Spectrum Of Dirac Operator Research Articles

Essential spectrum of one-dimensional Dirac operators with delta-interactions

В заметке дается описание существенного спектра одномерных операторов Дирака с сингулярными потенциалами, имеющими носители на дискретных множествах в $\mathbb{R}$, в терминах предельных операторов.

Metal or insulator? Dirac operator spectrum in holographic QCD

The lattice studies in QCD demonstrate the nontrivial localization behavior of the eigenmodes of the 4D Euclidean Dirac operator considered as Hamiltonian of 4+1 dimensional disordered system. We use the holographic viewpoint to provide the conjectural explanation of these properties. The delocalization of all modes in the confined phase is related to the θ=π - like phenomena when the fermions are delocalized on domain walls. It is conjectured that the localized modes separated by mobility edge from the rest of the spectrum in deconfined QCD correspond to the near-horizon region in the holographic dual.

Eigenvalues of one-dimensional non-self-adjoint Dirac operators and applications

We analyze eigenvalues emerging from thresholds of the essential spectrum of one-dimensional Dirac operators perturbed by complex and non-symmetric potentials. In the general non-self-adjoint setting we establish the existence and asymptotics of weakly coupled eigenvalues and Lieb-Thirring inequalities. As physical applications we investigate the damped wave equation and armchair graphene nanoribbons.

Mobility edge and Black Hole Horizon

We conjecture that the mobility edge in the 4D Euclidean Dirac operator spectrum in QCD in the deconfined phase found in the lattice studies corresponds to the near black hole (BH) horizon region in the holographic dual. We present some evidences both from the field theory side and from the worldsheet theory of long open string.

Spherical $$\Pi $$ Π -Type Operators in Clifford Analysis and Applications

The \(\Pi \)-operator (Ahlfors–Beurling transform) plays an important role in solving the Beltrami equation. In this paper we define two \(\Pi \)-operators on the n-sphere. The first spherical \(\Pi \)-operator is shown to be an \(L^2\) isometry up to isomorphism. To improve this, with the help of the spectrum of the spherical Dirac operator, the second spherical \(\Pi \) operator is constructed as an isometric \(L^2\) operator over the sphere. Some analogous properties for both \(\Pi \)-operators are also developed. We also study the applications of both spherical \(\Pi \)-operators to the solution of the spherical Beltrami equations.

Monte Carlo simulations of random non-commutative geometries

Random non-commutative geometries are introduced by integrating over the space of Dirac operators that form a spectral triple with a fixed algebra and Hilbert space. The cases with the simplest types of Clifford algebra are investigated using Monte Carlo simulations to compute the integrals. Various qualitatively different types of behaviour of these random Dirac operators are exhibited. Some features are explained in terms of the theory of random matrices but other phenomena remain mysterious. Some of the models with a quartic action of symmetry-breaking type display a phase transition. Close to the phase transition the spectrum of a typical Dirac operator shows manifold-like behaviour for the eigenvalues below a cut-off scale.

An Explicit Formula for the Dirac Multiplicities on Lens Spaces

We present a new description of the spectrum of the (spin-) Dirac operator D on lens spaces. Viewing a spin lens space L as a locally symmetric space $$\Gamma \backslash {\text {Spin}}(2m)/{\text {Spin}}(2m-1)$$ and exploiting the representation theory of the $${\text {Spin}}$$ groups, we obtain explicit formulas for the multiplicities of the eigenvalues of D in terms of finitely many integer operations. As a consequence, we present conditions for lens spaces to be Dirac isospectral. Tackling classic questions of spectral geometry, we prove with the tools developed that neither spin structures nor isometry classes of lens spaces are spectrally determined by giving infinitely many examples of finite families of Dirac isospectral lens spaces. These results are complemented by examples found with the help of a computer.

On the spectra of Dirac operators

On the spectra of Dirac operators

Anderson transition and multifractals in the spectrum of the Dirac operator of quantum chromodynamics at high temperature

We investigate the Anderson transition found in the spectrum of the Dirac operator of Quantum Chromodynamics (QCD) at high temperature, studying the properties of the critical quark eigenfunctions. Applying multifractal finite-size scaling we determine the critical point and the critical exponent of the transition, finding agreement with previous results, and with available results for the unitary Anderson model. We estimate several multifractal exponents, finding also in this case agreement with a recent determination for the unitary Anderson model. Our results confirm the presence of a true Anderson localization-delocalization transition in the spectrum of the quark Dirac operator at high-temperature, and further support that it belongs to the 3D unitary Anderson model class.

On Dirac equation with a potential and critical Sobolev exponent

In this paper we consider a critical Dirac equation with a potential on a compact spin manifold. We prove the existence of solutions based on the analysis of the spectrum of Dirac operator with a potential and the dual variational method.

Schnol’s Theorem and Spectral Properties of Massless Dirac Operators with Scalar Potentials

The spectra of massless Dirac operators are of essential interest e.g. for the electronic properties of graphene, but fundamental questions such as the existence of spectral gaps remain open. We show that the eigenvalues of massless Dirac operators with suitable real-valued potentials lie inside small sets easily characterised in terms of properties of the potentials, and we prove a Schnol'-type theorem relating spectral points to polynomial boundedness of solutions of the Dirac equation. Moreover, we show that, under minimal hypotheses which leave the potential essentially unrestrained in large parts of space, the spectrum of the massless Dirac operator covers the whole real line; in particular, this will be the case if the potential is nearly constant in a sequence of regions.

Quark mass anomalous dimension and Λ MS ¯ $$ {\varLambda}_{\overline{\mathrm{MS}}} $$ from the twisted mass Dirac operator spectrum

We investigate whether it is possible to extract the quark mass anomalous dimension and its scale dependence from the spectrum of the twisted mass Dirac operator in Lattice QCD. The answer to this question appears to be positive, provided that one goes to large enough eigenvalues, sufficiently above the non-perturbative regime. The obtained results are compared to continuum perturbation theory. By analyzing possible sources of systematic effects, we find the domain of applicability of the approach, extending from an energy scale of around 1.5 to 4 GeV. The lower limit is dictated by physics (non-perturbative effects at low energies), while the upper bound is set by the ultraviolet cut-off of present-day lattice simulations. The information about the scale dependence of the anomalous dimension allows also to extract the value of the $$ {\varLambda}_{\overline{\mathrm{MS}}} $$ -parameter of 2-flavour QCD, yielding the value 303(13)(25) MeV, where the first error is statistical and the second one systematic. We use gauge field configuration ensembles generated by the European Twisted Mass Collaboration (ETMC) with 2 flavours of dynamical twisted mass quarks, at 4 lattice spacings in the range between around 0.04 and 0.08 fm.

Absolutely continuous spectrum of Dirac operators with square-integrable potentials

We show that the absolutely continuous part of the spectral function of the one-dimensional Dirac operator on a half-line with a constant mass term and a real, square-integrable potential is strictly increasing throughout the essential spectrum (−∞, −1] ∪ [1, ∞). The proof is based on estimates for the transmission coefficient for the full-line scattering problem with a truncated potential and a subsequent limiting procedure for the spectral function. Furthermore, we show that the absolutely continuous spectrum persists when an angular momentum term is added, thus also establishing the result for spherically symmetric Dirac operators in higher dimensions.

Universality and the QCD Anderson Transition

We study the Anderson-type transition previously found in the spectrum of the QCD quark Dirac operator in the high-temperature, quark-gluon plasma phase. Using finite size scaling for the unfolded level spacing distribution, we show that in the thermodynamic limit there is a genuine mobility edge, where the spectral statistics changes from Poisson to Wigner-Dyson statistics in a nonanalytic way. We determine the correlation length critical exponent ν and find that it is compatible with that of the unitary Anderson model.

Conformal positive mass theorems for asymptotically flat manifolds with inner boundary

Inspired by Witten's insightful spinor proof of the positive mass theorem, in this paper, we use the spinor method to derive higher dimensional type conformal positive mass theorems on asymptotically flat spin manifolds with inner boundary, which states that under a condition about the plus (minus) relation between the scalar curvatures of the original and the conformal metrics in addition with some boundary condition, we will get the associated positivity of their ADM masses. The rigidity part of the plus part is used in the proof of black hole uniqueness theorems. They are related with quasi-local mass and the spectrum of Dirac operator.

Topological susceptibility from the twisted mass Dirac operator spectrum

We present results of our computation of the topological susceptibility with $N_f=2$ and $N_f=2+1+1$ flavours of maximally twisted mass fermions, using the method of spectral projectors. We perform a detailed study of the quark mass dependence and discretization effects. We make an attempt to confront our data with chiral perturbation theory and extract the chiral condensate from the quark mass dependence of the topological susceptibility. We compare the value with the results of our direct computation from the slope of the mode number. We emphasize the role of autocorrelations and the necessity of long Monte Carlo runs to obtain results with good precision. We also show our results for the spectral projector computation of the ratio of renormalization constants $Z_P/Z_S$.

Generalized Fuzzy Torus and its Modular Properties

We consider a generalization of the basic fuzzy torus to a fuzzy torus with non-trivial modular parameter, based on a finite matrix algebra. We discuss the modular properties of this fuzzy torus, and compute the matrix Laplacian for a scalar field. In the semi-classical limit, the generalized fuzzy torus can be used to approximate a generic commutative torus represented by two generic vectors in the complex plane, with generic modular parameter $\tau$. The effective classical geometry and the spectrum of the Laplacian are correctly reproduced in the limit. The spectrum of a matrix Dirac operator is also computed.

Chiral condensate from the twisted mass Dirac operator spectrum

AbstractWe present the results of our computation of the dimensionless chiral condensater0Σ1/3withNf= 2 andNf= 2 + 1 + 1 flavours of maximally twisted mass fermions. The condensate is determined from the Dirac operator spectrum, applying the spectral projector method proposed by Giusti and Lüscher. We use 3 lattice spacings and several quark masses at each lattice spacing to perform the chiral and continuum extrapolations. We study the effect of the dynamical strange and charm quarks by comparing our results forNf= 2 andNf= 2 + 1 + 1 dynamical flavours.

Eigenvalues of Type r of the Basic Dirac Operator on Kähler Foliations

Abstract. In this paper, we prove that on a Kahler spin foliatoin of codimension q = 2n,any eigenvalue λ of type r (r ∈ {1,··· ,[ n+12 ]}) of the basic Dirac operator D b satisfies theinequality λ 2 ≥ r4r−2 inf M σ ∇ , where σ ∇ is the transversal scalar curvature of F. 1. IntroductionOn a K¨ahler spin foliation (M,F) of codimension q = 2n, any eigenvalue λ ofthe basic Dirac operator D b satisfies(1.1) λ 2 ≥n+14ninf M K σ if n is odd [6,7],n4(n−1)inf M K σ if n is even [4],where K σ = σ ∇ + |κ| 2 with the transversal scalar curvature σ ∇ and the meancurvature form κ of F. In the limiting cases, F is minimal. For the point foliation,see [9,10]. Since the limiting cases of (1.1) are minimal, the inequalities (1.1) yieldthe following:(1.2) λ 2 ≥n+14ninf M σ ∇ if n is odd,n4(n−1)inf M σ ∇ if n is even.In this paper, we give an estimate of the eigenvalues λ of type r of the basic Diracoperator D b on a K¨ahler spin foliation. Recently, G. Habib and K. Richardson [5]proved that the spectrum of the basic Dirac operator does not change with respectto a change of bundle-like metric. And the existence of a bundle-like metric suchthat δ

Essential Spectra of One-Dimensional Dirac Operators

The present paper estimates or explicitly calculates the essential spectra of one-dimensional Dirac operators L by introducing two one-parameter pencils of Sturm–Liouville operators τ 1(λ) and τ 2(λ) for \({\lambda \in \mathbb{R}}\), considering spectral problems of τ j (λ) with a new spectral parameter, and examining the sets \({\{\lambda \in \mathbb{R} :\,\,\, 0 \notin \sigma_{\rm e}(\tau_j(\lambda))\}}\) for j = 1, 2. Applications are given to illustrate the scope of applicability.