Discrete Dirac Research Articles

Inverse scattering properties with Levinson formula for cubic eigenparameter-dependent discrete Dirac problem

UDC 517.9 We present some properties of zeros of the Jost function and of the scattering function. Then the uniqueness of the kernel and the continuity of the scattering function with appropriate Levinson-type formula are investigated for the inverse discrete Dirac problem based on the cubic eigenparameter-dependent boundary condition.

Quantum entropy couples matter with geometry

Abstract We propose a theory for coupling matter fields with discrete geometry on higher-order networks, i.e. cell complexes. The key idea of the approach is to associate to a higher-order network the quantum entropy of its metric. Specifically we propose an action having two contributions. The first contribution is proportional to the logarithm of the volume associated to the higher-order network by the metric. In the vacuum this contribution determines the entropy of the geometry. The second contribution is the quantum relative entropy between the metric of the higher-order network and the metric induced by the matter and gauge fields. The induced metric is defined in terms of the topological spinors and the discrete Dirac operators. The topological spinors, defined on nodes, edges and higher-dimensional cells, encode for the matter fields. The discrete Dirac operators act on topological spinors, and depend on the metric of the higher-order network as well as on the gauge fields via a discrete version of the minimal substitution. We derive the coupled dynamical equations for the metric, the matter and the gauge fields, providing an information theory principle to obtain the field theory equations in discrete curved space.

An efficient direct‐forcing immersed boundary method for flow around a pair of spheres

AbstractThe numerical study of flow around a pair of spheres and a square array of spheres is investigated by using a direct‐forcing immersed boundary method. Using high resolution three‐dimensional computations, we analyzed the flow around several configurations: a sphere, a pair of spheres in a tandem arrangement with center‐to‐center streamwise ratio L/D ranging from 1 to 6, and a square array with 9 spheres in a uniform arrangement. In the latter case, we explore the ratio of array diameter (DG) to sphere diameter (D) at 4, 5, 6 and 7. The center‐to‐center streamwise and transverse pitch is the same, varied from L/D = 1.5, 2, 2.5 to 3, and they were arranged in a square periodic array to allow uniform distribution within the array. Based on the effective direct‐forcing immersed boundary projection method, the fractional time marching methodology is applied for solving four field variables involving three velocities and one pressure component. The pressure Poisson equation is advanced in space by using the fast Fourier transform (FFT) and a tridiagonal matrix algorithm (TDMA), effectively solving for the diagonally dominant tridiagonal matrix equations. A direct‐forcing immersed boundary method is involved to treat the interfacial terms by adding the appropriate sources as force function at the boundary, separating the phases. Geometries featuring the stationary solid obstacles in the flow are embedded in the Cartesian grid with special discretizations near the embedded boundary using a discrete Dirac delta function to ensure the accuracy of the solution in the cut cells. An important characteristic of flow over the multiple spheres is devised by comparing with the drag and lift coefficients, as well as vortex shedding.

INTERACTION BETWEEN A BEYOND-BAND DISCRETE SOLITON AND A DIRAC SOLITON IN BINARY WAVEGUIDE ARRAYS

We investigate the intricate interactions between a beyond-band discrete soliton (BBDS) and Dirac soliton in a binary waveguide array (BWA). Through comprehensive analysis, we demonstrate that the behavior of these soliton interactions remains largely unaffected by the presence of central peaks or dips at the center of each soliton, or the initial phase differences between them. Our research highlights the critical role of the BBDS intensity in determining the nature of its interaction with Dirac solitons. We reveal that low-intensity BBDS exhibit consistent attraction towards Dirac solitons across multiple encounters, while high-intensity BBDSs show an initial phase of attraction followed by a repulsion phase, causing a subsequent divergence. These findings not only enhance our understanding of soliton interactions on a fundamental level but also set the stage for the development of innovative optical communication and signal processing technologies leveraging the distinctive properties of BBDSs and Dirac solitons.

Remarks on discrete Dirac operators and their continuum limits

We discuss possible definitions of discrete Dirac operators, and discuss their continuum limits. It is well known in the lattice field theory that the straightforward discretization of the Dirac operator introduces unwanted spectral subspaces, and it is known as the fermion doubling . In oder to overcome this difficulty, two methods were proposed. The first one is to introduce a new term, called the Wilson term , and the second one is the KS-fermion model or the staggered fermion model . We discuss mathematical formulations of these, and study their continuum limits.

A new digital image shifting method by considering the geometry of camera sensors

Digital image shifting occurs when the pixels needs to be moved, the algorithm behind this is interpolation. Current digital image shifting algorithm utilizing interpolation has a limitation: it treats all images as the same type of matrix and does not consider how the images are formed this could cause problems in some sensitive imaging scenarios such as microcopy and precision optical metrology. The goal of our research is to propose a new approach to implement the digital image shifting method. We based our new method off of digital camera sensors, specifically the CMOS sensor. When light (a continuous signal) hits the CMOS sensor, the sensor transforms the light into a discrete signal and its energy into electric energy. Then, this discrete signal then generates the pixels of the image. Our algorithm is similar: The signal is convoluted with the rectangular function, generating a continuous signal, and then we multiply the discrete dirac comb function to generate a discrete signal. After that, we only have to manipulate the discrete dirac comb function to obtain the desired result. Our conducted simulations on the computer proves our method correct, hopefully more experiments can be done in the industrial setting.

The mass of simple and higher-order networks

We propose a theoretical framework that explains how the mass of simple and higher-order networks emerges from their topology and geometry. We use the discrete topological Dirac operator to define an action for a massless self-interacting topological Dirac field inspired by the Nambu–Jona-Lasinio model. The mass of the network is strictly speaking the mass of this topological Dirac field defined on the network; it results from the chiral symmetry breaking of the model and satisfies a self-consistent gap equation. Interestingly, it is shown that the mass of a network depends on its spectral properties, topology, and geometry. Due to the breaking of the matter–antimatter symmetry observed for the harmonic modes of the discrete topological Dirac operator, two possible definitions of the network mass can be given. For both possible definitions, the mass of the network comes from a gap equation with the difference among the two definitions encoded in the value of the bare mass. Indeed, the bare mass can be determined either by the Betti number β 0 or by the Betti number β 1 of the network. We provide numerical results on the mass of different networks, including random graphs, scale-free, and real weighted collaboration networks. We also discuss the generalization of these results to higher-order networks, defining the mass of simplicial complexes. The observed dependence of the mass of the considered topological Dirac field with the topology and geometry of the network could lead to interesting physics in the scenario in which the considered Dirac field is coupled with a dynamical evolution of the underlying network structure.

Novel finite difference approach to discretize the symplectic dirac operator

Symplectic Dirac operator is an intertwining differential operator. Discretising symplectic Dirac operator gives a new direction to study the quantum space. The construction of discrete symplectic Dirac operator requires the theory of discrete symplectic Clifford analysis or the concept of discrete symplectic connections, which are not explained in literature. In this work, a discretization approach for symplectic Dirac operator is suggested by considering the forward and backward basis vectors on symplectic Clifford algebra. The suggested discrete symplectic Dirac operator is Ds=Ds++Ds- where the Ds+ and Ds- are the forward and backward discrete symplectic Dirac operators, respectively. The new discrete symplectic Dirac operator gives the factorization of discrete Laplacian on symplectic spaces. Further, we establish commutation relations involving forward and backward discrete symplectic Dirac operators in the representation theory.

Dirac signal processing of higher-order topological signals

Higher-order networks can sustain topological signals which are variables associated not only to the nodes, but also to the links, to the triangles and in general to the higher dimensional simplices of simplicial complexes. These topological signals can describe a large variety of real systems including currents in the ocean, synaptic currents between neurons and biological transportation networks. In real scenarios topological signal data might be noisy and an important task is to process these signals by improving their signal to noise ratio. So far topological signals are typically processed independently of each other. For instance, node signals are processed independently of link signals, and algorithms that can enforce a consistent processing of topological signals across different dimensions are largely lacking. Here we propose Dirac signal processing, an adaptive, unsupervised signal processing algorithm that learns to jointly filter topological signals supported on nodes, links and triangles of simplicial complexes in a consistent way. The proposed Dirac signal processing algorithm is formulated in terms of the discrete Dirac operator which can be interpreted as ‘square root’ of a higher-order Hodge Laplacian. We discuss in detail the properties of the Dirac operator including its spectrum and the chirality of its eigenvectors and we adopt this operator to formulate Dirac signal processing that can filter noisy signals defined on nodes, links and triangles of simplicial complexes. We test our algorithms on noisy synthetic data and noisy data of drifters in the ocean and find that the algorithm can learn to efficiently reconstruct the true signals outperforming algorithms based exclusively on the Hodge Laplacian.

An inverse scattering problem for eigenparameter-dependent discrete Dirac system with Levinson formula

In this paper, we have considered an inverse scattering problem for discrete Dirac operator L with eigenparameter dependent boundary condition. At the outset, the Jost solution and the Jost function of L are given. Then the zeros of the Jost function and some properties of the scattering function are examined. After finding the scattering dataset and the main equation for the operator L, the uniqueness of the kernel, continuity of the scattering function and the appropriate Levinson type formula are obtained.

Dirac gauge theory for topological spinors in 3+1 dimensional networks

Gauge theories on graphs and networks are attracting increasing attention not only as approaches to quantum gravity but also as models for performing quantum computation. Here we propose a Dirac gauge theory for topological spinors in dimensional networks associated to an arbitrary metric. Topological spinors are the direct sum of 0-cochains and 1-cochains defined on a network and describe a matter field defined on both nodes and links of a network. Recently in Bianconi (2021 J. Phys. Complex. 2 035022) it has been shown that topological spinors obey the topological Dirac equation driven by the discrete Dirac operator. In this work we extend these results by formulating the Dirac equation on weighted and directed dimensional networks which allow for the treatment of a local theory. The commutators and anti-commutators of the Dirac operators are non vanishing an they define the curvature tensor and magnetic field of our theory respectively. This interpretation is confirmed by the non-relativistic limit of the proposed Dirac equation. In the non-relativistic limit of the proposed Dirac equation the sector of the spinor defined on links follows the Schrödinger equation with the correct giromagnetic moment, while the sector of the spinor defined on nodes follows the Klein–Gordon equation and is not negligible. The action associated to the proposed field theory comprises of a Dirac action and a metric action. We describe the gauge invariance of the action under both Abelian and non-Abelian transformations and we propose the equation of motion of the field theory of both Dirac and metric fields. This theory can be interpreted as a limiting case of a more general gauge theory valid on any arbitrary network in the limit of almost flat spaces.

Continuum limits for discrete Dirac operators on 2D square lattices

We discuss the continuum limit of discrete Dirac operators on the square lattice in {mathbb {R}}^2 as the mesh size tends to zero. To this end, we propose the most natural and simplest embedding of ell ^2({mathbb {Z}}_h^d) into L^2({mathbb {R}}^d), which enables us to compare the discrete Dirac operators with the continuum Dirac operators in the same Hilbert space L^2({mathbb {R}}^2)^2. In particular, we prove that the discrete Dirac operators converge to the continuum Dirac operators in the strong resolvent sense. Potentials are assumed to be bounded and uniformly continuous functions on {mathbb {R}}^2 and allowed to be complex matrix-valued. We also prove that the discrete Dirac operators do not converge to the continuum Dirac operators in the norm resolvent sense. This is closely related to the observation that the Liouville theorem does not hold in discrete complex analysis.

Discrete Dirac reduction of implicit Lagrangian systems with abelian symmetry groups

<abstract><p>This paper develops the theory of discrete Dirac reduction of discrete Lagrange–Dirac systems with an abelian symmetry group acting on the configuration space. We begin with the linear theory and, then, we extend it to the nonlinear setting using retraction compatible charts. We consider the reduction of both the discrete Dirac structure and the discrete Lagrange–Pontryagin principle, and show that they both lead to the same discrete Lagrange–Poincaré–Dirac equations. The coordinatization of the discrete reduced spaces relies on the notion of discrete connections on principal bundles. At last, we demonstrate the method obtained by applying it to a charged particle in a magnetic field, and to the double spherical pendulum.</p></abstract>

Some fractional Dirac systems

In this work, including $\alpha\epsilon(0,1)$; we examined the Dirac system in the frame which includes$\ \alpha$ order right and left Reimann-Liouville fractional integrals and derivatives with exponential kernels, and the Dirac system which includes $\alpha$ order right and left Caputo fractional integrals and derivatives with exponential kernels. Furthermore, we have given some definitions and properties for discrete exponential kernels and their associated fractional sums and fractional differences, and we have studied discrete fractional Dirac systems.

On the discrete Dirac spectrum of a point electron in the zero-gravity Kerr–Newman spacetime

The discrete spectrum of the Dirac operator for a point electron in the maximal analytically extended Kerr–Newman spacetime is determined in the zero-G limit (zGKN), under some restrictions on the electrical coupling constant and on the radius of the ring-singularity of the zGKN spacetime. The spectrum is characterized by a triplet of integers, associated with winding numbers of orbits of dynamical systems on cylinders. A dictionary is established that relates the spectrum with the known hydrogenic Dirac spectrum. Numerical illustrations are presented. Open problems are listed.

Numerical Study of an Indicator Function for Front-Tracking Methods

In this paper, we present a detailed derivation and numerical investigation of an indicator function for front-tracking methods. We use the discrete Dirac delta function to construct an indicator function from a set of Lagrangian points and solve the resulting discrete Poisson equation with the zero Dirichlet boundary condition using an iterative method. We present several computational tests to investigate the effect of parameters such as distance between points, uniformity of the distance, and types of the Dirac delta functions on the indicator function.

Clifford algebra from quantum automata and unitary Wilson fermions

We introduce a spacetime discretization of the Dirac equation that has the form of a quantum automaton and that is invariant upon changing the representation of the Clifford algebra, like the Dirac equation itself. Our derivation follows Dirac's original one: We required that the square of the discrete Dirac scheme be what we define as an acceptable discretization of the Klein-Gordon equation. In contrast to standard lattice gauge theory in discrete time, in which unitarity needs to be proven, we show that the quantum automaton delivers naturally unitary Wilson fermions for any choice of Wilson's parameter.

Dual-Band Helical Edge States and Discrete Dirac Vortices in Solid-State Elastic Waves

Dual-band equipment has unique advantages as regards ensuring the stability of wireless communication and improving efficiency. With the development of topological photonics and phononics, electromagnetic and mechanical wave transmission with simultaneous advantages, e.g., antireflection, broadband, and nondispersion characteristics, has been simultaneously realized. This outcome has greatly improved elastic wave manipulation performance in solids, which are important media for wireless communication. Here, using a plate phononic crystal (PnC), a dual-mode elastic topological insulator is theoretically designed and experimentally realized, and the quantum spin Hall effect is observed under a dual band. Using discrete Kekul\'e modulation to construct the PnC multidomain junction, dual-mode and dual-band Dirac vortexes, i.e., a type of topological corner state, are obtained in a single PnC. These transmitted and localized topological states have the same excellent properties as their antecedents, which are obtained with single-band materials. The implementation mechanisms are diverse and applicable to multiple modes and broad frequencies, offering antijamming, low-loss, high-efficiency, and high-capacity signal processing components. The findings of this study will aid development of solid-state topology PnCs, as well as future topological-phononic integrated circuits with high performance and multifunctionality.

Scattering properties and dispersion estimates for a one‐dimensional discrete Dirac equation

AbstractWe derive dispersion estimates for solutions of a one‐dimensional discrete Dirac equations with a potential. In particular, we improve our previous result, weakening the conditions on the potential. To this end we also provide new results concerning scattering for the corresponding perturbed Dirac operators which are of independent interest. Most notably, we show that the reflection and transmission coefficients belong to the Wiener algebra.

Discrete pseudo‐differential operators in hypercomplex analysis

In this paper we discuss discrete pseudo‐differential operators in the context of hypercomplex analysis. We will provide the definitions and establish the necessary calculus which will be used to discuss questions like boundedness and compactness of operators. In the end, we will discuss well‐known examples from hypercomplex analysis as well as give new examples including discrete Dirac operators with non‐constant coefficients.