Degenerate Points Research Articles

Acoustic Tunneling Study for Hexachiral Phononic Crystals Based on Dirac-Cone Dispersion Properties

Acoustic tunneling is an essential property for phononic crystals in a Dirac-cone state. By analyzing the linear dispersion relations for the accidental degeneracy of Bloch eigenstates, the influence of geometric parameters on opening the Dirac-cone state and the directional band gaps’ widths are investigated. For two-dimensional hexachiral phononic crystals, for example, the four-fold accidental degenerate Dirac point emerges at the center of the irreducible Brillouin zone (IBZ). The Dirac cone properties and the band structure inversion problem are discussed. Finally, to verify acoustic transmission properties near the double-Dirac-cone frequency region, the numerical calculation of the finite-width phononic crystal structure is carried out, and the acoustic transmission tunneling effect is proved. The results enrich and expand the manipulating method in the topological insulator problem for hexachiral phononic crystals.

Quantum phase transition revealed by the exceptional point in a Hopfield-Bogoliubov matrix

We use the exceptional point in Hopfield-Bogoliubov matrix to find the phase transition points in the bosonic system. In many previous jobs, the excitation energy vanished at the critical point. It can be stated equivalently that quantum critical point is obtained when the determinant of Hopfield-Bogoliubov matrix vanishes. We analytically obtain the Hopfield-Bogoliubov matrix corresponding to the general quadratic Hamiltonian. For single-mode system the appearance of the exceptional point in Hopfield-Bogoliubov matrix is equivalent to the disappearance of the determinant of Hopfield-Bogoliubov matrix. However, in multi-mode bosonic system, they are not equivalent except in some special cases. For example, in the case of perfect symmetry, that is, swapping any two subsystems and keeping the total Hamiltonian invariable, the exceptional point and the degenerate point coincide all the time when the phase transition occurs. When the exceptional point and the degenerate point do not coincide, we find a significant result. With the increase of two-photon driving intensity, the normal phase changes to the superradiant phase, then the superradiant phase changes to the normal phase, and finally the normal phase changes to the superradiant phase.

Observation of Quadratic (Charge‐2) Weyl Point Splitting in Near‐Infrared Photonic Crystals

AbstractWeyl points are point degeneracies that occur in momentum space of 3D periodic materials and are associated with a quantized topological charge. Here, the splitting of a quadratic (charge‐2) Weyl point into two linear (charge‐1) Weyl points in a 3D micro‐printed photonic crystal is observed experimentally via Fourier‐transform infrared spectroscopy. Using a theoretical analysis rooted in symmetry arguments, it is shown that this splitting occurs along high‐symmetry directions in the Brillouin zone. This micro‐scale observation and control of Weyl points is important for realizing robust topological devices in the near‐infrared.

Transition from degeneracy to coalescence: Theorem and applications

Exceptional point (EP) is exclusive for non-Hermitian system and distinct from that at a degeneracy point (DP), supporting intriguing dynamics, which can be utilized to probe quantum phase transition and prepare eigenstates in a Hermitian many-body system. In this work, we investigate the transition from DP for a Hermitian system to EP driven by non-Hermitian terms. We present a theorem on the existence of transition between DP and EP for a general system. The obtained EP is robust to the strength of non-Hermitian terms. We illustrate the theorem by an exactly solvable quasi-one-dimensional model, which allows the existence of transition between fully degeneracy and exceptional spectra driven by non-Hermitian tunnelings in real and k spaces, respectively. We also study the EP dynamics for generating coalescing edge modes in Su-Schrieffer-Heeger-like models. This finding reveals the ubiquitous connection between DP and EP.

Spin-orbit-coupled exciton-polariton condensates in lead halide perovskites.

Spin-orbit coupling (SOC) is responsible for a range of spintronic and topological processes in condensed matter. Here, we show photonic analogs of SOCs in exciton-polaritons and their condensates in microcavities composed of birefringent lead halide perovskite single crystals. The presence of crystalline anisotropy coupled with splitting in the optical cavity of the transverse electric and transverse magnetic modes gives rise to a non-Abelian gauge field, which can be described by the Rashba-Dresselhaus Hamiltonian near the degenerate points of the two polarization modes. With increasing density, the exciton-polaritons with pseudospin textures undergo phase transitions to competing condensates with orthogonal polarizations. Unlike their pure photonic counterparts, these exciton-polaritons and condensates inherit nonlinearity from their excitonic components and may serve as quantum simulators of many-body SOC processes.

Nonmonotonic quantum phase gathering in curved spintronic circuits

Spin carriers propagating along quantum circuits gather quantum spin phases depending on the circuit's size, shape, and spin-orbit coupling (SOC) strength. These phases typically grow monotonically with the SOC strength, as found in Rashba quantum wires and rings. In this work we show that the spin-phase gathering can be engineered by geometric means, viz. by the geometric curvature of the circuits, to be non-monotonic. We demonstrate this peculiar property by using one-dimensional polygonal models where flat segments alternate with highly curved vertices. The complex interplay between dynamic and geometric spin-phase components -- triggered by a series of emergent spin degeneracy points -- leads to bounded, global spin phases. Moreover, we show that the particulars of the spin-phase gathering have observable consequences in the Aharonov-Casher conductance of Rashba loops, a connection that passed unnoticed in previous works.

Topological acoustic triple point

Acoustic phonon is a classic example of triple degeneracy point in band structure. This triple point always appears in phonon spectrum because of the Nambu–Goldstone theorem. Here, we show that this triple point can carry a topological charge {mathfrak{q}} that is a property of three-band systems with space-time-inversion symmetry. The charge {mathfrak{q}} can equivalently be characterized by the skyrmion number of the longitudinal mode, or by the Euler number of the transverse modes. We call triple points with nontrivial {mathfrak{q}} the topological acoustic triple point (TATP). TATP can also appear at high-symmetry momenta in phonon and spinless electron spectrums when Oh or Th groups protect it. The charge {mathfrak{q}} constrains the nodal structure and wavefunction texture around TATP, and can induce anomalous thermal transport of phonons and orbital Hall effect of electrons. Gapless points protected by the Nambu–Goldstone theorem form a new platform to study the topology of band degeneracies.

Localised spatial structures in the Thomas model

The Thomas model is a system of two reaction–diffusion equations which was originally proposed in the context of enzyme kinetics. It was subsequently realised that it offers a plausible chemical mechanism for the generation of coat markings on mammals. To that end previous investigations have focused on establishing the conditions for the Turing instability and on following the associated patterns as the bifurcation parameter increases through the instability.In this paper we use modern ideas from the theory of dynamical systems to systematically investigate the formation of localised structures in this model. The Turing instability is found to exhibit a degeneracy at which it changes from being subcritical to supercritical (or vice versa). Associated with such degenerate points is a heteroclinic connection which is a crucial requirement for the generation of localised patterns.Localised solutions containing a single ‘spike’ are associated with the so-called Belyakov–Devaney transition. Localised solutions containing either multiple ‘peaks’ or multiple ‘holes’ are associated with homoclinic snaking bifurcations. These structures are investigated using continuation software. The snaking bifurcations are found to collapse when the system crosses the Belyakov–Devaney transition. The isolated spike solutions collapse at a codimension two heteroclinic connection to create the so-called collapsed snaking.We show that the temporal stability of the localised solutions depends upon the presence of Hopf bifurcations which in turn are controlled by the diffusion ratio. For small values of the ratio only solutions in the spike region are destabilised by a Hopf bifurcation. For larger values of the diffusion ratio Hopf bifurcations destabilise most of the localised patterns.

Encyclopedia of emergent particles in three-dimensional crystals

The past decade has witnessed a surge of interest in exploring emergent particles in condensed matter systems. Novel particles, emerged as excitations around exotic band degeneracy points, continue to be reported in real materials and artificially engineered systems, but so far, we do not have a complete picture on all possible types of particles that can be achieved. Here, via systematic symmetry analysis and modeling, we accomplish a complete list of all possible particles in time-reversal-invariant systems. This includes both spinful particles such as electron quasiparticles in solids, and spinless particles such as phonons or even excitations in electric–circuit and mechanical networks. We establish detailed correspondence between the particle, the symmetry condition, the effective model, and the topological character. This obtained encyclopedia concludes the search for novel emergent particles and provides concrete guidance to achieve them in physical systems.

Influence of attenuation on the generation of optical vortices in multihelicoidal optical fibers

In this paper we have studied influence of attenuation on conversion processes of the fundamental mode (FM) in multihelicoidal optical fibers (MHF) in the vicinity of the point of accidental spectrum degeneracy within the framework of the scalar approximation. To this end, we have obtained expressions for modes of the MHF, which consist of the FM and an optical vortex (OV), and shown that conversion of the FM into the OV takes place. The difference in the attenuation coefficients for the partial fields of MHF’s modes leads to deterioration in the conversion process even with an ideal system’s tuning. At sufficiently large values of attenuation coefficients the conversion of the incoming FM into the vortex vanishes. Also we have shown the presence of exceptional point (EP) in the spectra of modes of the MHF and demonstrated enhanced sensitivity of the fiber in the vicinity of the EP to perturbations.

Eightfold Degenerate Fermions in Two Dimensions.

Multifold degenerate fermions have attracted a lot of research interest in condensed matter physics and materials science, but always lack in two dimensions. In this Letter, from symmetry analysis and lattice model construction, we demonstrate that eightfold degenerate fermions can be realized in two-dimensional systems. In nonmagnetic materials with negligible spin-orbit coupling, the gray magnetic space groups together with SU(2) spin rotation symmetry can protect the two-dimensional eightfold degenerate fermions on a certain high-symmetry axis in the Brillouin zone, no matter whether the system is centrosymmetric or noncentrosymmetric. In antiferromagnetic materials, the eightfold degenerate fermions can also be protected by certain "spin space groups." Furthermore, by first-principles electronic structure calculations, we predict that the paramagnetic phase of the monolayer LaB_{8} on a suitable substrate is a two-dimensional eightfold degenerate as well as Dirac node-line semimetal. Especially, the eightfold degenerate points are close to the Fermi level, which makes monolayer LaB_{8} a good platform to study the exotic physical properties of two-dimensional eightfold degenerate fermions.

Equation of State and Composition of Proto-Neutron Stars and Merger Remnants with Hyperons

Finite-temperature equation of state (EoS) and the composition of dense nuclear and hypernuclear matter under conditions characteristic of neutron star binary merger remnants and supernovas are discussed. We consider both neutrino free-streaming and trapped regimes which are separated by a temperature of a few MeV. The formalism is based on covariant density functional (CDF) theory for the full baryon octet with density-dependent couplings, suitably adjusted in the hypernuclear sector. The softening of the EoS with the introduction of the hyperons is quantified under various conditions of lepton fractions and temperatures. We find that Λ, Ξ−, and Ξ0 hyperons appear in the given order with a sharp density increase at zero temperature at the threshold being replaced by an extended increment over a wide density range at high temperatures. The Λ hyperon survives in the deep subnuclear regime. The triplet of Σs is suppressed in cold hypernuclear matter up to around seven times the nuclear saturation density, but appears in significant fractions at higher temperatures, T≥20 MeV, in both supernova and merger remnant matter. We point out that a special isospin degeneracy point exists where the baryon abundances within each of the three isospin multiplets are equal to each other as a result of (approximate) isospin symmetry. At that point, the charge chemical potential of the system vanishes. We find that under the merger remnant conditions, the fractions of electron and μ-on neutrinos are close and are about 1%, whereas in the supernova case, we only find a significant fraction (∼10%) of electron neutrinos, given that in this case, the μ-on lepton number is zero.

Qualitative Analysis of a Single-Species Model with Distributed Delay and Nonlinear Harvest

In this paper, a single-species population model with distributed delay and Michaelis-Menten type harvesting is established. Through an appropriate transformation, the mathematical model is converted into a two-dimensional system. Applying qualitative theory of ordinary differential equations, we obtain sufficient conditions for the stability of the equilibria of this system under three cases. The equilibrium A1 of system is globally asymptotically stable when br−c>0 and η<0. Using Poincare-Bendixson theorem, we determine the existence and stability of limit cycle when br−c>0 and η>0. By computing Lyapunov number, we obtain that a supercritical Hopf bifurcation occurs when η passes through 0. High order singularity of the system, such as saddle node, degenerate critical point, unstable node, saddle point, etc, is studied by the theory of ordinary differential equations. Numerical simulations are provided to verify our main results in this paper.

Novel family of topological semimetals with butterflylike nodal lines

In recent years, the exotic properties of topological semimetals (TSMs) have attracted great attention and significant efforts have been made in seeking for new topological phases and material realization. In this work, we propose a new family of TSMs which harbors an unprecedented nodal line (NL) landscape consisting of a pair of concentric intersecting coplanar ellipses (CICE) at half-filling. Meanwhile, the CICE at half-filling guarantees the presence of a second pair of CICE beyond half-filling. Both CICEs are linked at four-fold degenerate points (FDPs) at zone boundaries. In addition, we identify the generic criteria for the existence of the CICE in a time reversal invariant {\it spinless} fermion system or a spinfull system with negligible spin-orbital coupling (SOC). Consequently, 9 out of 230 space groups (SGs) are feasible for hosting CICE whose location centers in the first Brillouin zone (BZ) are identified. We provide a simplest model with SG $Pbam$ (No. 55) which exhibits CICE, and the exotic intertwined drumhead surface states, induced by double-band-inversions. Finally, we propose a series of material candidates that host butterfly-like CICE NLs, such as, ZrX$_2$ (X=P,As), GeTe$_5$Tl$_2$, CYB$_2$ and Al$_2$Y$_3$.

Triply degenerate point in three-dimensional spinless systems

We study the possibility of triply-degenerate points (TPs) that can be stabilized in spinless crystalline systems. Based on an exhaustive search over all 230 space groups, we find that the spinless TPs can exist at both high-symmetry points and high-symmetry paths, and they may have either linear or quadratic dispersions. For TPs located at high-symmetry points, they all share a common minimal set of symmetries, which is the point group $T$. The TP protected solely by the $T$ group is chiral and has a Chern number of $\pm2$. By incorporating additional symmetries, this TP can evolve into chiral pseudospin-1 point, linear TP without chirality, or quadratic contact TP. For accidental TPs residing on a high-symmetry path, they are not chiral but can have either linear or quadratic dispersions in the plane normal to the path. We further construct effective $k\cdot p$ models and minimal lattice models for characterizing these TPs. Distinguished phenomena for the chiral TPs are discussed, including the extensive surface Fermi arcs and the chiral Landau bands.

Coupling microwave photons to topological spin textures in Cu2OSeO3

Topologically protected nanoscale spin textures, known as magnetic skyrmions, possess particle-like properties and feature emergent magnetism effects. In bulk cubic heli-magnets, distinct skyrmion resonant modes are already identified using a technique like ferromagnetic resonance in spintronics. However, direct light-matter coupling between microwave photons and skyrmion resonance modes has not been demonstrated yet. Utilising two distinct cavity systems, we realise to observe a direct interaction between the cavity resonant mode and two resonant skyrmion modes, the counter-clockwise gyration and breathing modes, in bulk Cu$_2$OSeO$_3$. For both resonant modes, we find the largest coupling strength at 57 K indicated by an enhancement of the cavity linewidth at the degeneracy point. We study the effective coupling strength as a function of temperature within the expected skyrmion phase. We attribute the maximum in effective coupling strength to the presence of a large number of skyrmions, and correspondingly to a completely stable skyrmion lattice. Our experimental findings indicate that the coupling between photons and resonant modes of magnetic skyrmions depends on the relative density of these topological particles instead of the pure spin number in the system.

A New Algorithm for the Single Source Weber Problem with Limited Distances

The single source Weber problem with limited distances (SSWPLD) is a continuous optimization problem in location theory. The SSWPLD algorithms proposed so far are based on the enumeration of all regions of [Formula: see text] defined by a given set of n intersecting circumferences. Early algorithms require [Formula: see text] time for the enumeration, but they were recently shown to be incorrect in case of degenerate intersections, that is, when three or more circumferences pass through the same intersection point. This problem was fixed by a modified enumeration algorithm with complexity [Formula: see text], based on the construction of neighborhoods of degenerate intersection points. In this paper, it is shown that the complexity for correctly dealing with degenerate intersections can be reduced to [Formula: see text] so that existing enumeration algorithms can be fixed without increasing their [Formula: see text] time complexity, which is due to some preliminary computations unrelated to intersection degeneracy. Furthermore, a new algorithm for enumerating all regions to solve the SSWPLD is described: its worst-case time complexity is [Formula: see text]. The new algorithm also guarantees that the regions are enumerated only once.

Topological characteristic of Weyl degeneracies in a reciprocal chiral metamaterials system

Being a research hotspot in the field of topological semimetals, Weyl points (WPs) are monopoles of Berry curvature in momentum space. In this paper, we report the existence of photonic Weyl degeneracies in a reciprocal chiral metamaterials system. Due to the flat dispersion relation of the bulk plasmon modes, Weyl degeneracies here lie right on the critical transition between the type-I and type-II WPs. The photonic ‘Fermi arc’ connects the projection of pairs of WPs at the interface between the metamaterials and vacuum. Despite the bulk equi-frequency surfaces have changed dramatically, the ‘Fermi arc’ always exists. In addition, numerical simulations of topologically protected ‘Fermi arc’ surface states show that the surface waves are not scattered or reflected by the presence of sharp corners. Notably, such metamaterials host either type-I, type-II WPs or triple degenerate points (TDPs) depending on the nonlocal response. Our work provides an ideal photonic platform for studying the closely relation between WPs and other exotic states.

Photoinduced spin-Hall resonance in a k3 -Rashba spin-orbit coupled two-dimensional hole system

We study the band structure modulation and spin-Hall effect of an irradiated two-dimensional heavy-hole system with ${k}^{3}$-Rashba spin-orbit coupling. We find that the band structure becomes anisotropic under the illumination by the light. Most remarkably, in the presence of linearly polarized light a pair of additional spin-degeneracy points, apart from the $\mathrm{\ensuremath{\Gamma}}$ point, emerge in the energy dispersion. The locations of these points are solely determined by the amplitude of the incident light. If this degeneracy occurs around the Fermi level, the spin-Hall conductivity exhibits a resonance. Away from the degeneracy points, the light rotates the average spin polarization. The possible effects of ${k}^{3}$-Dresselhaus spin-orbit coupling are also discussed.

Manipulating Second Harmonic Generation in Higher‐Order Topological Photonic Crystals

AbstractHigher‐order topological photonic crystals (TPCs) have attracted much attention in the past several years since they do not obey the conventional bulk‐boundary correspondence in topological insulating phases. This characteristic offers a new dimension with which to trap and manipulate the flow of light. In this work, 2D higher‐order TPCs for engineering optical second harmonic generation (SHG) are designed. In the TPCs, a topological corner‐state is achieved and it is demonstrated that the high localization of photons at the corner‐state can significantly promote optical SHG. In addition, topological edge‐states are obtained by breaking the Dirac degeneracy point near the SHG frequency. They are also topologically protected and immune to bending resistance. Therefore, the enhanced SHG signal from the corner‐state can be manipulated precisely for transmitting through the designed interface with relatively low loss due to the topological‐protection edge waves. This design strategy combines the advantages of corner‐state and edge‐state in higher‐order TPCs, demonstrating potential applications of topological photonic devices, such as optical communication and optical computing technologies.