Low-lying Eigenmodes Research Articles

The acoustics of plasma-sprayed, thick-thermal-barrier coatings

Plasma-sprayed, thick-thermal-barrier coatings are very porous, approximately 2 mm thick, coatings used to insulate metal engine parts from high-temperature combustion processes. They are grown such that their elastic constants and density increase with depth; at the surface the elastic constants take small values, especially the shear modulus which is very small, but rapidly increase to those typical of the metal substrate. A Thomson–Haskell matrix technique is used to find numerically the eigenmode(s) along each vertical line through the coating and substrate, at each station along the direction of propagation. Using the lowest eigenmode, calculated at each station, a JWKB approximation to propagation in a coating with variable thickness is calculated, and from this the changes in travel time caused by varying degress of spalling at the surface are worked out. Using the two or three lowest eigenmodes for a coating of uniform thickness, a coupled-mode calculation is used to determine to what degree an interfacial inclusion couples the lowest eigenmode into the next higher one. In both cases the goal is to calculate the changes caused by the defects so that their presence may be detected. [Work supported by NSF.]

The low-lying Dirac eigenmodes from domain wall fermions

We calculate the low-lying eigenvalues and eigenvectors of the hermitian domain wall Dirac operator on various gauge backgrounds by Ritz minimization. The mass dependence of these eigenvalues is studied to extract the physical 4 dimensional λ, whose spectral density is related to 〈 ψ ψ〉 through the Banks-Casher relation, and δ m, which represents the effects of the residual chiral symmetry breaking in domain wall formalism on a per eigenmode basis. The topological structure of the underlying gauge field is examined by measuring the Γ 5 matrix elements between the low-lying eigenmodes.

Lattice QCD at finite temperature: Evidence for calorons from the eigenvectors of the Dirac operator

We analyze the eigenvalues and eigenvectors of the staggered Dirac operator in quenched lattice QCD in the vicinity of the deconfinement phase transition using the Lüscher-Weisz gauge action. The spectral and localization properties of the low-lying eigenmodes show characteristic differences between the Z 3 sectors above the critical temperature T c . These findings can be interpreted in terms of calorons.

Efficient computation of low-lying eigenmodes of non-Hermitian Wilson-Dirac type matrices

A polynomial transformation for non-Hermitian matrices is presented, which provides access to wedge-shaped spectral windows. For Wilson-Dirac type matrices this procedure not only allows the determination of the physically interesting low-lying eigenmodes but also provides a substantial acceleration of the eigenmode algorithm employed.

Zero-modes of the QED Neuberger Dirac operator

We consider 4 d compact lattice QED in the quenched approximation. First, we briefly summarize the spectrum of the staggered Dirac operator and its connection with random matrix theory. Afterwards we present results for the low-lying eigenmodes of the Neuberger overlap-Dirac operator. In the strong coupling phase we find exact zero-modes. Subsequently we discuss possibly related topological excitations of the U(1) lattice gauge theory.

Low-lying fermion modes of Nf = 2 improved Wilson fermions

We present preliminary results for the topological charge and susceptibility determined from the low-lying eigenmodes of the Wilson-Dirac operator. These modes have been computed on dynamical configurations with N f = 2 non-perturbatively improved Wilson fermions. We compare our results with the eigenmodes of fermions in the quenched approximation.

Low-lying eigenmodes of the Wilson–Dirac operator and correlations with topological objects

The probability density of low-lying eigenvectors of the hermitian Wilson–Dirac operator H( κ)= γ 5 D W( κ) is examined. Comparisons in position and size between eigenvectors, topological charge and action density are made. We do this for standard Monte-Carlo generated SU(3) background fields and for single instanton background fields. Both hot and cooled SU(3) background fields are considered. An instanton model is fitted to eigenmodes and topological charge density and the sizes and positions of these are compared.

Giant eigenproblems from lattice gauge theory on CRAY T3E systems

The determination of physical properties of flavor singlet objects like the η′ meson by computer simulation requires the computation of functionals of the inverse fermionic matrix M −1. So far, only stochastic methods could cope with the enormous size of M. In this paper, we introduce an alternative approach which is based on the computation of a subset of low-lying eigenmodes of the fermionic matrix. The high quality of this ‘truncated eigenmode approximation’ (TEA) is demonstrated by comparison with the pion correlator, a flavor octet quantity, which is readily computable through a linear system of equations. We show that TEA can successfully approximate the flavor singlet η′ correlator. We find that the systematic error of the method is tolerable. As the determination of the chosen subset of 300 eigenmodes requires about 3.5 Tflops-hours CPU-time per canonical ensemble and at least 15 GBytes of memory, the power of high-end supercomputers like the CRAY T3E is indispensable.

Evidence against instanton dominance of topological charge fluctuations in QCD

The low-lying eigenmodes of the Dirac operator associated with typical gauge field configurations in QCD encode, among other low-energy properties, the physics behind the solution to the ${\mathrm{U}}_{A}(1)$ problem (i.e., the origin of the ${\ensuremath{\eta}}^{\ensuremath{'}}$ mass), the nature of spontaneous chiral symmetry breaking, the physics of string-breaking, quark-antiquark pair production, and the Okubo-Zweig-Iizuka (OZI) rule. Moreover, the space-time chiral structure of these eigenmodes reflects the space-time topological structure of the underlying gauge field. We discuss initial evidence from lattice QCD on the local chiral structure of low Dirac eigenmodes which suggests that topological charge fluctuations of the QCD vacuum are not instanton dominated. Such a conclusion would support Witten's arguments that topological charge is produced by confinement-related gauge fluctuations rather than instantons.

Low fermionic eigenmode dominance in QCD on the lattice

We demonstrate the utility of a spectral approximation to fermion loop operators using low-lying eigenmodes of the Hermitian Dirac-Wilson matrix, $Q={\ensuremath{\gamma}}_{5}M.$ The investigation is based on a total of 400 full QCD vacuum configurations, with two degenerate flavors of dynamical Wilson fermions at $\ensuremath{\beta}=5.6,$ at two different sea quark masses. The spectral approach is highly competitive for accessing both topological charge and disconnected diagrams, on large lattices and small quark masses. We propose suitable partial summation techniques that provide sufficient saturation for estimating $\mathrm{Tr}{Q}^{\ensuremath{-}1},$ which is related to the topological charge. In the effective mass plot of the ${\ensuremath{\eta}}^{\ensuremath{'}}$ meson we achieved a consistent early plateau formation, by ground state projecting the connected piece of its propagator.

Short distance current correlators: Comparing lattice simulations to the instanton liquid

Point to point correlators of currents are computed in quenched QCD using a chiral lattice fermion action, the overlap action. I compare correlators made of exact quark propagators with correlators restricted to low (less than 500 MeV) eigenvalue eigenmodes of the Dirac operator. In many cases they show a qualitative resemblance (typically at small values of the quark mass and distances larger than 0.4 fm) and they differ qualitatively at larger quark masses or at very short distance. Lattice results are in qualitative agreement (and in the difference of vector and axial vector channels, quantitative agreement) with the expectations of instanton liquid models. The scalar channel shows the effects of a quenched finite volume zero mode artifact, a negative correlator.

LOW DIRAC EIGENMODES AND THE TOPOLOGICAL AND CHIRAL STRUCTURE OF THE QCD VACUUM

Several lattice calculations which probe the chiral and topological structure of QCD are discussed. The results focus attention on the low-lying eigenmodes of the Dirac operator in typical gauge field configurations.

Exact zero-modes of the compact QED Dirac operator

We calculate the low-lying eigenmodes of the Neuberger overlap-Dirac operator for 4d compact lattice QED in the quenched approximation. In the strong coupling phase we find exact zero-modes, quite similar as in non-Abelian lattice QCD. Subsequently we make an attempt to identify responsible topological excitations of the U(1) lattice gauge theory.

Rayleigh wave propagation in a curved elastic waveguide

A JWKB asymptotic expansion describing in‐plane elastic waves is used to approximate a Rayleigh‐like wave guided within an elastic waveguide whose curvature is small and changes slowly over a wavelength. The two lowest eigenmodes in a curved guide, taken together, constitute the Rayleigh‐like wave. It is shown that this wave lies in the shadows of four closely spaced, virtual caustics. If the curvature becomes too large, the virtual caustic closest to the outer surface enters the guide, after which Rayleigh‐like propagation ceases. This wave is dispersive: The dispersion arises both from the confinement within the guide and from the curvature. Propagation in both a thin and a thick guide are examined. Propagation into an environment of increasing curvature is studied, for both waveguides, to exhibit the influence of nearby caustics.

Generic representation of active cavity VCSEL eigenmodes by optimized waist Gauss-Laguerre modes

Gain-guided eigenmodes in open VCSEL cavities are constructed by superposition of paraxial (i.e., Gauss-Laguerre) (GL) modes, employing the effective cavity hard mirror equivalent for the DBRs. A generic round-trip matrix is obtained analytically for simple gain profiles, including finite mirror diameter losses, diffraction spreading and aperture scattering effects. Diagonalization yields the full range of stable, unstable, and steady-state complex eigenmodes and gain eigenvalues. More importantly, it is demonstrated that in cases of interest the lower order cavity eigenmodes can be approximated by pure GL modes with optimum waist size prescribed through a variational principle. A simple analytic relation is thus obtained for the mode waist for a variety of laterally open cavities. The theory is confirmed by comparison with experimental results. The GL eigenmode properties account for wavelength blue-shifting, increasing density threshold current and increasing differentiation in modal losses with decreasing current aperture. They also yield the correct aperture placement effects in the cavity standing wave. Diffraction and scattering losses are shown to dominate over mirror losses at small cavity apertures.

Geometric Gaussianity and non-Gaussianity in the cosmic microwave background

In this paper, the Gaussianity of eigenmodes and non-Gaussianity in the cosmic microwave background (CMB) temperature fluctuations in the two smallest compact hyperbolic (CH) models are investigated. First, it is numerically found that the expansion coefficients of low-lying eigenmodes on the two CH manifolds behave as if they are Gaussian random numbers in almost all places. Next, the non-Gaussianity of the temperature fluctuations in the $(l,m)$ space in these models is studied. Assuming that the initial fluctuations are Gaussian, the real expansion coefficients ${b}_{\mathrm{lm}}$ of the temperature fluctuations in the sky are found to be distinctively non-Gaussian. In particular, the cosmic variances are found to be much larger than for Gaussian models. On the other hand, the anisotropic structure is vastly erased if one averages the fluctuations at a number of different observation points because of the Gaussian pseudorandomness of the eigenmodes. Thus the dominant contribution to the two-point correlation functions comes from the isotropic terms described by the angular power spectra ${C}_{l}.$ Finally, topological quantities, the total length and the genus of isotemperature contours are investigated. The variances of total length and genus at high and low threshold levels are found to be considerably larger than that of Gaussian models while the means almost agree with them.

Computation of eigenmodes on a compact hyperbolic 3-space

Measurements of cosmic microwave background (CMB) anisotropy are ideal experiments for discovering the non-trivial global topology of the universe. To evaluate the CMB anisotropy in multiply-connected compact cosmological models, one needs to compute the eigenmodes of the Laplace - Beltrami operator. Using the direct boundary element method, we obtain numerically the low-lying eigenmodes on a compact hyperbolic 3-space called the Thurston manifold which is the second smallest in the known compact hyperbolic 3-manifolds. The computed eigenmodes are expanded in terms of eigenmodes on the unit three-dimensional pseudosphere. We find numerically that the expansion coefficients behave as Gaussian pseudo-random numbers for low-lying eigenmodes. The observed Gaussianity in the CMB fluctuations can partially be attributed to the Gaussian pseudo-randomness of the expansion coefficients assuming that the Gaussian pseudo-randomness is the universal property of the compact hyperbolic spaces.

Observation of Bound States and Counterrotating Lower Hybrid Eigenmodes in the Auroral Ionosphere

Definitive evidence from a plasma wave interferometer carried on a sounding rocket is presented for the observation of counterrotating electric field structures observed at frequencies above and below the local lower hybrid resonance frequency, ${f}_{\mathrm{LHR}}$, of the ambient plasma. These counterrotating structures were observed within the context of a lower hybrid solitary structure (LHSS), and were found to rotate in a right-handed (left-handed) sense around the geomagnetic field at frequencies above (below) ${f}_{\mathrm{LHR}}$, in accord with previous predictions for and observations of electric field eigenmodes within a density depletion. The presence of a narrow band feature below ${f}_{\mathrm{LHR}}$ indicates the presence of trapped waves, and represents clear evidence of a bound state of lower hybrid waves in the auroral ionosphere.

Localized magnon mode of in-plane magnetic vortices in easy-plane magnets

We consider localized magnon modes of magnetic vortices in two-dimensional classical magnets, with exchange or single-ion easy-plane anisotropy stronger than the critical value required to stabilize in-plane vortices. A discrete lattice ansatz for the structure of a magnon mode on one vortex is analyzed. Its lowest eigenmodes are found to be identical with modes obtained from numerical diagonalization for ferro- (FM) and antiferromagnets (AFM), showing that the ansatz is exact. For the AFM model, one mode is found to be localized, with frequency reaching a size-independent asymptotic limit. A continuum treatment leads to an effective Schr\"odinger problem that requires a small-radius cutoff due to the singularity of the vortex core. We find, however, that it is not possible to choose this cutoff consistently to recover the exact local mode frequency from the ansatz. This suggests that strong discrete lattice effects not well described by the usual continuum approximation always appear from the core of in-plane vortices.

Topological structure of the SU(3) vacuum and exceptional eigenmodes of the improved Wilson-Dirac operator

We present a study of the instanton size and spatial distributions in pure SU(3) gauge theory using under-relaxed cooling. We also investigate the low-lying eigenmodes of the (improved) Wilson-Dirac operator, in particular, the appearance of zero-modes and their space-time localisation with respect to instantons in the underlying gauge field.