Logarithmic Singularity Research Articles

Resonances in the generalized LC model of granular metal–dielectric nanocomposites

LC networks in the form of spatial lattices with randomly arranged inductive and capacitive bonds, which serve as an adequate theoretical model for describing plasmon resonances in disordered granular metal–dielectric composites, have been considered. Earlier, such networks were theoretically studied only in the case when the values of all L- and C-bonds were equal. This approximation is not applicable when the sizes of metallic granules and dielectric gaps between them appreciably fluctuate. In the present work, a generalized model making it possible to consider networks with arbitrary values of L and C is developed. Using this generalized model, the spectral density of resonant states for networks with different values of inductive and capacitive bonds is studied. It is shown that, in the case of a discrete set of values of L- and/or C-bonds, the typical peaks in the density of states are split and, in the case of a continuous distribution of these values, they are smoothened. Anticrossing of frequencies due to dipole–dipole interaction of single resonant clusters is studied, and a logarithmic singularity in the spectral state density at the ends of the spectrum for smooth distributions of L and C is found.

Rough estimates and binomial approximations for Crocco's equation in the boundary problems

In order to solve the Crocco boundary problems known as the typical one and the uniform one, binomials (as approximants of exact solutions) and integral identities have been used. The extent of the closeness of the exact solution to its approximation was estimated using the φ(0) value. The solution of the typical Crocco boundary problem was proved to have a logarithmic singularity of the derivative at φ=0. Crocco's equation was found to provide both necessary and sufficient conditions for the minimum of a positive distribution being vortex in dφ/dh. The uniform Crocco boundary problem was demonstrated to be equivalent to the two typical Crocco boundary problems with a common critical point.

How does surface tension affect energy release rate of cracks loaded in Mode I?

The role of surface stress on fracture of elastic solids has been studied by two group of researchers with different conclusions. One group concludes that surface stress has no effect on energy release rate except that the singularity of the crack tip field is stronger than the usual inverse square root singularity dictated by linear elastic fracture mechanics. The other group concludes that the singularity of the stress field reduces to a logarithmic singularity, thus implying that the local energy release rate is zero. In this letter we resolve this paradox by examining the solution of a special case where surface stress is isotropic and independent of surface strain. We show that surface tension resists crack growth by lowering the applied energy release rate while retaining the inversely square root singularity of the elastic crack tip field. We demonstrate this idea by solving a perturbation problem where the capillary length is much smaller than the crack length. A closed form expression for the local energy release rate is obtained in the limit of small surface tension.

Approximating the Ising model on fractal lattices of dimension less than two

.We construct periodic approximations to the free energies of Ising models on fractal lattices of dimension smaller than two, in the case of a zero external magnetic field, based on the combinatorial method of Feynman and Vdovichenko. We show that the procedure is applicable to any fractal obtained by the removal of sites from a periodic two-dimensional lattice. As a first application, we compute estimates for the critical temperatures of many different Sierpinski carpets and we compare them to known Monte Carlo estimates. The results show that our method is capable of determining the critical temperature with, possibly, arbitrary accuracy and paves the way for determination of any fractal of dimension less than two. Critical exponents are more difficult to determine since the free energy of any periodic approximation still has a logarithmic singularity at the critical point implying . We also compute the correlation length as a function of the temperature and extract the relative critical exponent. We find for all periodic approximations, as expected from universality.

Displaced logarithmic profile of the velocity distribution in the boundary layer of a turbulent flow over an unbounded flat surface

It is shown that the Reynolds equations for a turbulent flow over an unbounded flat surface in the presence of a constant pressure-gradient lead to a displaced logarithmic profile of the velocity distribution; the displaced logarithmic profile is obtained by assuming a constant production rate of turbulence energy. The displacement height measured on the (vertical) axis perpendicular to the surface is either positive or negative. For a positive displacement height the boundary layer exhibits an inversion, while for a negative displacement height the boundary layer is a direct one. In an inversion boundary layer the logarithmic velocity profile is disrupted into two distinct branches separated by a logarithmic singularity. The viscosity transforms this logarithmic singularity into a sharp edge, governed by a generalized Reynolds number. The associated temperature distribution is calculated, and the results are discussed in relation to meteorological boundary-layer jets and stratified layers. The effects of gravitation and atmospheric thermal or fluid-mixture concentration gradients (“external forcings”) are also considered; it is shown that such circumstances may lead to various modifications of the boundary layers. A brief presentation of a similar situation is described for a circular pipe.

On the approximation by product rules of weakly singular double integrals over the square

Double integrals with algebraic and/or logarithmic singularities are of interest in the application of boundary element method, e.g. linear theory of the aerodynamics of slender bodies of revolution and in many other fields, for example computational electromagnetics. Therefore, the numerical evaluation of such type of integrals deserves attention. In this connection we propose here product interpolatory rules based on suitable Jacobi zeros, giving numerical tests to show the goodness of the proposed algorithm as well as from a point of view of convergence also for the simplicity of implementation.

Problems of polygonal inclusions in orthotropic materials with due consideration on the stresses at corners

Two-dimensional elastic Green’s functions in an orthotropic medium are obtained and expressed in a more explicit and detailed form than those shown in the preceding publications. The stress field associated with a line segment loaded by a uniform traction is found by a direct integration, and it has logarithmic singularities at both ends of the line segment. The basic solutions for the line segment are exploited to solve problems of polygonal inclusions. Stress distributions of circular, triangular, and square inclusions subjected to uniform dilatational eigenstrains in titanium single crystals are presented to show the effectiveness and accuracy of the present approach. The coefficients characterizing the strength of singularity for the stresses at corners are also numerically determined and discussed.

Surface states in a 3D topological insulator: The role of hexagonal warping and curvature

We explore a combined effect of hexagonal warping and a finite effective mass on both the tunneling density of electronic surface states and the structure of Landau levels of 3D topological insulators. We find the increasing warping to transform the square-root van Hove singularity into a logarithmic one. For moderate warping, an additional logarithmic singularity and a jump in the tunneling density of surface states appear. By combining the perturbation theory and the WKB approximation, we calculate the Landau levels in the presence of hexagonal warping. We predict that due to the degeneracy removal, the evolution of Landau levels in the magnetic field is drastically modified.

Ratner’s property and mild mixing for smooth flows on surfaces

Let${\mathcal{T}}=(T_{t}^{f})_{t\in \mathbb{R}}$be a special flow built over an IET$T:\mathbb{T}\rightarrow \mathbb{T}$of bounded type, under a roof function$f$with symmetric logarithmic singularities at a subset of discontinuities of$T$. We show that${\mathcal{T}}$satisfies the so-called switchable Ratner’s property which was introduced in Fayad and Kanigowski [On multiple mixing for a class of conservative surface flows.Invent. Math.to appear]. A consequence of this fact is that such flows are mildly mixing (before, they were only known to be weakly mixing [Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations.J. Mod. Dynam.3(2009), 35–49] and not mixing [Ulcigrai. Absence of mixing in area-preserving flows on surfaces.Ann. of Math.(2)173(2011), 1743–1778]). Thus, on each compact, connected, orientable surface of genus greater than one there exist flows that are mildly mixing and not mixing.

Isospin breaking decayη(1405)→f0(980)π0→3π

There are attempts in the literature to theoretically explain the large breaking of isotopic invariance in the decay $\eta(1405)$\,$\to$\,$f_0(980)\pi^0$\,$\to$\,$ 3\pi$ by the mechanism containing the logarithmic (triangle) singularity, i.e., as being due to the transition $\eta(1405)\to(K^*\bar K+\bar K^*K)\to(K^+K^-+K^0\bar K^0)\pi^0\to f_0(980)\pi^0\to3\pi$. The corresponding calculations were fulfilled for a hypothetic case of the stable $K^*$ meson. Here, we show that the account of the finite width of the $K^*$ ($\Gamma_{K^*\to K\pi}\approx50$ MeV) smoothes the logarithmic singularities in the amplitude and results in the suppression of the calculated decay width $\eta(1405)\to f_0(980) \pi^0\to3\pi$ by the factor of $6-8$ as compared with the case of $\Gamma_{K^*\to K\pi}$\,=\,0. We also analyze the difficulties related with the assumption of the dominance of the $\eta(1405)\to(K^*\bar K+\bar K^*K)\to K\bar K\pi$ decay mechanism and discuss the possible dynamics of the decay $\eta(1405)\to\eta\pi \pi$. The decisive improvement of the experimental data on the $K\bar K$, $ K\pi$, $\eta\pi$, and $\pi\pi$ mass spectra in the decay of the resonance structure $\eta(1405/1475)$ to $K\bar K\pi$ and $\eta\pi\pi$, and on the shape of the resonance peaks themselves in the $K\bar K\pi$ and $\eta\pi\pi $ decay channels is necessary for the further establishing the $\eta(1405)\to3 \pi$ decay mechanism.

Positive amplitudes in the amplituhedron

The all-loop integrand for scattering amplitudes in planar $$ \mathcal{N}=4 $$ SYM is determined by an “amplitude form” with logarithmic singularities on the boundary of the amplituhedron. In this note we provide strong evidence for a new striking property of the superamplitude, which we conjecture to be true to all loop orders: the amplitude form is positive when evaluated inside the amplituhedron. The statement is sensibly formulated thanks to the natural “bosonization” of the superamplitude associated with the amplituhedron geometry. However this positivity is not manifest in any of the current approaches to scattering amplitudes, and in particular not in the cellulations of the amplituhedron related to on-shell diagrams and the positive grassmannian. The surprising positivity of the form suggests the existence of a “dual amplituhedron” formulation where this feature would be made obvious. We also suggest that the positivity is associated with an extended picture of amplituhedron geometry, with the amplituhedron sitting inside a co-dimension one surface separating “legal” and “illegal” local singularities of the amplitude. We illustrate this in several simple examples, obtaining new expressions for amplitudes not associated with any triangulations, but following in a more invariant manner from a global view of the positive geometry.

Difference Between the Partial Self-Inductance at DC and at HF

The difference of the partial self-inductance of a single conductor is determined between two extremes of the frequency range: at dc and at HF (infinite frequency). They are first calculated separately for a finite large length. Both contain the same logarithm(length) singularity, but this disappears in their difference, which only involves the logarithm of the ratio of the equivalent radius and the geometrical mean distance. We have applied this to some common conductor cross sections: circle, ellipse, rectangle, and diamond and to the limit of zero thickness of these cross sections. In the latter case, a general expression is derived.

A numerical framework for singular limits of a class of reaction diffusion problems

We present a numerical framework for solving localized pattern structures of reaction–diffusion type far from the Turing regime. We exploit asymptotic structure in a set of well established pattern formation problems to analyze a singular limit model that avoids time and space adaptation typically associated to full numerical simulations of the same problems. The singular model involves the motion of a curve on which one of the chemical species is concentrated. The curve motion is non-local with an integral equation that has a logarithmic singularity. We generalize our scheme for various reaction terms and show its robustness to other models with logarithmic singularity structures. One such model is the 2D Mullins–Sekerka flow which we implement as a test case of the method. We then analyze a specific model problem, the saturated Gierer–Meinhardt problem, where we demonstrate dynamic patterns for a variety of parameters and curve geometries.

New insights into the problem with a singular drift term in the complex Langevin method

The complex Langevin method aims at performing path integral with a complex action numerically based on complexification of the original real dynamical variables. One of the poorly understood issues concerns occasional failure in the presence of logarithmic singularities in the action, which appear, for instance, from the fermion determinant in finite density QCD. We point out that the failure should be attributed to the breakdown of the relation between the complex weight that satisfies the Fokker-Planck equation and the probability distribution associated with the stochastic process. In fact, this problem can occur in general when the stochastic process involves a singular drift term. We show, however, in a simple example that there exists a parameter region in which the method works although the standard reweighting method is hardly applicable.

On quadrature of highly oscillatory integrals with logarithmic singularities

In this paper a quadrature rule is discussed for highly oscillatory integrals with logarithmic singularities. At the same time, its error depends on the frequency ω and the computation of its moments are given. The new rule is implemented by interpolating f at Chebyshev nodes and singular point where the interpolation polynomial satisfies some conditions. Numerical experiments conform the efficiency for obtaining the approximations.

Erratum: Spin waves in a triangular lattice antiferromagnet: Decays, spectrum renormalization, and singularities [Phys. Rev. B79, 144416 (2009)

Spin-waves in triangular lattice antiferromagnet: decays, spectrum renormalization, and singularities A. L. Chernyshev Department of Physics, University of California, Irvine, California 92697, USA and Max-Planck-Institut f¨ ur Physik komplexer Systeme, 01187 Dresden, Germany M. E. Zhitomirsky Commissariat a ` l’Energie Atomique, DSM/INAC/SPSMS, F-38054 Grenoble, France (Dated: June 12, 2015) We present a comprehensive study of the dynamical properties of the quantum Heisenberg an- tiferromagnet on a triangular lattice within the framework of spin-wave theory. The distinct fea- tures of spin-wave excitations in the triangular-lattice antiferromagnet are (i) finite lifetime at zero temperature due to spontaneous two-magnon decays, (ii) strong renormalization of magnon ener- gies e k with respect to the harmonic result, and (iii) logarithmic singularities in the decay rate Γ k . Quantum corrections to the magnon spectrum are obtained using both the on-shell and off- shell solutions of the Dyson equation with the lowest-order magnon self-energy. At low-energies magnon excitations remain well-defined albeit with the anomalous decay rate Γ k ∝ k 2 at k → 0 and Γ k ∝ |k − Q AF | 7/2 at k → Q AF . At high energies, magnons are heavily damped with the decay rate reaching (2Γ k /e k ) ∼ 0.3 for the case S = 1/2. The on-shell solution shows logarithmic singularities in Γ k with the concomitant jump-like discontinuities in Re[e k ] along certain contours in the momen- tum space. Such singularities are even more prominent in the magnon spectral function A(k, ω). Although the off-shell solution removes such log-singularities, the decay rates remain strongly en- hanced. We also discuss the role of higher-order corrections and show that such singularities may lead to complete disappearance of the spectrum in the vicinity of certain k-points. The kinematic conditions for two-magnon decays are analyzed for various generalizations of the triangular-lattice antiferromagnet as well as for the XXZ model on a kagom´e lattice. Our results suggest that decays and singularities in the spin-wave spectra must be ubiquitous in all these systems. In addition, we give a detailed introduction in the spin-wave formalism for noncollinear Heisenberg antiferromagnets and calculate several quantities for the triangular-lattice model including the ground-state energy and the sublattice magnetization. PACS numbers: 75.10.Jm, 75.30.Ds, 78.70.Nx I. INTRODUCTION Heisenberg antiferromagnet (HAF) on a triangular lat- tice has been a focus of much attention as one of the basic model systems in which geometric frustration and low dimensionality are expected to yield new physical phenomena. Although available experimental realiza- tions of the triangular-lattice antiferromagnet 1–6 are de- scribed by such a model only approximately, either due to anisotropies or because of additional interactions, the ideal nearest-neighbor Heisenberg antiferromagnet on a triangular lattice given by ˆ = J H X S i · S j , hiji remains the principal reference point. The semiclassical S 1 triangular-lattice HAF or- ders in the so-called 120 ◦ structure, see Fig. 1. Histori- cally, it was anticipated that enhanced quantum fluctua- tions destroy the long-range antiferromagnetic order for the spin-1/2 model. 7 However, calculations of quantum corrections within the spin-wave theory have suggested that the 120 ◦ magnetic structure remains stable even for S = 1/2. 8–11 The early numerical results for small clusters were less conclusive, some supporting magnet- ically disordered state 12 and some confirming the spin- wave results. 13 Since the quantum Monte Carlo suffers from the infamous sign problem when applied to frus- trated models, it is not until the Green’s function Monte Carlo work 14 that the magnetically ordered ground state of the spin-1/2 triangular-lattice HAF has been gener- ally agreed upon. More recent series-expansion 15 and density-matrix renormalization group (DMRG) studies 16 have confirmed the stability of the 120 ◦ spin structure for the case of S = 1/2 and yielded the value of ordered mo- ments hSi ≈ 0.20, close to the previous result. 14 Gradually, it has been recognized that the truly dis- tinct physics of the quantum triangular-lattice antiferro- magnet concerns its excitation spectrum and the thermo- dynamic properties. The anomalous behavior of the lat- ter has been discovered earlier by the high-temperature series-expansion study. 17 The temperature dependence of such quantities as entropy or susceptibility, exhibits significant differences between the triangular- and the square-lattice models: upon lowering temperature down to about J/2 the square-lattice antiferromagnet shows strong signs of ordering, while the triangular-lattice one does not. More recently, developments in the series- expansion method have allowed to calculate the excita-

Axisymmetric problems of a penny-shaped crack at the interface of a bi-material under shear and compression

This paper studies an axisymmetric problem of a penny-shaped interface crack of a bi-material subjected to compression and radial shear at the crack surfaces in cylindrical coordinate system. The Hankel transform technique is applied to convert the problem to dual integral equations. Closed-form solutions for mode-I and II stress intensity factors are obtained for arbitrary radial shear loading at the crack surfaces. Obtained results indicate that no oscillation singularity occurs. Explicit expressions for the stresses and displacements in the whole bi-material elastic space are derived for constant and linear radial shear stress acting on the crack surfaces, respectively. Under uniform radial shear stress, normal and shear stresses have a usual square-root singularity behind and ahead of the crack front, respectively, and the induced normal stress exhibits a logarithmic singularity at the crack center. Under linearly-distributed radial shear stress, the logarithmic singularity at the crack center disappears, and the square-root singularity is still present for the normal and shear stresses near the crack front. The induced mode-I stress intensity factors depend on Dundurs’ parameter and on applied shear loading. Numerical results for a penny-shaped interface crack in Al2O3/PMMA bi-material are presented to show the influence of the bi-material properties on the stress distribution.

The Singular Hill Equation and Generalized Lindemann–Stieltjes Method

UDC 517.9 Based on the Lindemann–Stieltjes method, we propose an approach to the solution of a singular Hill equation. We consider Hill equations with logarithmic and fractional power singularities. In the space of parameters, we find resonance zones and compute the Floquet exponent. Bibliography :1 2titles. Illustrations :4 figures.

Logarithmic singularities and maximally supersymmetric amplitudes

The dual formulation of planar N = 4 super-Yang-Mills scattering amplitudes makes manifest that the integrand has only logarithmic singularities and no poles at infinity. Recently, Arkani-Hamed, Bourjaily, Cachazo and Trnka conjectured the same singularity properties hold to all loop orders in the nonplanar sector as well. Here we conjecture that to all loop orders these constraints give us the key analytic information contained in dual conformal symmetry. We also conjecture that to all loop orders, while N = 8 supergravity has poles at infinity, at least at four points it has only logarithmic singularities at finite locations. We provide nontrivial evidence for these conjectures. For the three-loop four-point N = 4 super-Yang-Mills amplitude, we explicitly construct a complete basis of diagram integrands that has only logarithmic singularities and no poles at infinity. We then express the complete amplitude in terms of the basis diagrams, with the coefficients determined by unitarity. We also give examples at three loops showing how to make the logarithmic singularity properties manifest via dlog forms. We give additional evidence at four and five loops supporting the nonplanar logarithmic singularity conjecture. Furthermore, we present a variety of examples illustrating that these constraints are more restrictive than dual conformal symmetry. Our investigations show that the singularity structures of planar and nonplanar amplitudes in N = 4 super-Yang-Mills are strikingly similar. While it is not clear how to extend either dual conformal symmetry or a dual formulation to the nonplanar sector, these results suggest that related concepts might exist and await discovery. Finally, we describe the singularity structure of N = 8 supergravity at three loops and beyond.

On the Order of Growth of the Solutions of Linear Differential Equations in the Vicinity of a Branching Point

Assume that the coefficients and solutions of the equation f (n)+p n−1(z)f (n−1)+. . .+p s+1(z)f (s+1)+ . . . + p 0(z)f = 0 have a branching point at infinity (e.g., a logarithmic singularity) and that the coefficients p j , j = s+1, . . . ,n−1, increase slower (in terms of the Nevanlinna characteristics) than p s (z). It is proved that this equation has at most s linearly independent solutions of finite order.