Degenerate Line Research Articles

Non-Local Problems for the Fractional Order Diffusion Equation and the Degenerate Hyperbolic Equation

This research explores nonlocal problems associated with fractional diffusion equations and degenerate hyperbolic equations featuring singular coefficients in their lower-order terms. The uniqueness of the solution is established using the energy integral method, while the existence of the solution is equivalently reduced to solving Volterra integral equations of the second kind and a fractional differential equation. The study focuses on a mixed domain where the parabolic section aligns with the upper half-plane, and the hyperbolic section is bounded by two characteristics of the equation under consideration and a segment of the x-axis. By utilizing the solution representation of the fractional-order diffusion equation, a primary functional relationship is derived between the traces of the sought function on the x-axis segment from the parabolic part of the mixed domain. An explicit solution form for the modified Cauchy problem in the hyperbolic section of the mixed domain is presented. This solution, combined with the problem’s boundary condition, yields a fundamental functional relationship between the traces of the unknown function, mapped to the interval of the equation’s degeneration line. Through the conjugation condition of the problem, an equation with fractional derivatives is obtained by eliminating one unknown function from two functional relationships. The solution to this equation is explicitly formulated. For a specific solution of the proposed problem, visualizations are provided for various orders of the fractional derivative. The analysis demonstrates that the derivative order influences both the intensity of the diffusion (or subdiffusion) process and the shape of the wave front.

Ultrahigh Néel Temperature Antiferromagnetism and Ultrafast Laser-Controlled Demagnetization in a Dirac Nodal Line MoB3 Monolayer.

Two-dimensional (2D) antiferromagnetic (AFM) materials boasting a high Néel temperature (TN), high carrier mobility, and fast spin response under an external field are in great demand for efficient spintronics. Herein, we theoretically present the MoB3 monolayer as an ideal 2D platform for AFM spintronics. The AFM MoB3 monolayer features a symmetry-protected, 4-fold degenerate Dirac nodal line (DNL) at the Fermi level. It demonstrates a high magnetic anisotropy energy of 865 μeV/Mo and an ultrahigh TN of 1050 K, one of the highest recorded for 2D AFMs. Importantly, we reveal the ultrafast demagnetization of AFM MoB3 under laser irradiation, which induces a rapid transition from a DNL semimetallic state to a metallic state on the time scale of hundreds of femtoseconds. This work presents an effective method for designing advanced spintronics using 2D high-temperature DNL semimetals and opens up a new idea for ultrafast modulation of magnetization in topological semimetals.

A topological Hund nodal line antiferromagnet

The interplay of topology, magnetism, and correlations gives rise to intriguing phases of matter. In this study, through state-of-the-art angle-resolved photoemission spectroscopy, density functional theory, and dynamical mean-field theory calculations, we visualize a fourfold degenerate Dirac nodal line at the boundary of the bulk Brillouin zone in the antiferromagnet YMn2Ge2. We further demonstrate that this gapless, antiferromagnetic Dirac nodal line is enforced by the combination of magnetism, space-time inversion symmetry, and nonsymmorphic lattice symmetry. The corresponding drumhead surface states traverse the whole surface Brillouin zone. YMn2Ge2 thus serves as a platform to exhibit the interplay of multiple degenerate nodal physics and antiferromagnetism. Interestingly, the magnetic nodal line displays a d-orbital dependent renormalization along its trajectory in momentum space, thereby manifesting Hund’s coupling. Our findings offer insights into the effect of electronic correlations on magnetic Dirac nodal lines, leading to an antiferromagnetic Hund nodal line.

Berry curvature inside a PT -symmetry protected exceptional surface

A three-dimensional non-Hermitian Hamiltonian with parity-time symmetry can exhibit a closed exceptional surface (EP surface) in momentum space, which is a non-Hermitian deformation of the degeneracy line. Since the degeneracy line lacks an internal space, the distributions of Berry curvature inside the EP surface become particularly intriguing. This paper studies the distributions taking a toruslike EP surface as an example. In a meridian cross section, the Berry connection exhibits a vortexlike field with only angular components, while the Berry curvature is perpendicular to this cross section; in a equatorial cross section, the Berry curvature forms a closed curve surrounding the central genus. Both Berry connection and curvature converge along the coplanar axis and diverge at the surface. We find the Berry flux depends on the radius of the integration region and is not quantized inside the EP torus. Approaching the surface, the Berry flux tends to infinity and the dynamical phase oscillates violently. We point out that the streamlines of Berry curvature can be used to estimate the zero or nonzero Berry flux. We generalize the above patterns to the case of EP surfaces with complex shapes and present a proposal of realizing the EP surface in an electrical circuit. Our research outcomes enhance the comprehension of EP surfaces and the topological characteristics of non-Hermitian systems with parity-time (PT) symmetry. Published by the American Physical Society 2024

Two-dimensional antiferromagnetic nodal-line semimetal and spin Hall effect in MnC4

Nodal-line semimetals, characterized by Dirac-like crossings along one dimensional k-space lines, represent a unique class of topological materials. In this study, we investigate the intriguing properties of room-temperature antiferromagnetic MnC4 and its nodal-line features both with and without spin–orbit coupling (SOC). In the absence of SOC, we identify a doubly degenerate Dirac-nodal line, robustly protected by a combination of time-reversal, mirror, and partial-translation symmetries. Remarkably, this nodal line withstands various external perturbations, including isotropic and anisotropic strain, and torsional deformations, due to the ionic-like bonding between Mn atoms and C clusters. With the inclusion of SOC, we observe a distinctive quasi-Dirac-nodal line that emerges due to the interplay between antiferromagnetism and SOC-induced spin-rotation symmetry breaking. Finally, we observed a robust spin Hall conductivity that aligns with the energy range where the quasi-nodal line appears. This study presents a compelling example of a robust symmetry-protected Dirac-nodal line antiferromagnetic monolayer, which has potential for applications in next-generation spintronic devices.

APPLICATION OF THE POTENTIAL THEORY TO SOLVING A MIXEDPROBLEM FOR A MULTIDIMENSIONAL ELLIPTIC EQUATION WITHONE SINGULAR COEFFICIENT

Potential theory has played a paramount role in both analysis and computation for boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially and fruitfully used. At the present time, fundamental solutions of a multidimensional elliptic equation with one degeneration line are already known. In this paper, we investigate the double- and simple-layer potentials for this kind of elliptic equations. Results from potential theory allow us to represent the solution of the boundary value problems in integral equation form. By using some properties of Gaussian hypergeometric function, we prove limiting theorems and derive integral equations concerning a densities of the double- and simple-layer potentials. The obtained results are applied to find a solution of the mixed problem for the multidimensional singular elliptic equation in the half-space

Triply degenerate nodal line and tunable contracted-drumhead surface state in a tight-binding model

The study of topological semimetals has been extended to more general topological nodal systems such as metamaterials and artificial periodic structures. Among various nodal structures, triply degenerate nodal line (TDNL) is rare and, hence, has received little attention. In this work, we have proposed a simple tight-binding (TB) model, which hosts a topological non-trivial TDNL. This TDNL not only has the drumhead surface states (DSSs) as usual nodal line systems but also has surface states that form a contracted-drumhead shape. The shape and area of this contracted drumhead can be tuned by the hopping parameters of the model. This provides an effective way to modulate surface states and their density of states, which can be important in future applications of topological nodal systems.

Conditional Well-Posedness of the Initial-Boundary Value Problem for a System of Inhomogeneous Mixed Type Equations with Two Degeneration Lines

Conditional Well-Posedness of the Initial-Boundary Value Problem for a System of Inhomogeneous Mixed Type Equations with Two Degeneration Lines

The Problem in the Unbounded Domain with the Frankl Condition on the Segment of the Degeneration Line and with a Missing Gellerstedt Condition for a Class of Mixed-Type Equations

The Problem in the Unbounded Domain with the Frankl Condition on the Segment of the Degeneration Line and with a Missing Gellerstedt Condition for a Class of Mixed-Type Equations

Charge density wave induced nodal lines in LaTe3

LaTe3 is a non-centrosymmetric material with time reversal symmetry, where the charge density wave is hosted by the Te bilayers. Here, we show that LaTe3 hosts a Kramers nodal line—a twofold degenerate nodal line connecting time reversal-invariant momenta. We use angle-resolved photoemission spectroscopy, density functional theory with an experimentally reported modulated structure, effective band structures calculated by band unfolding, and symmetry arguments to reveal the Kramers nodal line. Furthermore, calculations confirm that the nodal line imposes gapless crossings between the bilayer-split charge density wave-induced shadow bands and the main bands. In excellent agreement with the calculations, spectroscopic data confirm the presence of the Kramers nodal line and show that the crossings traverse the Fermi level. Furthermore, spinless nodal lines—completely gapped out by spin-orbit coupling—are formed by the linear crossings of the shadow and main bands with a high Fermi velocity.

Kramers nodal lines and Weyl fermions in SmAlSi

Kramers nodal lines (KNLs) have recently been proposed theoretically as a special type of Weyl line degeneracy connecting time-reversal invariant momenta. KNLs are robust to spin orbit coupling and are inherent to all non-centrosymmetric achiral crystal structures, leading to unusual spin, magneto-electric, and optical properties. However, their existence in in real quantum materials has not been experimentally established. Here we gather the experimental evidence pointing at the presence of KNLs in SmAlSi, a non-centrosymmetric metal that develops incommensurate spin density wave order at low temperature. Using angle-resolved photoemission spectroscopy, density functional theory calculations, and magneto-transport methods, we provide evidence suggesting the presence of KNLs, together with observing Weyl fermions under the broken inversion symmetry in the paramagnetic phase of SmAlSi. We discuss the nesting possibilities regarding the emergent magnetic orders in SmAlSi. Our results provide a solid basis of experimental observations for exploring correlated topology in SmAlSi

Squeezing Properties of Degenerate High-Order Hyper-Raman Lines Emitted by Two-Level System

Squeezing Properties of Degenerate High-Order Hyper-Raman Lines Emitted by Two-Level System

Hermitian and non-Hermitian Weyl physics in synthetic three-dimensional piezoelectric phononic beams

Recently, the evolution of the Weyl point (WP) caused by the introduction of nonhermiticity into Weyl semimetals has aroused great research interest. We consider elastic flexural wave propagation in a phononic beam containing piezoelectric materials and introduce nonhermiticity through active regulation of external circuits. Considering a synthetic parameter space constituted by the one-dimensional Bloch wave vector and two geometrical parameters, we demonstrate that a double WP (DWP) arises at the band crossing. Then we study its evolution from the hermitic to nonhermitic situation under the effect of the active piezoelectric materials. We find that the DWP in the hermitic case evolves into a Weyl degenerate line and a Weyl hollow ring as concerns the real and imaginary parts of the Weyl frequencies, respectively. The formation mechanisms of the DWPs, lines, and rings are explained through the Hamiltonian of the system. Further, we observe the changes of the DWP and degenerate line in the transmission spectra of finite structures. Finally, we discuss the synthetic Fermi arc interface states through the analysis of the reflected phase vortices. In this paper, we provide insights into the high-dimensional Hermitian and non-Hermitian physics in elastic wave systems using synthetic dimensions.

On the unique solvability of a nonlocal boundary value problem with the poincaré condition

As is known, it is customary in the literature to divide degenerate equations of mixed type into equations of the first and second kind. In the case of an equation of the second kind, in contrast to the first, the degeneracy line is simultaneously the envelope of a family of characteristics, i.e. is itself a characteristic, which causes additional difficulties in the study of boundary value problems for equations of the second kind. In this paper, in order to establish the unique solvability of one nonlocal problem with the Poincaré condition for an elliptic-hyperbolic equation of the second kind developed a new principle extremum, which helps to prove the uniqueness of resolutions as signed problem. The existence of a solution is realized by reducing the problem posed to a singular integral equation of normal type, which known by the Carleman-Vekua regularization method developed by S.G. Mikhlin and M.M. Smirnov equivalently reduces to the Fredholm integral equation of the second kind, and the solvability of the latter follows from the uniqueness of the solution delivered problem.

Dirichlet’s problem for a third-order parabolic hyperbolic type equation of the second kind

Equations of mixed type, the degeneracy line of which is the envelope of a family of characteristics, therefore, is itself also a characteristic, in the literature it is customary to call equations of mixed type of the second kind, which causes additional difficulties in the study of boundary value problems for equations of the second kind. In the present work, a boundary value problem for a homogeneous equation of parabolic-hyperbolic type of the third order of the second kind is investigated. Necessary and sufficient conditions for the existence and uniqueness of a generalized solution of the problem are found. In some special cases, the representation of the solution of the problem under study is written out explicitly.

Stacking-induced magnetic frustration and spiral spin liquid

Like the twisting control in magic-angle twisted bilayer graphene, the stacking control is another mechanical approach to manipulate the fundamental properties of solids, especially the van der Waals materials. We explore the stacking-induced magnetic frustration and the spiral spin liquid on a multilayer triangular lattice antiferromagnet where the system is built from ABC stacking with competing intralayer and interlayers couplings. By combining the nematic bond theory and the self-consistent Gaussian approximation, we establish the phase diagram for this ABC-stacked multilayer magnet. It is shown that the system supports a wide regime of spiral spin liquid with multiple degenerate spiral lines in the reciprocal space, separating the low-temperature spiral order and the high-temperature featureless paramagnet. The transition to the spiral order from the spiral spin liquid regime is first order. We further show that the spiral-spin-liquid behavior persists even with small perturbations such as further neighbor intralayer exchanges. The connection to the ABC-stacked magnets, the effects of Ising or planar spin anisotropy, and the outlook on the stacking-engineered quantum magnets are discussed.

Degenerate line modes in the surface and bulk phonon spectra of orthorhombic NaMgF3 perovskite

Degenerate bulk-line phonon modes have been widely reported in various crystal system types; however, degenerate surface-line phonon modes have only been reported in monoclinic crystal systems, such as SnIP with space group P2/c (No. 13). Herein, we propose that degenerate surface-line phonon modes can also emerge in solids with orthorhombic structures. Based on first-principle calculations and symmetry analysis, we propose that orthorhombic NaMgF3 fluoroperovskite with space group Pnma (No. 62) is a material candidate with degenerate line states in both the bulk phonon mode and the (010) surface phonon mode. We discovered four closed nodal loops (two type-I and two hybrid-type) on the ky = 0 plane in the bulk phonon mode, all of which coexisted with Dirac points on the Z–U and X–U paths. Moreover, we discovered symmetry-projected doubly degenerate nodal lines along the X¯–U¯ surface path in the (010) surface phonon mode. The proposed degenerate surface-line phonons in NaMgF3 is quite clean and protected by symmetries, which will aid future experimental detection.

Tricomi Problem for a Differential–Difference Equation of Mixed Type with Parallel Degeneration Lines

Tricomi Problem for a Differential–Difference Equation of Mixed Type with Parallel Degeneration Lines

Nonlinear band structure of cold atoms with interaction-dependent dispersion

Band structure is an important tool to characterize the physical properties of periodic systems. In this work, we investigate the nonlinear spectrum and current density in cold atoms with interaction-dependent dispersion (IDD). The results reveal that IDD causes the nonlinear loop structure, which means that the dynamical stability is destroyed, resulting in the loss of superfluidity. More interestingly, the repulsive IDD induces a more complex deformation of spectrum. Different from the previous studies, the conversion between positive and negative curvature loop structures appears in the system. Furthermore, there will be a pair of triple degeneracy points on the lower band (n=2nc) and a double degeneracy line between them, which will uncover new physical properties, e.g., nonlinear Landau-Zener tunneling. Since the nonlinear interaction of cold atoms is highly controllable through Feshbach resonance, this fascinating phenomenon is expected to be realized in cold atomic systems in the near future.

Photonic Dirac nodal-line semimetals realized by a hypercrystal

Recently, the gapless Dirac/Weyl nodal semimetals with linear dispersion and topologically protected modes degeneracy are rapidly growing frontiers of topological physics. Especially, type-I, type-II, and critical type-III nodal semimetals are discovered according to the tilt angles of the Dirac/Weyl cones. Here, by introducing hyperbolic metamaterials into one-dimensional photonic crystals, we design the "hyper-crystal" and study the photonic four-fold degenerate Dirac nodal line semimetals (DNLSs) with two types of perpendicularly polarized waves. Moreover, the flexibly controlled photonic DNLSs using the phase compensation effect of hyperbolic dispersion are studied. Our results not only demonstrate a new platform to realize the various photonic DNLSs, where the optical polarization plays the role of electron spin in electronic DNLSs, but also may pave a novel way to explore the abundant Dirac/Weyl physics in the simple classical wave systems.