Degenerate Points Research Articles

A bulk-edge correspondence between the second Chern number and the spectral flow

A one-parameter family of semi-quantum Hamiltonians admitting the time-reversal symmetry is defined on the product S2(r1)×S2(r2) of two-spheres of respective radii r1 and r2, and a corresponding one-parameter family of quantum Hamiltonians is defined in terms of the angular momentum operators associated with respective spheres. The second Chern number of the eigen-vector bundle associated with a positive eigenvalue of the semi-quantum Hamiltonian is a piecewise constant function of the parameter with jumps at singular parameter values leading to the emergence of total degeneracy points for the eigenvalues. Correspondingly, the quantum Hamiltonian has edge-state eigenvalues which, as functions of the parameter, pass the zero energy level at the singular parameter values and the bulk-state eigenvalues prove to correspond to eigenvalues of the semi-quantum Hamiltonian. It is shown that the second-Chern number c2(T) at a non-singular parameter value T is in one-to-one correspondence with the spectral flow as a function of the end point of the interval (−∞,T). The bulk-edge correspondence is also shown to hold between the linearized semi-quantum and quantum systems, where the semi-quantum Hamiltonian is linearized at each of the degeneracy points of the eigenvalues and quantized to define linear quantum Hamiltonians. On this level, the spectral flow of each linear quantum Hamiltonian with a parameter μ proves to coincide with the index of the linear quantum Hamiltonian with μ=0.

Center conditions to find certain degenerate centers with characteristic directions

We consider the two-dimensional autonomous systems of differential equations where the origin is a monodromic degenerate singular point, i.e., with null linear part. In this work we give two heuristic procedures to obtain some center conditions (perhaps not necessary) for certain degenerate centers at the origin although they have characteristic directions.

Equivalent resolvents of Douglas-Rachford splitting and other operator splitting algorithms: a unified degenerate proximal point analysis

We introduce a generalized proximal point algorithm, and perform a detailed convergence analysis with the focus on the case of degenerate metric. The degeneracy leads to a well-defined resolvent form restricted to a reduced dimensional space. This approach unifies the algorithmic structures of Douglas-Rachford splitting and other related operator splitting schemes. Various aspects of these algorithms, in particular, the convergence of Douglas-Rachford splitting in terms of the solution itself and Chambolle-Pock algorithm under the limit setting, are investigated or revisited by the unified degenerate proximal point analysis.

Design of High Precision Temperature Sensor with Current Gain Compensation Technology for On-Chip Application

The integration of on-chip temperature sensors within various systems, industrial Internet of Things (IoT), and wireless sensor networks is greatly facilitated by their small size, cost-effectiveness, and capability to provide direct digital output. However, the diverse application scenarios pose challenges in designing these sensors. On one hand, real-time clock calibration demands high-precision temperature sensors, while on-chip heat management emphasizes compactness and low-voltage operation. Additionally, streamlining the calibration cost for mass production holds significant practical value. Addressing these challenges, this study systematically investigates on-chip complementary metal-oxide-semiconductor (CMOS) temperature sensors based on distinct signal domains processed by temperature readout circuits. Specifically, the research commences by analyzing the issues of several degeneracy points in the front-end circuit of a bipolar junction transistor (BJT) temperature sensor with current gain compensation technology. To address the intricate design challenges in advanced technologies and calibration complexities in industrial applications, dynamic component matching, current gain compensation, and chopper stabilization are harnessed. A novel dynamic current gain canceling technique for temperature readout is introduced, enhancing temperature measurement accuracy without incurring additional power consumption or area overhead. Ultimately, an all-digital CMOS temperature sensor is realized using the SMIC 55 nm CMOS process. Occupying a mere 0.29 mm2 of core area, the design operates efficiently across a wide supply voltage range of 1.2 V to 3.6 V. Covering a temperature spectrum from −40 °C to 125 °C, the sensor demonstrates a calibration error of just ±0.7 °C. This achievement is attributed to the incorporation of the proposed dynamic current gain compensation technique.

Requisites on material viscoelasticity for exceptional points in passive dynamical systems

Recent progress in non-Hermitian physics and the notion of exceptional point (EP) degeneracies in elastodynamics have led to the development of novel metamaterials for the control of elastic wave propagation, hypersensitive sensors, and actuators. The emergence of EPs in a parity-time symmetric system relies on judiciously engineered balanced gain and loss mechanisms. Creating gain requires complex circuits and amplification mechanisms, making engineering applications challenging. Here, we report strategies to achieve EPs in passive non-Hermitian elastodynamic systems with differential loss derived from viscoelastic materials. We compare different viscoelastic material models and show that the EP emerges only when the frequency-dependent loss-tangent of the viscoelastic material remains nearly constant in the frequency range of operation. This type of loss tangent occurs in materials that undergo stress-relaxation over a broad spectrum of relaxation times, for example, materials that follow the Kelvin–Voigt fractional derivative (KVFD) model. Using dynamic mechanical analysis, we show that a few common viscoelastic elastomers, such as polydimethylsiloxane and polyurethane rubber, follow the KVFD behavior such that the loss tangent becomes almost constant after a particular frequency. The material models we present and the demonstration of the potential of a widely available material system in creating EPs pave the way for developing non-Hermitian metamaterials with hypersensitivity to perturbations or enhanced emissivity.

Photonic Dirac cone and topological transition in a moving dielectric slab

The moving media theory is applied to a photonic confined structure which is a continuous dielectric slab waveguide with the uniaxial anisotropy and without the discrete translational symmetry. The moving effect not only brings about non-reciprocity to the whole photonic band structure in the co-moving and counter-moving directions, but also leads to the topological transition of local degenerate points within the band diagram. We demonstrate through calculation that the type-II Dirac point can be turned into type-I Dirac point when the uniaxial slab is moving over certain speed. Our results provide a new approach to regulate the topology of degeneracy for two-dimensional photonic bands in the continuous translational symmetry condition.

Level crossings, attractor points and complex multiplication

We study the complex structure moduli dependence of the scalar Laplacian eigenmodes for one-parameter families of Calabi-Yau n-folds in ℙn+1. It was previously observed that some eigenmodes get lighter while others get heavier as a function of these moduli, which leads to eigenvalue crossing. We identify the cause for this behavior for the torus. We then show that at points in a sublocus of complex structure moduli space where Laplacian eigenmodes cross, the torus has complex multiplication. We speculate that the generalization to arbitrary Calabi-Yau manifolds could be that level crossing is related to rank one attractor points. To test this, we compute the eigenmodes numerically for the quartic K3 and the quintic threefold, and match crossings to CM and attractor points in these varieties. To quantify the error of our numerical methods, we also study the dependence of the numerical spectrum on the quality of the Calabi-Yau metric approximation, the number of points sampled from the Calabi-Yau variety, the truncation of the eigenbasis, and the distance from degeneration points in complex structure moduli space.

Experimental Realization of a Three-Dimensional Dirac Semimetal Phase with a Tunable Lifshitz Transition in Au_{2}Pb.

Three-dimensional Dirac semimetals are an exotic state of matter that continue to attract increasing attention due to the unique properties of their low-energy excitations. Here, by performing angle-resolved photoemission spectroscopy, we investigate the electronic structure of Au_{2}Pb across a wide temperature range. Our experimental studies on the (111)-cleaved surface unambiguously demonstrate that Au_{2}Pb is a three-dimensional Dirac semimetal characterized by the presence of a bulk Dirac cone projected off-center of the bulk Brillouin zone (BZ), in agreement with our theoretical calculations. Unusually, we observe that the bulk Dirac cone is significantly shifted by more than 0.4eV to higher binding energies with reducing temperature, eventually going through a Lifshitz transition. The pronounced downward shift is qualitatively reproduced by our calculations indicating that an enhanced orbital overlap upon compression of the lattice, which preserves C_{4} rotational symmetry, is the main driving mechanism for the Lifshitz transition. These findings not only broaden the range of currently known materials exhibiting three-dimensional Dirac phases, but also show a viable mechanism by which it could be possible to switch on and off the contribution of the degeneracy point to electron transport without external doping.

Flux-periodic oscillations in proximitized core–shell nanowires

Flux-periodic oscillations of the superconducting gap in proximitized core–shell nanowires are explored. Periodicity of oscillations in the energy spectrum of a cylindrical nanowire is compared with nanowires having hexagonal and square cross-section geometry, along with the effects of Zeeman and Rashba spin–orbit interaction. A transition between h/e and h/2e periodicity is found and shown to be dependent on the chemical potential, with correspondence to degeneracy points of the angular momentum quantum number. For a thin shell of a square nanowire, solely h/e periodicity is found in the infinite wire spectrum and shown to result from energy separation between the lowest groups of excited states.

Catalog of noninteracting tight-binding models with two energy bands in one dimension

We classify Hermitian tight-binding models describing noninteracting electrons on a one-dimensional periodic lattice with two energy bands. To do this, we write a generalized Rice-Mele model with two orbitals per unit cell, including all possible complex-valued long-range hoppings consistent with Hermicity. We then apply different forms of time-reversal, charge-conjugation and chiral symmetry in order to constrain the parameters, resulting in an array of possible models in different symmetry classes. For each symmetry class, we define a single, canonical form of the Hamiltonian and identify models that are related to the canonical form by an off-diagonal unitary transformation in the atomic basis. The models have either symmorphic or nonsymmorphic nonspatial symmetries (time $T$, chiral and charge-conjugation). The nonsymmorphic category separates into two types of state of matter: an insulator with a $\mathbb{Z}_2$ topological index in the absence of nonsymmorphic time-reversal symmetry or, in the presence of nonsymmorphic time-reversal symmetry, a metallic state. The latter is an instance of Kramer's degeneracy with one degeneracy point in the Brillouin zone as opposed to no degeneracy points in symmorphic systems with $T^2 = 1$ and two in symmorphic systems with $T^2 = - 1$.

On the well-posed variational principle in degenerate point particle systems using embeddings of the symplectic manifold

Abstract A methodology on making the variational principle well-posed in degenerate systems is constructed. In the systems including higher-order time derivative terms being compatible with Newtonian dynamics, we show that a set of position variables of a coordinate system of a given system has to be fixed on the boundaries and that such systems are always Ostrogradski stable. For these systems, Frobenius integrability conditions are derived in explicit form. Relationships between integral constants indicated from the conditions and boundary conditions in a given coordinate system are also investigated by introducing three fundamental correspondences between Lagrange and Hamilton formulation. Based on these ingredients, we formulate problems that have to be resolved to realize the well-posedness in the degenerate systems. To resolve the problems, we compose a set of embeddings that extract a subspace holding the symplectic structure of the entire phase space in which the variational principle should be well-posed. Using these embeddings, we establish a methodology to set appropriate boundary conditions that the well-posed variational principle demands. Finally, we apply the methodology to examples and summarize this work as a three-step procedure such that one can use just by following it.

New hybrid quadrature schemes for weakly singular kernels applied to isogeometric boundary elements for 3D Stokes flow

This work proposes four novel hybrid quadrature schemes for the efficient and accurate evaluation of weakly singular boundary integrals (1/r kernel) on arbitrary smooth surfaces. Such integrals appear in boundary element analysis for several partial differential equations including the Stokes equation for viscous flow and the Helmholtz equation for acoustics. The proposed quadrature schemes apply a Duffy transform-based quadrature rule (Duffy, 1982) to surface elements containing the singularity and classical Gaussian quadrature to the remaining elements. Two of the four schemes additionally consider a special treatment for elements near to the singularity, where refined Gaussian quadrature and a new moment-fitting quadrature rule are used.The hybrid quadrature schemes are systematically studied on flat B-spline patches and on NURBS spheres considering two different sphere discretizations: An exact single-patch sphere with degenerate control points at the poles and an approximate discretization that consist of six patches with regular elements. The efficiency of the quadrature schemes is further demonstrated in boundary element analysis for Stokes flow, where steady problems with rotating and translating curved objects are investigated in convergence studies for both, mesh and quadrature refinement. Much higher convergence rates are observed for the proposed new schemes in comparison to classical schemes.

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.

Analytically integrable system orbitally equivalent to a semi-quasihomogeneous system

For perturbations of integrable non-Hamiltonian quasi-homogeneous planar vector field whose origin is a non-degenerate singular point, orbital linearization and analytic integrability are equivalent. We show a class of analytically integrable vector fields whose origin is a degenerate singular point which is orbitally equivalent to a semi-quasi-homogeneous system, that is, it is not orbital equivalent to its lowest-degree quasi-homogeneous term.

Emerging Weyl Semimetal States in Ternary TaPxAs1−x Alloys: Insights from Electronic and Topological Analysis

AbstractThis study presents a thorough analysis of the electronic structures of the TaPxAs1−x series of compounds, which are of significant interest due to their potential as topological materials. Using a combination of first principles and Wannier‐based tight‐binding methods, this study investigates both the bulk and surface electronic structures of the compounds for varying compositions (x = 0, 0.25, 0.50, 0.75, 1), with a focus on their topological properties. By using chirality analysis, (111) surface electronic structure analysis, and surface Fermi arcs analysis, it is established that the TaPxAs1−x compounds exhibit topologically nontrivial behavior, characterized as Weyl semimetals (WSMs). The effect of spin–orbit coupling (SOC) on the topological properties of the compounds is further studied. In the absence of SOC, the compounds exhibit linearly dispersive fourfold degenerate points in the first Brillouin zone (FBZ) resembling Dirac semimetals. However, the introduction of SOC induces a phase transition to WSM states, with the number and position of Weyl points (WPs) varying depending on the composition of the alloy. For example, TaP has 12 WPs in the FBZ. The findings provide novel insights into the electronic properties of TaPxAs1−x compounds and their potential implications for the development of topological materials for various technological applications.

Local regularity for nonlinear elliptic and parabolic equations with anisotropic weights

AbstractThe main purpose of this paper is to capture the asymptotic behaviour for solutions to a class of nonlinear elliptic and parabolic equations with the anisotropic weights consisting of two power-type weights of different dimensions near the degenerate or singular point, especially covering the weighted p-Laplace equations and weighted fast diffusion equations. As a consequence, we also establish the local Hölder estimates for their solutions in the presence of single power-type weights.

Superimposition of topological charges between vortex beams and singular points of photonic crystal slabs

The bound state in the continuum (BIC) is a singular point of polarization in the momentum space of a periodic structure. It has been demonstrated that vortex beams can be generated by utilizing the polarization vortex around BIC based on the geometric phase in momentum space. In this letter, we propose a reflective photonic crystal (PhC) to generate a vortex beam and demonstrate the superimposition of topological charges in the momentum space between PhC and the vortex beams. In addition to BICs, we demonstrate that other singularities like degenerate points can also be used to generate vortex beams.

Theory of Caroli–de Gennes–Matricon analogs in full-shell hybrid nanowires

Full-shell nanowires are hybrid nanostructures consisting of a semiconducting core encapsulated in an epitaxial superconducting shell. When subject to an external magnetic flux, they exhibit the Little-Parks (LP) phenomenon of flux-modulated superconductivity, an effect connected to the physics of Abrikosov vortex lines in type-II superconductors. We show theoretically that full-shell nanowires can host subgap states that are a variant of the Caroli-de Gennes-Matricon (CdGM) states in vortices. These CdGM analogs are shell-induced Van Hove singularities in core subbands. We elucidate their structure, parameter dependence and behavior in tunneling spectroscopy through a series of models of growing complexity. Using microscopic numerical simulations, we show that CdGM analogs exhibit a characteristic skewness towards higher flux values inside non-zero LP lobes resulting from the interplay of three ingredients. First, the orbital coupling to the field shifts the energy of the CdGM analogs proportionally to the flux and to their generalized angular momentum. Second, CdGM analogs coalesce into degeneracy points at flux values for which their corresponding radial wavefunctions are threaded by an integer multiple of the flux quantum. And third, the average radii of all CdGM-analog wavefunctions inside the core are approximately equal and are controlled by the electrostatic band bending at the core/shell interface. As the average radius moves away from the interface, the degeneracy points shift towards larger fluxes from the center of the LP lobes, causing the skewness. This analysis provides a transparent interpretation of the nanowire spectrum that allows to extract microscopic information by measuring the number and skewness of CdGM analogs. Moreover, it allows to derive an efficient Hamiltonian of the full-shell nanowire in terms of a modified hollow-core model at the average radius.

Lasing at a stationary inflection point

The concept of lasers based on the frozen mode regime in active periodic optical waveguides with a 3rd-order exceptional point of degeneracy (EPD) is advanced. The frozen mode regime in a lossless and gainless waveguide is associated with a stationary inflection point (SIP) in the Bloch dispersion relation, where three Bloch eigenmodes coalesce forming the frozen mode. As a practical example, we consider an asymmetric serpentine optical waveguide (ASOW). An ASOW operating near the SIP frequency displays a large group delay of a non-resonant nature that scales as the cube of the waveguide length, leading to a strong gain enhancement when active material is included. Therefore, a laser operating in the close vicinity of an SIP has a gain threshold that scales as a negative cube of the waveguide length. We determine that this scaling law is maintained in the presence of small distributed losses, such as radiation associated with waveguide bends and roughness. In addition, we show that although gain causes a distortion in the modes coalescing at the SIP, the properties of the frozen mode are relatively resistant to such small perturbations and we still observe a large degree of exceptional degeneracy for gain values that bring the system above threshold. Finally, our study also reveals that lasing near an SIP is favored over lasing near a photonic band edge located in close proximity to the SIP. In particular, we observe that an SIP-induced lasing in an ASOW displays lower gain threshold compared to lasing near the photonic regular band edge (RBE), even though the SIP resonance has a lower quality factor than the RBE resonance.

Enhanced eigenvector sensitivity and algebraic classification of sublattice-symmetric exceptional points

Exceptional points (EPs) are degeneracy of non-Hermitian Hamiltonians, at which the eigenvalues, along with their eigenvectors, coalesce. Their orders are given by the Jordan decomposition. Here, we focus on higher-order EPs arising in fermionic systems with a sublattice symmetry, which restricts the eigenvalues of the Hamitlonian to appear in pairs of $\lbrace E, -E\rbrace $. Thus, a naive prediction might lead to only even-order EPs at zero energy. However, we show that odd-order EPs can exist and exhibit enhanced sensitivity in the behaviour of eigenvector-coalescence in their neighbourhood, depending on how we approach the degenerate point. The odd-order EPs can be understood as a mixture of higher- and lower-valued even-order EPs. Such an anomalous behaviour is related to the irregular topology of the EPs as the subspace of the Hamiltonians in question, which is a unique feature of the Jordan blocks. The enhanced eigenvector sensitivity can be described by observing how the quantum distance to the target eigenvector converges to zero. In order to capture the eigenvector-coalescence, we provide an algebraic method to describe the conditions for the existence of these EPs. This complements previous studies based on resultants and discriminants, and unveils heretofore unexplored structures of higher-order exceptional degeneracy.