Academic literature on the topic 'Crossing symmetric dispersion relation'

We review recent theoretical progress in the understanding and prediction of novel topological semimetals. Topological semimetals define a class of gapless electronic phases exhibiting topologically stable crossings of energy bands. Different types of topological semimetals can be distinguished on the basis of the degeneracy of the band crossings, their codimension (e.g., point or line nodes), and the crystal space group symmetries on which the protection of stable band crossings relies. The dispersion near the band crossing is a further discriminating characteristic. These properties give rise to a wide range of distinct semimetal phases such as Dirac or Weyl semimetals, point or line node semimetals, and type I or type II semimetals. In this review we give a general description of various families of topological semimetals, with an emphasis on proposed material realizations from first-principles calculations. The conceptual framework for studying topological gapless electronic phases is reviewed, with a particular focus on the symmetry requirements of energy band crossings, and the relation between the different families of topological semimetals is elucidated. In addition to the paradigmatic Dirac and Weyl semimetals, we pay particular attention to more recent examples of topological semimetals, which include nodal line semimetals, multifold fermion semimetals, and triple-point semimetals. Less emphasis is placed on their surface state properties, their responses to external probes, and recent experimental developments.

ABSTRACT Long-term statistical simulations of the past evolution of high-inclination Centaurs showed that their orbits tend to be polar with respect to the Solar system’s invariable plane over a large semimajor axis range in trans-neptunian space. Here, we lay the analytical foundation of the study of the inclination pathways of planet-crossing asteroids that explains these findings. We show that the Tisserand relation partitions the inclination–semimajor axis parameter space of the three-body problem into distinct regions depending on the asteroid’s Tisserand parameter T or equivalently its orbital inclination I∞ far from the planet. The Tisserand relation shows that asteroids with I∞ > 110° (T < −1) cannot be injected inside the planet’s orbit. Injection on to retrograde orbits and high-inclination prograde orbits occurs inside the inclination corridor 45° ≤ I∞ ≤ 110° (−1 ≤ T ≤ 2). Inclination dispersion across the inclination pathway for moderate and high inclinations is explained by the secular perturbations from the planet and is smallest for polar orbits. When a planet-crossing asteroid temporarily leaves the inclination pathway, its long-term evolution still depends on its Tisserand parameter as evidenced by its eccentricity dispersion. Simulations of asteroid orbits using the equations of motion with Neptune as the perturbing planet confirm these results for moderate to high inclinations, forward and backward in time because the Tisserand relation is time-independent. The Tisserand inclination pathways will provide important constraints on comet delivery from the outer Solar system as well as on the possible presence of unknown planets in trans-neptunian space.

Spin wave propagation over long distances in confined ferromagnetic strip lines exhibiting magnetoelasticity opens up promising perspectives for device applications. Domain walls as natural spin wave channels increase the spin wave propagation distance. We calculate the magnetic and elastic modes through micromagnetic simulations and the dispersion relation of strip lines containing a Néel wall. We show that at the crossing points in the dispersion relation, two behaviors are observed: an anticrossing gap when a strong coupling is present or a gapless point when the magnetoelastic feedback cycle is not fulfilled. For the Néel wall-confined magnetic mode, the magnetic and elastic waves oscillate independently forming a gapless crossing point. For the domain modes, both behaviors are found. We discuss the gap existence based on the symmetry of the eigenmodes.

Abstract This paper suggests the use of a simple square root approximation to the dispersion relation of axially-symmetric electron surface waves on cylindrical plasmas. The point is not merely to substitute the exact expression for the dispersion relation which involves a number of Bessel functions with a more tractable analytical approximant, but to cast the dispersion relation in a form useful in the comparison with other waves, such as water surface gravity waves and the associated tide-rip effect. The square root form of the dispersion relation is also of help in the analysis of surfactron plasmas, as it directly predicts a linear roll-off of electron density in the discharge.

The dispersion relations of energy bands in solids are characterized by their density of states, but a given density of states may originate from various band structures. We show how a spherically symmetric dispersion can be constructed for any one-band density of states. This method is applied to one-, two- and three-dimensional systems. It also serves to establish that any one-band spectrum with finite bandwidth can be obtained from a properly scaled dispersion relation in the limit of infinite dimensions.

A novel method of tuning the crossing angle, that is, the phase-matching condition, in condensed-phase CARS measurements has been presented. The method enables us to tune the phase-matching condition by simply tilting the sample cell with no need for adjusting the crossing angle of excitation laser beams over the range of about 600 cm−1.To this end, a relation between the crossing angle and the frequency difference between the two excitation lasers, which is associated with the Raman shift, has been derived in the normal dispersion region and has been experimentally examined.

This paper presents a method of analysing the dispersion relation and field shape of any type of wave field for which the dispersion relation is transcendental. The method involves replacing each transcendental term in the dispersion relation by a finite-product polynomial. The finite products chosen must be consistent with the low-frequency, low-wavenumber limit; but the method is nevertheless accurate up to high frequencies and high wavenumbers. Full details of the method are presented for a non-trivial example, that of anti-symmetric elastic waves in a layer; the method gives a sequence of polynomial approximations to the dispersion relation of extraordinary accuracy over an enormous range of frequencies and wavenumbers. It is proved that the method is accurate because certain gamma-function expressions, which occur as ratios of transcendental terms to finite products, largely cancel out, nullifying Runge’s phenomenon. The polynomial approximations, which are unrelated to Taylor series, introduce no spurious branches into the dispersion relation, and are ideal for numerical computation. The method is potentially useful for a very wide range of problems in wave theory and stability theory.

This paper presents a technique, based on a deferred approach to a limit, for analysing the dispersion relation for propagation of long waves in a curved waveguide. The technique involves the concept of an analytically satisfactory pair of Bessel functions, which is different from the concept of a numerically satisfactory pair, and simplifies the dispersion relations for curved waveguide problems. Details are presented for long elastic waves in a curved layer, for which symmetric and antisymmetric waves are strongly coupled. The technique gives high-order corrections to a widely used approximate dispersion relation based a kinematic hypothesis, and determines rigorously which of its coefficients are exact.

In this thesis, we develop new methods for the S matrix bootstrap in the context of 2-2 scattering amplitudes and four-point correlators in conformal field theories (CFTs). For 2-2 scattering in quantum field theories, we consider manifestly three-channel crossing symmetric dispersion relation (CSDR), unlike the two-channel symmetric fixed-t dispersion relation. We give simple derivations of certain known positivity conditions for effective field theories, including the null constraints, which lead to two-sided bounds and derive a general set of new nonperturbative inequalities. We derive the analyticity domain of the CSDR analogously to the Lehmann-Martin ellipse. We present a fascinating correspondence between an area of mathematics called geometric function theory (GFT) and the scattering amplitudes focusing on the case with O(N) global symmetry. We obtain two-sided bounds on Wilson coefficients of physical Pion amplitudes via positivity and GFT. Then we consider Bell correlations in light-by-light (LbyL) scattering at low energies. The known contributions in the Standard Model (SM) lead to Bell violation at all scattering angles except for a small transverse region, leading to a fine-tuning problem. Incorporating a light axion/axion-like particle (ALP) removes this problem and constrains the axion-coupling--axion-mass parameter space. In the second part of the thesis, we consider CSDR for Mellin amplitudes of scalar four-point correlators in conformal field theories. This allows us to rigorously set up the nonperturbative Polyakov bootstrap for the conformal field theories in Mellin space, fixing the contact term ambiguities in previous work. Our framework allows us to connect with the conceptually rich picture of the Polyakov blocks being identified with Witten diagrams in anti-de Sitter space. We also give two-sided bounds for Wilson coefficients for effective field theories in anti-de Sitter space. The derivation of the Polyakov bootstrap allows rigorous epsilon expansion solely from bootstrap principles.

The characteristics of planar waveguides with nonlinear layers has been extensively studied due to their potential application as all-optical switching and signal-processing element [1-3]. Here we present a generalized form of nonlinear dispersion equations for various nonlinear waveguide structures with Kerr-type films. By using the relation of the interface electric fields, we obtain the generalized form of nonlinear dispersion equations as an analytic form. The “generalized” means that the equation can represent most of nonlinear waveguide structures with up to five linear and/or nonlinear layers. For example, we apply the generalized dispersion equation to 4-types of nonlinear waveguide structures: I)the waveguides with a nonlinear core layer, II)the waveguides with two nonlinear core layers, III)the planar structures having a linear core with nonlinear surrounding media, and IV)the five-layer nonlinear waveguides having symmetric or asymmetric structures.

The appearance of modulational sidebands building up from noise has been reported when an intense cw or quasi-cw propagates in a fiber in the anomalous dispersion regime [1]. The modulation transforms the input wave into a train of pulses with ultra-high repetition rate. This process may be stimulated by seeding incoherently (i.e., by means of a different weak detuned laser) the modulational instability (MI) [2]. However the experiments and the early theory on MI [3] have led to the diffuse but erroneous belief that in the presence of MI the input wave becomes a train of solitons. On the contrary (temporally) periodic wave solutions of the nonlinear Schroedinger (NLS) equation have shown that the propagation is periodic also in space (a phenomenon known as Fermi-Pasta-Ulam recurrence [4]), leading to the formation of complex spatiotemporal patterns [5-7]. We show here that the nonlinear dynamics of modulated waves, which includes in principle the interaction of an infinite number of Fourier modes, is essentially locked to the simple interaction between three modes: the pump and the first symmetric sidebands. In this case a simple integrable one-dimensional equivalent oscillator model [8-9] enables one to unfold the role of a coherent modulation at the input in the generation of the spatio-temporal patterns. This suggests also the possibility of new experiments in which the pulse train and switching among two logical state is controlled by the input phase relation between the pump and the sidebands.

Automated driving requires a large number of sensors. They are used to detect environmental influences and regulate vehicle guidance. Many of these sensors have to fulfil requirements regarding position and functional installation space. This creates a conflict of objectives between aesthetically appealing integration into the design and functional integration into trim parts or an aerodynamic concept. In addition, the perception of the sensors by vehicle users and other road users could favour the recognition of an automated vehicle (AV). Numerous studies showed the fundamental importance of a vehicle's exterior, especially with regard to the various semantic aspects of the resulting product effect. This is particularly apparent with regard to the communication between the driver or AV and other road users. Therefore, in this paper we consider the preconditioning of the user in relation to the degree of integration of sensor technology for automated driving. For this purpose, sensor configurations differing in shape and colour are applied to a concept car and examined in an eye-tracking study with regard to their influence on the perception of passengers and other road users. This will provide designers and engineers with insights into the design of sensor clusters in the exterior design of future vehicles.METHODSIn a first step, existing studies were systematically examined to determine the influence of the exterior design on the perception of the vehicle. In addition to the basic idea of appeal, the preconditioning by the design was examined in particular. It has been shown that the exterior design has an influence on the user's driving style, the spatial perception and the brand affiliation recognition. In more recent studies, the influence of so-called external HMIs (eHMI) on the perception of an automated vehicle has been increasingly investigated. Studies on the influence of more or less exposed sensors for automated driving on the perception of users or pedestrians are not known.In a second step, an eye-tracking study was therefore designed. In the study, the test persons were sequentially shown renderings in ¾ front and ¾ rear perspective of a concept car with different sensor configurations on the screen. The relevant sensor configurations are based on a previous study in which we conducted a position analysis and derived integration strategies for sensors. The stimulus patterns differ specifically in the sub-forms of structure (position), shape and colour. The test persons evaluated each of the stimulus patterns by means of a questionnaire with regard to their subjective impression in the use cases "crossing the street as a pedestrian" and "getting into the vehicle and being driven home". System trust, the perception of safety, recognisability and the judgement of appeal were assessed. Absolute dwell time, relative dwell time and fixation were recorded via eye tracking.RESULTSThe statistical evaluation of the results has shown that the sensors already have an influence on system trust and the perception of safety of the users and other road users in both use cases considered. In particular, for the configuration without visible sensors and additively integrated sensors, significant differences emerged as expected. As expected, there was a high degree of dispersion in the opinion towards liking the design of the users. Areas of interest were derived from the eye-tracking data via heat maps. The more accurate understanding of the perception of the degree of integration of sensors into the exterior design can support the work of designers by giving them the freedom to realise significantly more innovative designs in their design proposals compared to previous vehicles.OUTLOOKExtending the investigations to much larger vehicles, especially trucks or robotic vehicles according to SAE Level 5 without passengers could show whether the perception of sensors can already be used as a salient feature in exterior design. Furthermore, it should be investigated whether there are interactions with eHMI on AVs. In addition, system trust and safety perception could be investigated in an extended virtual reality test design with a variety of other relevant use cases such as: "oncoming AV at right-before-left intersection" to show influences of immersion. Also, the influence of time and the associated habituation factor certainly plays a role. For this reason, the effects of road users getting used to the sight with sensors should also be considered.ACKNOWLEDGMENTSThis research was supported by Federal Ministry for Economic Affairs and Climate Action in the national research project RUMBA.