl cone transitions in driven semimetals

Periodically driven systems provide tunable platforms to realize interesting Floquet topological phases and phase transitions. In electronic systems with Weyl dispersions, the band crossings are topologically protected even in the presence of time-periodic perturbations. This robustness permits various routes to shift and tilt the Weyl spectra in the momentum and energy space using circularly polarized light of sufficient intensity. We show that type-II Weyl fermions, in which the Weyl dispersions are tilted with the appearance of pocket-like Fermi surfaces, can be induced in driven Dirac semimetals and line node semimetals. Under a circularly polarized drive, both semimemtal systems immediately generate Weyl node pairs whose types can be further controlled by the driving amplitude and direction. The resultant phase diagrams demonstrate experimental feasibilities.

The index for Fredholm elements in a Banach algebra via a trace II

We show that the index defined via a trace for Fredholm elements in a Banach algebra has the property that an index zero Fredholm element can be decomposed as the sum of an invertible element and an element in the socle. We identify the set of index zero Fredholm elements as an upper semiregularity with the Jacobson property. The Weyl spectrum is then characterized in terms of the index.

General note on the theorem of Stampfli

It is well known that for every bounded operator A in $L(H)$ , there exists a compact operator K in $K(H)$ such that the Weyl spectrum $\sigma_{W}(A)$ of the operator A coincides with the spectrum $\sigma(A+K)$ of the perturbed operator A. In this work, we show the extension of this relation by the use of Kato’s decomposition to the set of semi-Fredholm operators.

Essential, Weyl and Browder spectra of unbounded upper triangular operator matrices

This paper is concerned with the spectral properties of the unbounded upper triangular operator matrix with diagonal domain. Some sufficient and necessary conditions are given under which the essential spectrum, the Weyl spectrum and the Browder spectrum of such operator matrix, respectively, coincide with the union of the essential spectrum, the Weyl spectrum and the Browder spectrum of its diagonal entries.

Weyl Spectral Identity and Biquasitriangularity

AbstractLet A and B be operators acting on infinite-dimensional complex Banach spaces. We say that the Weyl spectral identity holds for the tensor product A⊗B if σw(A⊗B) = σw(A)·σ(B)∪σ(A)·σw(B), where σ(·) and σw(·) stand for the spectrum and the Weyl spectrum, respectively. Conditions on A and B for which the Weyl spectral identity holds are investigated. Especially, it is shown that if A and B are biquasitriangular (in particular, if the spectra of A and B have empty interior), then the Weyl spectral identity holds. It is also proved that if A and B are biquasitriangular, then the tensor product A ⊗ B is biquasitriangular.