Eigenvalues Of Dirac Operator Research Articles

Variational methods in relativistic quantum mechanics

This review is devoted to the study of stationary solutions of lin- ear and nonlinear equations from relativistic quantum mechanics, involving the Dirac operator. The solutions are found as critical points of an energy func- tional. Contrary to the Laplacian appearing in the equations of nonrelativistic quantum mechanics, the Dirac operator has a negative continuous spectrum which is not bounded from below. This has two main consequences. First, the energy functional is strongly indefinite. Second, the Euler-Lagrange equations are linear or nonlinear eigenvalue problems with eigenvalues lying in a spectral gap (between the negative and positive continuous spectra). Moreover, since we work in the space domain R 3 , the Palais-Smale condition is not satisfied. For these reasons, the problems discussed in this review pose a challenge in the Calculus of Variations. The existence proofs involve sophisticated tools from nonlinear analysis and have required new variational methods which are now applied to other problems. In the first part, we consider the fixed eigenvalue problem for models of a free self-interacting relativistic particle. They allow to describe the localized state of a spin-1/2 particle (a fermion) which propagates without changing its shape. This includes the Soler models, and the Maxwell-Dirac or Klein- Gordon-Dirac equations. The second part is devoted to the presentation of min-max principles al- lowing to characterize and compute the eigenvalues of linear Dirac operators with an external potential, in the gap of their essential spectrum. Many con- sequences of these min-max characterizations are presented, among them a new kind of Hardy-like inequalities and a stable algorithm to compute the eigenvalues. In the third part we look for normalized solutions of nonlinear eigenvalue problems. The eigenvalues are Lagrange multipliers, lying in a spectral gap. We review the results that have been obtained on the Dirac-Fock model which is a nonlinear theory describing the behavior of N interacting electrons in an external electrostatic field. In particular we focus on the problematic definition of the ground state and its nonrelativistic limit. In the last part, we present a more involved relativistic model from Quan- tum Electrodynamics in which the behavior of the vacuum is taken into ac- count, it being coupled to the real particles. The main interesting feature of this model is that the energy functional is now bounded from below, providing us with a good definition of a ground state.

INVESTIGATION OF SPONTANEOUS CHIRAL-SYMMETRY BREAKING FROM LATTICE QCD WITH MASSLESS QUARKS

We apply the probability distribution function method to the study of chiral properties of QCD with quarks in the exact massless limit. A relation among the chiral condensate, zeros of the Bessel function and eigenvalue of Dirac operator is also given. The chiral condensate in this limit can be measured with small number of eigenvalues of the massless Dirac operator and without any ambiguous mass extrapolation. Results for lattice QCD with Kogut-Susskind quarks are shown.

MATRIX MODELS AND QCD WITH CHEMICAL POTENTIAL

The random matrix model approach to quantum chromodynamics (QCD) with nonvanishing chemical potential is reviewed. The general concept using global symmetries is introduced, as well as its relation to field theory, the so-called epsilon regime of chiral perturbation theory (∊χPT). Two types of matrix model results are distinguished: phenomenological applications leading to phase diagrams, and an exact limit of the QCD Dirac operator spectrum matching with ∊χPT. All known analytic results for the spectrum of complex and symplectic matrix models with chemical potential are summarised for the symmetry classes of ordinary and adjoint QCD, respectively. These include correlation functions of Dirac operator eigenvalues in the complex plane for real chemical potential, and in the real plane for imaginary isospin chemical potential. Comparisons of these predictions to recent lattice simulations are also discussed.

A new chiral two-matrix theory for Dirac spectra with imaginary chemical potential

We solve a new chiral random two-matrix theory by means of biorthogonal polynomials for any matrix size N. By deriving the relevant kernels we find explicit formulas for all ( n , k ) -point spectral (mixed or unmixed) correlation functions. In the microscopic limit we find the corresponding scaling functions, and thus derive all spectral correlators in this limit as well. We extend these results to the ordinary (non-chiral) ensembles, and also there provide explicit solutions for any finite size N, and in the microscopic scaling limit. Our results give the general analytical expressions for the microscopic correlation functions of the Dirac operator eigenvalues in theories with imaginary baryon and isospin chemical potential, and can be used to extract the tree-level pion decay constant from lattice gauge theory configurations. We find exact agreement with previous computations based on the low-energy effective field theory in the two special cases where comparisons are possible.

EIGENVALUE ESTIMATES OF THE BASIC DIRAC OPERATOR ON A RIEMANNIAN FOLIATION

On a foliated Riemannian manifold with a transverse spin structure, we give a lower bound of the square for the eigenvalues of the basic Dirac operator in terms of the smallest eigenvalue of the basic Yamabe operator and of the norm of an appropriate endomorphism of the normal bundle $Q$ of $\mathcal F$. We study the limiting case.

Density profiles of small Dirac operator eigenvalues for two color QCD at nonzero chemical potential compared to matrix models

We investigate the eigenvalue spectrum of the staggered Dirac matrix in two color QCD at finite chemical potential. The profiles of complex eigenvalues close to the origin are compared to a complex generalization of the chiral Gaussian Symplectic Ensemble, confirming its predictions for weak and strong non-Hermiticity. They differ from the QCD symmetry class with three colors by a level repulsion from both the real and imaginary axis.

Dirac Operator on Fuzzy Sphere with U(1) Dirac Monopole Background**The project supported by National Natural Science Foundation of China under Grant No. 90103010

We treat the Dirac operator on the Fuzzy sphere with the help of the generalized coherent state. It is shown that the derivatives can be constructed by Moyal product with symbol of the operator. We obtain the eigenvalue of the free fermion Dirac operator as same as the result by [Hajime Aoki, Satoshi Iso, and Kelichi Nagao, Phys. Rev. D67 (2003) 065018]. Meanwhile, we also give the eigenvalue of Dirac operator with U(1) Dirac monopole background.

Chiral condensate of lattice QCD with massless quarks from the probability distribution function method

We apply the probability distribution function method to the study of chiral properties of QCD with quarks in the exact massless limit. A relation among the chiral condensate, the zeros of the Bessel function, and the eigenvalue of the Dirac operator is also given. The chiral condensate in this limit can be measured with a small number of eigenvalues of the massless Dirac operator and without any ambiguous mass extrapolation. The results for SU(3) gauge theory with quenched Kogut-Susskind quarks on the ${10}^{4}$ lattice are shown.

Lattice QCD in the epsilon-regime and random matrix theory

In the epsilon-regime of QCD the main features of the spectrum of the low-lying eigenvalues of the (euclidean) Dirac operator are expected to be described by a certain universality class of random matrix models. In particular, the latter predict the joint statistical distribution of the individual eigenvalues in any topological sector of the theory. We compare some of these predictions with high-precision numerical data obtained from lattice QCD for a range of lattice spacings and volumes. While no complete matching is observed, the results agree with theoretical expectations at volumes larger than about 5 fm^4.

Absence of eigenvalues of Dirac operators with potentials diverging at in‐finity

AbstractWe investigate the non‐existence of eigenvalues of Dirac operators α · p +m(r)β + V(x) in the Hilbert space L2(R3)4 with a variable massm(r) and a matrix‐valued function V(x), which may decay or diverge at infinity. As a result we show that there are no eigenvalues of Dirac operators for a large class of m(r) and V(x) such that |m(r)| ≪ |V(x)| →∞ as r → ∞ and V(x) is positive or negative definite at infinity. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Microscopic correlation functions for the QCD Dirac operator with chemical potential.

A chiral random matrix model with complex eigenvalues is solved as an effective model for QCD with nonvanishing chemical potential. The new correlation functions derived from it are conjectured to predict the local fluctuations of complex Dirac operator eigenvalues at zero virtuality. The parameter measuring the non-Hermiticity of the random matrix is related to the chemical potential. In the phase with broken chiral symmetry all spectral correlations are calculated for finite matrix size N and in the large-N limit at weak and strong non-Hermiticity. The derivation uses the orthogonality of the Laguerre polynomials in the complex plane.

Quenched finite volume logarithms

Quenched chiral perturbation theory is used to compute the first finite volume correction to the chiral condensate. The correction diverges logarithmically with the four-volume $V$. We point out that with dynamical quarks one can obtain both the chiral condensate and the pion decay constant from the distributions of the lowest Dirac operator eigenvalues.

Residual chiral symmetry breaking in domain-wall fermions

We study the effective quark mass induced by the finite separation of the domain walls in the domain-wall formulation of chiral fermion as the function of the size of the fifth dimension ( L s ), the gauge coupling (β) and the physical volume ( V). We measure the mass by calculating the small eigenvalues of the hermitian domain-wall Dirac operator ( H DWF( m 0 = 1.8)) in the topologically-nontrivial quenched SU(3) gauge configurations. We find that the induced quark mass is nearly independent of the physical volume, decays exponentially as a function of L s , and has a strong dependence on the size of quantum fluctuations controlled by β. The effect of the choice of the lattice gluon action is also studied.

Residual chiral symmetry breaking in domain-wall fermions

We study the effective quark mass induced by the finite separation of the domain walls in the domain-wall formulation of chiral fermion as the function of the size of the fifth dimension ($L_s$), the gauge coupling $\beta$ and the physical volume $V$. We measure the mass by calculating the small eigenvalues of the hermitian domain-wall Dirac operator ($H_{\rm DWF}(m_0))$ in the topologically-nontrivial quenched SU(3) gauge configurations. We find that the induced quark mass is nearly independent of the physical volume, decays exponentially as a function of $L_s$, and has a strong dependence on the size of quantum fluctuations controlled by $\beta$. The effect of the choice of the lattice gluon action is also studied.

Massive random matrix ensembles atβ=1and 4: QCD in three dimensions

The zero momentum sectors in effective theories of three dimensional QCD coupled to pseudoreal (two colors) and real (adjoint) quarks in a classically parity-invariant manner have alternative descriptions in terms of orthogonal and symplectic ensembles of random matrices. Using this correspondence, we compute finite-volume QCD partition functions and correlation functions of Dirac operator eigenvalues in a presence of finite quark masses of the order of the smallest Dirac eigenvalue. These novel correlation functions, expressed in terms of quaternion determinants, are reduced to conventional results for the Gaussian ensembles in the quenched limit.

Eigenvalue distributions of the QCD Dirac operator

We compute by Monte Carlo methods the individual distributions of the kth smallest Dirac operator eigenvalues in QCD, and compare them with recent analytical predictions. We do this for both massless and massive quarks in an SU(3) gauge theory with staggered fermions. Very precise agreement is found in all cases. As a simple by-product we also extract the microscopic spectral density of the Dirac operator in SU(3) gauge theory with dynamical massive fermions for N f =1 and 2, and obtain high-accuracy agreement with analytical expressions.

Domain-wall induced quark masses in topologically nontrivial backgrounds

In the domain-wall formulation of chiral fermion, the finite separation between domain-walls ($L_s$) induces an effective quark mass ($m_{\rm eff}$) which complicates the chiral limit. In this work, we study the size of the effective mass as the function of $L_s$ and the domain-wall height $m_0$ by calculating the smallest eigenvalue of the hermitian domain-wall Dirac operator in the topologically-nontrivial background fields. We find that, just like in the free case, $m_{\rm eff}$ decreases exponentially in $L_s$ with a rate depending on $m_0$. However, quantum fluctuations amplify the wall effects significantly. Our numerical result is consistent with a previous study of the effective mass from the Gell-Mann-Oakes-Renner relation.

Low-lying eigenvalues of the QCD Dirac operator at finite temperature

We compute the low-lying spectrum of the staggered Dirac operator above and below the finite temperature phase transition in both quenched QCD and in dynamical four flavor QCD. In both cases we find, in the high temperature phase, a density with close to square root behavior, $\rho(\lambda) \sim (\lambda-\lambda_0)^{1/2}$. In the quenched simulations we find, in addition, a volume independent tail of small eigenvalues extending down to zero. In the dynamical simulations we also find a tail, decreasing with decreasing mass, at the small end of the spectrum. However, the tail falls off quite quickly and does not seem to extend to zero at these couplings. We find that the distribution of the smallest Dirac operator eigenvalues provides an efficient observable for an accurate determination of the location of the chiral phase transition, as first suggested by Jackson and Verbaarschot.

Massive chiral random matrix ensembles atβ=1and 4: QCD Dirac operator spectra

The zero momentum sectors in effective theories of quantum chromodynamics coupled to pseudoreal (two colors) and real (adjoint) quarks have alternative descriptions in terms of chiral orthogonal and symplectic ensembles of random matrices. Using this correspondence, we compute correlation functions of Dirac operator eigenvalues within a sector with an arbitrary topological charge in the presence of finite quark masses of the order of the smallest Dirac eigenvalue. These novel correlation functions, expressed in terms of Pfaffians, interpolate between known results for the chiral and quenched limits as quark masses vary.

Looking for effects of topology in the Dirac spectrum of staggered fermions

We classify SU(3) gauge field configurations in different topological sectors by the smearing technique. In each sector we compute the distribution of low lying eigenvalues of the staggered Dirac operator. In all sectors we find perfect agreement with the predictions for the sector of topological charge zero. The smallest Dirac operator eigenvalues of staggered fermions at presently realistic lattice couplings are thus insensitive to gauge field topology. On the smeared configurations, 4ν eigenvalues go to zero in agreement with the index theorem.