Triple Degeneracy Research Articles

Spin resolved topological bulk state in acoustics

Extremely rare topologically protected acoustic energy sink is presented in this article. Acoustic topological phenomena are generally described using quantum anomalous hall effects (QAHE), quantum valley hall effects (QVHE), and quantum spin hall effects (QSHE) where spin orbit coupling is predominant. Topological edge states are demonstrated by bulk-boundary distinction when the bulk is insulated. In this article topological acoustic conductor and its phenomena are theoretically demonstrated where the boundaries are insulated. This is exactly opposite to the behavior of a topological acoustic insulator. Phenomena presented in this article could not be explained by any of the trio Quantum Hall effects. To explain the phenomenon phononic crystals or PnCs were designed to obtain accidental triple degeneracies, resulting a Dirac-like cone at the Γ point (k→=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overrightarrow{k}=0$$\\end{document}). The phenomenon is microarchitecture and microrotation field independent. Here time reversal symmetry or the space inversion symmetry is not broken, and the degenerated ‘Deaf band’ dominates the local dispersion with a syncline top band. This scenario results in continuously changing ‘up spin’ and ‘down spin’ of the wave energy in the media and remain trapped without specific preferential direction of wave transport. The spin was found to generate the spin angular momentum, causing the switching in geometric phase from 0-2π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0-2\\pi$$\\end{document} in cyclic pattern, keeping the energy trapped inside the bulk media.

A simultaneous eigen valued expression to model the three coupled electronic states with triply and doubly degenerate seams together II

AbstractIn part I a novel and yet complicated mathematical model to predict the presence of triply degenerate seam was presented and was illustrated its efficiency using some physical models. Here in this part, a discussion on the the model's ability to account for the triple degeneracy found in the potential energy surface is presented in addition to a discussion on the derivatives and its discontinuities and infiniteness at a specific point. Analytical evaluation of derivatives is possible in principle but in practice, numerical approximation to the exact derivatives is the only way out to solve this problem. Two examples (one being physical and the other being chemically realistic) are taken to the probe the above mentioned properties of the proposed simultaneous model. Integration of model function gives finite values everywhere irrespective of the point of degeneracy being present or not. This is just opposite to derivative evaluation.

Intrinsic Triple Degeneracy Point Bounded by Nodal Surfaces in Chiral Photonic Crystal.

In periodic systems, band degeneracies are typically protected and classified by spatial symmetries. However, in photonic systems, the Γ point at zero frequency is an intrinsic degeneracy due to the polarization degree of freedom of electromagnetic waves. For chiral photonic crystals, such an intrinsic degeneracy carries ±2 chiral topological charge while having linear band dispersions, different from the general perception of charge-2 nodes being associated with quadratic dispersions. Here, we show that these topological characters originate from the spin-1 Weyl point at zero frequency node of triple degeneracy, due to the existence of an electrostatic flat band. Such a topological charge at zero frequency is usually buried in bulk band projections and has never been experimentally observed. To address this challenge, we introduce space-group screw symmetries in the design of chiral photonic crystal, which makes the Brillouin zone boundary an oppositely charged nodal surface enclosing the Γ point. As a result, the emergent Fermi arcs are forced to connect the projections of these topological singularities, enabling their experimental observation. The number of Fermi arcs then directly reveals the embedded topological charge at zero frequency.

Thermal Transport and Thermoelectric Properties of Rb2PdX6 (X=Cl, Br) from First‐principles Study

AbstractVacancy‐ordered double perovskites are characterized by stable structure, non‐toxicity, low thermal conductivity, and large Seebeck coefficient, which are potential thermoelectric materials. In this work, the thermal transport and thermoelectric properties of Rb2PdX6(X=Cl, Br) are theoretically investigated. We find that the calculated thermal conductivity is underestimated from the phonon Boltzmann transport equation (PBTE) since the mean free path of a large number of phonons is less than the Ioffe‐Regel limit. The room‐temperature thermal conductivities of Rb2PdCl6 and Rb2PdBr6 are 0.42 W m−1 K−1 and 0.22 W m−1 K−1 based on two‐channel phonon transport mode, respectively. The thermal conductivity of Rb2PdBr6 is almost unchanged with the temperature. Rb2PdX6(X=Cl, Br) have large Seebeck coefficient for p‐type doping due to triple degeneracy in the VBM and the thermoelectric property of p‐type doping is better than that for n‐type doping. The ab initio scattering and transport program (AMSET) is used to calculate the electron scattering rate. At 700 K, the maximum ZT of p‐type doping Rb2PdCl6 and Rb2PdBr6 can reach to 0.51 and 1.31, respectively. Replacement of Cl by Br results in lower thermal conductivity and smaller bandgap, and thus larger the maximum ZT value. This work demonstrates that Rb2PdBr6 is a potential high‐performance p‐type thermoelectric perovskite material.

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.

Hidden breathing kagome topology in hexagonal transition metal dichalcogenides

A kagome lattice, formed by triangles of two different directions, is known to have many emergent quantum phenomena. Under the breathing anisotropy of bond strengths, this lattice can become a higher-order topological insulator (HOTI), which hosts topologically protected corner states. Experimental realizations of HOTI on breathing kagome lattices have been reported for various artificial systems, but not for simple natural materials with an electronic breathing kagome lattice. Here we prove that a breathing kagome lattice and HOTI are hidden inside the electronic structure of hexagonal transition metal dichalcogenides (h-TMD). Due to the trigonal prismatic symmetry, $s{p}^{2}$-like hybrid $d$ orbitals create an electronic kagome lattice with anisotropic intersite and on-site hopping interactions. We demonstrate that HOTI h-TMD triangular nanoflakes host topologically protected corner states, which could be quantum-mechanically entangled with triple degeneracy. Because $\mathrm{h}\text{\ensuremath{-}}\mathrm{TM}{\mathrm{D}}_{\mathrm{S}}$ are easily synthesizable and stable with a large bandgap at ambient conditions, our findings open an avenue for quantum physics based on simple condensed matter systems.

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.

Structure-insensitive zero refraction effect based on a two-dimensional annular photonic crystal

The zero refraction (ZR) effect based on a two-dimensional square lattice air-ring photonic crystal (PC) provides a powerful mean for beam steering. However, most previous studies have reported ZR performance mainly for transverse-magnetic polarization. This paper proposes a typical non-accidental triple degeneracy at Dirac-like point for transverse-electric polarization, which in case of the structure-insensitive parameters. As compared to conventional PCs, the proposed PC exhibits a totally different performance attributed to its robust structure, which greatly reduces the difficulty of processing technology.

Merit functions and measurement schemes for single parameter depolarization models.

Mueller polarized bi-directional scattering distribution functions (pBSDFs) are 4 × 4 matrix-valued functions which depend on acquisition geometry. A widely used backscattering pBSDF model proposed by Priest and Meier [Opt. Eng.41, 988 (2002)10.1117/1.1467360] is a weighted sum between a Fresnel matrix and an ideal depolarizer. This work's main contribution is relating the relative weight between an ideal depolarizer and Fresnel matrix to a single depolarization parameter. Rather than a 16-dimensional matrix norm, this parameter can form a one-dimensional merit function. Then, instead of a full Mueller matrix measurement, a scheme for pBSDF fitting to only two polarimetric measurements is introduced. Depolarization can be mathematically expressed as the incoherent addition of coherent states [J. Opt. Soc. Am. A30, 691 (2013)10.1364/JOSAA.30.000691]. This work shows that, for a Mueller matrix to be in the span of a Fresnel matrix and an ideal depolarizer, the weights in the incoherent addition are triply degenerate. This triple degeneracy is observed in five different colored opaque plastics treated with nine different surface textures and measured at varying acquisition geometries and wavebands.

Investigation of the thermoelectric properties of Lithium-Aluminium-Silicide (LiAlSi) compound from first-principles calculations

Using Density Functional Theory (DFT) and Boltzmann’s transport theory, we determined the band structure and thermoelectric properties in Lithium-Aluminium-Silicide compound. The band-structure calculations showed a triple point degeneracy at Γ point due to the energy difference between the Lithium-Aluminium ligand formation. With the valence band maximum at Γ and the conduction band minimum at X, we observed an indirect bandgap along the Γ→X transition. Observation of the transport properties of LiAlSi at temperatures below 500K, showed several shoulders or minor pre-peaks which are more pronounced in the low energy range and might be due to intrinsic magnetization of the constituent elements. The Seebeck coefficient has the highest value of 325 μVK−1 in the p-type region, while the n-type doping has a maximum value of 297 μVK−1. In detail, the calculated results of Seebeck coefficient, thermal and electrical conductivity, power factor and electronic fitness function are in this work.

Optical conductivity of triple point fermions

As a low-energy effective theory on non-symmorphic lattices, we consider a generic triple point fermion Hamiltonian, which is parameterized by an angular parameter λ. We find strong λ dependence in both Drude and interband optical absorption of these systems. The deviation of the T 2 coefficient of the Drude weight from Dirac/Weyl fermions can be used as a quick way to optically distinguish the triple point degeneracies from the Dirac/Weyl degeneracies. At the particular λ = π/6 point, we find that the ‘helicity’ reversal optical transition matrix element is identically zero. Nevertheless, deviating from this point, the helicity reversal emerges as an absorption channel.

Exceptional points make an astroid in non-Hermitian Lieb lattice: Evolution and topological protection

An astroid-shaped loop of exceptional points (EPs), comprising four cusps, is found to spawn from the triple degeneracy point in the Brillouin zone (BZ) of a Lieb lattice with nearest-neighbor hoppings when non-Hermiticity is introduced. The occurrence of the EP loop is due to the realness of the discriminant which is guaranteed by the non-Hermitian chiral symmetry. The EPs at the four cusps involve the coalescence of three eigenstates, which is the combined result of the non-Hermitian chiral symmetry and mirror-T symmetry. The EP loop is exactly an astroid in the limit of an infinitesimal non-Hermiticity. The EP loop expands from the $\phantom{\rule{4pt}{0ex}}M$ point with increasing non-Hermiticity and splits into two EP loops at a critical non-Hermiticity. The further increase of non-Hermiticity contracts the two EP loops towards and finally to two EPs at the $\phantom{\rule{4pt}{0ex}}X$ and $\phantom{\rule{4pt}{0ex}}Y$ points in the BZ, accompanied by the emergence of Dirac-like cones. The two EPs vanish at a larger non-Hermiticity. The EP loop disappears and several discrete EPs are found to survive when next-nearest hoppings are introduced to break the non-Hermitian chiral symmetry. A topological invariant called the discriminant number is used to characterize their robustness against perturbations. Both discrete EPs and those on the EP loop(s) are found to show anisotropic asymptotic behaviors. Finally, the experimental realization of the Lieb lattice using a coupled waveguide array is discussed.

Topological character of three-dimensional nexus triple point degeneracies

Recently a generic class of three-dimensional band structures was identified that host two-fold line degeneracies meeting at three-fold or triple point degeneracies, which resist the usual topological characterization of isolated point degeneracies as in Dirac/Weyl semimetals. For these so-called ``Nexus" fermions which lie beyond Dirac/Weyl fermions, we lay out several concepts to characterize the wavefunction geometry and spell out its topology. Our approach is based on an understanding of the analyticity properties of Nexus wavefunctions building on a two-dimensional analogue studied recently by us. We use this to write down a homological classification of various Nexus triple point degeneracies in three dimensions.

Thermoelectric transport properties in chalcogenides ZnX (X=S, Se): From the role of electron-phonon couplings

Electron-phonon coupling (EPC) is a key factor for thermoelectric properties of materials. In this paper, the thermoelectric properties of zinc-blende chalcogenides (p-type) ZnS and ZnSe have been studied through full evaluation of EPC from first-principles, including the influences on both electrical and thermal transport. We find that the polar longitudinal optical phonon scattering is the dominant mechanism for electrical transport. Due to the triple degeneracy near the valence band maximum, the inter-band scattering also has detrimental contributions to the electrical conductivities. For phonon transport, it shows that the lattice thermal conductivity can be reduced by the electron-phonon scattering significantly at high carrier concentrations (e.g., at 300 K with 1021 cm−3 of hole, the reduction is ∼24.9% for ZnS and ∼28.4% for ZnSe, respectively). Finally, the p-type thermoelectric figure of merit (ZT) of two systems have been obtained, which are 0.129 for ZnS and 0.141 for ZnSe, at 700 K with their respective optimal hole concentrations. Our work provides a complete and in-depth study of thermoelectric properties in chalcogenides ZnX from the role of EPC. The results suggest EPC plays an important role on the thermoelectric properties and thus full evaluation of EPC is necessary especially for polar materials.

Hexagonal wurtzite MnO in ferromagnetic state: A magnetic topological spin-gapless Weyl semimetal

Magnetic topological materials have attracted increasingly attentions in recent years due to their exotic electronic behaviors emerging from the couplings of topological, magnetic, and crystalline symmetries. In this work, based on the first-principles calculations, we propose that hexagonal wurtzite MnO is a magnetic topological spin-gapless semi-half-metal with two pairs of type-I Weyl fermions near the Fermi level in ferromagnetic state, which is a promising candidate material in spintronic and piezoelectric applications. In the absence of spin-orbit coupling (SOC), it hosts one triple degeneracy point (TP) in the irreducible Brillouin zone. Owing to weak SOC, the TP splits into two type-I Weyl points that are very close to each other. The Fermi arc surface states connecting the projected Weyl points with opposite chirality are observed. Our results therefore provide a wonderful platform to study the interplay of magnetism and topology.

Quasi-triply-degenerate states and zero refractive index in two-dimensional all-dielectric photonic crystals.

We introduce the concept of a quasi-triply-degenerate state (QTDS) and demonstrate its relation to an effective zero refractive index (ZRI) in a two-dimensional (2D) square lattice photonic crystal (PC) of all dielectric pillars. A QTDS is characterized by a triple band structure (TBS), wherein two of the bands manifest a linear dispersion around the Γ-point, i.e. a Dirac-like cone, while the third is a flat zero refractive index (ZRI) band with a frequency that is degenerate with one of the other bands. Significantly, we find that while triple degeneracy of the bands is not observed, the three bands approach one another so close that the observable properties of PCs adapted to the QTDS frequency perform as expected of a ZRI material. We closely examine the ZRI band at the Γ-point and show that by varying the PC material and structure parameters, the ZRI band behavior extends over a wide range of dielectric refractive indices enabling materials made with polymeric constituents. Moreover, the ZRI characteristics are robust and tolerant over a range of frequencies. Furthermore, the computational screening we employ to identify QTDS parameters enables the rational design of low-loss 2D ZRI materials for a broad range of photonic applications, including distributing a common reference phase, cloaking and focusing light.

High-degeneracy points protected by site-permutation symmetries

Space-group symmetries dictate the energy degeneracy of quasiparticles (e.g., electronic, photonic) in crystalline structures. For spinless systems, there can only be double or triple degeneracies protected by these symmetries, while other degeneracies are usually taken as accidental. In this Rapid Communication we show that it is possible to design higher degeneracies exploring site permutation symmetries. These design principles are shown to be satisfied in previously studied lattices, and new structures are proposed with three, four, and five degeneracy points for spinless systems. The results are general and apply to different quasiparticle models. Here, we focus on a tight-binding approach for the electronic case as a proof of principle. The resulting high-degeneracy points are protected by the site-permutation symmetries, yielding pseudospin-1 and -2 Dirac fermions. The strategy proposed here can be used to design lattices with high-degeneracy points in electronic (e.g., metal-organic frameworks), photonic, phononic, magnonic, and cold-atom systems.

Body-centered-cubic structure and weak anharmonic phonon scattering in tungsten

It was recently found that the anharmonic phonon–phonon scattering in tungsten is extremely weak at high frequencies, leading to a predominance of electron–phonon scattering and consequently anomalous phonon transport behaviors. In this work, we calculate the phonon linewidths of W along high-symmetry directions from first principles. We find that the weak phonon–phonon scattering can be traced back to two factors. The first is the triple degeneracy of the phonon branches at the P and H points, a universal property of elemental body-centered-cubic (bcc) structures. The second is a relatively isotropic character of the phonon dispersions. When both are met, phonon–phonon scattering rates must vanish at the P and H points. The weak phonon–phonon scattering feature is also applicable to Mo and Cr. However, in other elemental bcc substances like Na, the isotropy condition is violated due to the unusually soft character of the lower transverse acoustic phonon branch along the Γ-N direction, opening emission channels and leading to much stronger phonon–phonon scattering. We also look into the distributions of electron mean-free paths (MFPs) at room temperature in tungsten, which can help engineer the resistivity of nanostructured W for applications such as interconnects.

Bound states in the continuum and Fano resonances in the Dirac cone spectrum

We consider light scattering by two dimensional arrays of high-index dielectric spheres arranged into the triangular lattice. It is demonstrated that in the case a triple degeneracy of resonant leaky modes in the Gamma-point the scattering spectra exhibit a complicated picture of Fano resonances with extremely narrow line-width. The Fan features are explained through coupled mode theory for a Dirac cone spectrum as a signature of optical bound states in the continuum (BIC). It is found that the standing wave in-Gamma BIC induces a ring of off-Gamma BICs due to different scaling laws for real and imaginary parts of the resonant eigenfrequencies in the Dirac cone spectrum. A quantitative theory of the spectra is proposed.

Floquet topological phase transition in the α−T3 lattice

We investigate topological characteristics of the photon-dressed band structure of $\alpha$-$\mathcal{T}_3$ lattice on being driven by off-resonant circularly polarized radiation. We obtain exact analytical expressions of the quasienergy bands over the first Brillouin zone. The broken time-reversal symmetry caused by the circularly polarized light lifts the triple point degeneracy completely at both the Dirac points. The gaps become unequal at $ {\bf K}$ and ${\bf K}^{\prime}$ (except at $\alpha=0$ and 1), which reveals the absence of inversion symmetry in the system. At $\alpha=1/\sqrt{2}$, the gap between flat and valence bands closes at ${\bf K}$, while that between conduction and flat bands closes at ${\bf K}^{\prime}$, thereby restoring a semimetalic phase. At the gap closing point ($\alpha=1/\sqrt{2}$) which is independent of the radiation amplitude, there is a reappearance of low-energy Dirac cones around ${\bf K}$ and ${\bf K}^{\prime}$ points. Under the influence of the circularly polarized radiation, the $\alpha$-$\mathcal{T}_3$ lattice is transformed from semimetal to a Haldane-like Chern insulator characterized by non-zero Chern number. The system undergoes a topological phase transition from $\mathcal{C} = 1 (-1)$ to $\mathcal{C}=2 (-2)$ at $\alpha =1/\sqrt{2}$, where $\mathcal{C}$ is the Chern number of the valence (conduction) band. This sets an example of a multiband system having larger Chern number. These results are supported by the appearance of chiral edge states in irradiated $\alpha$-$\mathcal{T}_3$ nanoribbon.