Logarithmic Singularity Research Articles

Ground states for the planar NLSE with a point defect as minimizers of the constrained energy

We investigate the ground states for the focusing, subcritical nonlinear Schr\"odinger equation with a point defect in dimension two, defined as the minimizers of the energy functional at fixed mass. We prove that ground states exist for every positive mass and show a logarithmic singularity at the defect. Moreover, up to a multiplication by a constant phase, they are positive, radially symmetric, and decreasing along the radial direction. In order to overcome the obstacles arising from the uncommon structure of the energy space, that complicates the application of standard rearrangement theory, we move to the study of the minimizers of the action functional on the Nehari manifold and then establish a connection with the original problem. An ad hoc result on rearrangements is given to prove qualitative features of the ground states.

First-order processes of the de Sitter QED in the rest frame vacuum

The processes in the first order of perturbations of de Sitter quantum electrodynamics (QED) in Coulomb gauge are studied but setting our recently proposed rest frame vacuum of the Dirac field instead of the traditional adiabatic one. This vacuum gives rise to a new phenomenology favouring the transitions between neutral states, i.e. the pair creation from a photon and the lepton creation from vacuum, while the transitions between charged states are inhibited. The probabilities per unit of volume and unit of time are derived in the conformal chart according to a new definition that holds even if the energy is no longer conserved. Surprisingly, these probabilities have new logarithmic singularities apart from the poles usually arising in perturbations. These singularities affect the probabilities in the critical positions in which the fermion momenta are either parallel or anti-parallel and the helicity is conserved. To remove all these singularities, a regularization is first performed to extract the logarithmic one before removing the poles, applying the usual reduction procedure. It is shown that the resulting reduced probabilities reach their maxima only in the critical positions, thus complying with the helicity conservation. The corresponding time-dependent probabilities in the physical local chart govern a simple model of fermion creation from vacuum, vacrightarrow gamma +e^++e^-rightarrow (e^++e^-)+e^++e^-, which rapidly reaches saturation because of the pair creation which is dominant.

Semi-analytic theory of multiphonon effects on the static structure factors of warm solids.

A semi-analytic formula for the temperature-dependent static structure factor S(k) of polycrystalline and amorphous solids applicable to the entire wavenumber (k) range is derived. The formula describes thermal diffuse scattering due to multiphonon processes entirely by a single kernel function without resorting to the standard perturbation expansion. It is analytically proven that S(k → 0) is determined from the one-phonon term, whereas the asymptotic limit S(k → ∞) = 1 can be reproduced through a Gaussian integral of the multiphonon term. The formula also reveals that an enhancement of the one-phonon scattering intensity at each Bragg point is expressed as a logarithmic singularity. Numerical examples for a face-centred cubic polycrystal near the melting point are shown. The present formula is computationally more efficient than other theoretical models, requiring only a one-dimensional integration to obtain S(k) once the elastic part of the structure factor and the Debye-Waller factor are given.

Singularity-free treatment of delta-function point scatterers in two dimensions and its conceptual implications

In two dimensions, the standard treatment of the scattering problem for a delta-function potential, , leads to a logarithmic singularity which is subsequently removed by a renormalization of the coupling constant . Recently, we have developed a dynamical formulation of stationary scattering (DFSS) which offers a singularity-free treatment of this potential. We elucidate the basic mechanism responsible for the implicit regularization property of DFSS that makes it avoid the logarithmic singularity one encounters in the standard approach to this problem. We provide an alternative interpretation of this singularity showing that it arises, because the standard treatment of the problem takes into account contributions to the scattered wave whose momentum is parallel to the detectors’ screen. The renormalization schemes used for removing this singularity has the effect of subtracting these unphysical contributions, while DFSS has a built-in mechanics that achieves this goal.

Triangle singularities in the Tcc+→D*+D0→π+D0D0 decay width

The values of the masses of the particles involved in the decay of $T^+_{cc}\to D^{*+}D^0\to\pi^+D^0D^0$ suggest that due to the final state interactions in the transition vertex $T^+_{cc}\to D^{*+}D^0 $ there may be triangle logarithmic singularities. We discuss their possible role and show that the tree approximation for calculating the decay widths $T^+_{cc}\to(D^{*+}D^0+D^{*0}D^+)\to\pi^+D^0D^0$, $\pi^0D^0 D^+$, $\gamma D^0D^+$ is quite sufficient at the current level of measurement accuracy.

Multiple-SLE κ connectivity weights for rectangles, hexagons, and octagons

In previous work, two of the authors determined, completely and rigorously, a solution space for a homogeneous system of 2N + 3 linear partial differential equations (PDEs) in 2N variables that arises in conformal field theory (CFT) and multiple Schramm–Löwner evolution (SLE κ ). The system comprises 2N null-state equations and three conformal ward identities that govern CFT correlation functions of 2N one-leg boundary operators or SLE κ partition functions. M Bauer et al conjectured a formula, expressed in terms of ‘pure SLE κ partition functions,’ for the probability that the growing curves of a multiple-SLE κ process join in a particular connectivity. In a previous article, we rigorously define certain elements of , which we call ‘connectivity weights,’ argue that they are in fact pure SLE κ partition functions, and show how to find explicit formulas for them in terms of Coulomb gas contour integrals. Our formal definition of the connectivity weights immediately leads to a method for finding explicit expressions for them. However, this method gives very complicated formulas where simpler versions may be available, and it is not applicable for certain values of κ ∈ (0, 8) corresponding to well-known critical lattice models in statistical mechanics. In this article, we determine expressions for all connectivity weights in for N ∈ {1, 2, 3, 4} (those with N ∈ {3, 4} are new) and for so-called ‘rainbow connectivity weights’ in for all . We verify these formulas by explicitly showing that they satisfy the formal definition of a connectivity weight. In appendix B, we investigate logarithmic singularities of some of these expressions, appearing for certain values of κ predicted by logarithmic CFT.

Complex analysis of divergent perturbation theory at finite temperature.

We investigate the convergence properties of finite-temperature perturbation theory by considering the mathematical structure of thermodynamic potentials using complex analysis. We discover that zeros of the partition function lead to poles in the internal energy and logarithmic singularities in the Helmholtz free energy that create divergent expansions in the canonical ensemble. Analyzing these zeros reveals that the radius of convergence increases at higher temperatures. In contrast, when the reference state is degenerate, these poles in the internal energy create a zero radius of convergence in the zero-temperature limit. Finally, by showing that the poles in the internal energy reduce to exceptional points in the zero-temperature limit, we unify the two main mathematical representations of quantum phase transitions.

Effective conductivity of random composites with shape-distributed near-spherical inclusions

While traditional and widely accepted mixing rules are based on the concept of a fixed inclusion shape, real particle composites are usually shape-distributed. For such composite materials, however, the analytical homogenization techniques are not well developed, while their numerical analysis becomes particularly complicated. We introduce and analyze two analytical models for the effective conductivity of a particular class of two-phase composites, namely those containing shape-distributed rigid inclusions. The models are derived with the use of the Bergman–Milton spectral density function approach. As we show, one of them, which involves the Gauss hypergeometric function, provides reasonable agreement with experimental data within the wide range of filling factors for ion-exchange resin beads in aqueous solutions. Another model, which is based on the uniform spectral density function, gives rise to a logarithmic singularity for the effective conductivity at the percolation threshold.

The logarithmic phase singularity in the inverted harmonic oscillator

A spacetime singularity is located at the center of a black hole and surrounded by an event horizon, separating spacetime into two disjunct regions: one of them accessible to an outside observer and one that is not. At the event horizon, a logarithmic phase singularity emerges in the mode functions of a massless scalar field, being characteristic for Hawking radiation emitted by the black hole. We demonstrate that related features are present in the elementary quantum system of an inverted harmonic oscillator. Central to our analysis are the energy eigenfunctions of this system and their phase space representation. At first glance, neither a horizon nor a logarithmic phase dependence are apparent. However, both features are hidden in phase space and revealed by a suitable coordinate transformation. In particular, we show that the Fourier transform of a logarithmic phase leads to an expression that is reminiscent of a specific quantum statistics, governing the reflection and transmission coefficients of the inverted harmonic oscillator.

A singularity analysis in Keer’s elastic indentation problem

A non-axisymmetric three-dimensional frictionless contact problem for an elastic half-space and a cylindrical indenter with the face in a wedge form is considered under the assumption of a circular region of contact. Explicit formulas are derived for the critical indenter displacement and contact force that are needed to ensure full contact over the entire contact region. It is shown that the contact pressure beneath the indenter maintains a logarithmic singularity, which is produced by the wedge edge and is similar to the singularity of the contact pressure in the corresponding two-dimensional contact problem for a wedge rigid indenter.

Boundedness and compactness of a class of integral operators with power and logarithmic singularity when pleq q

In this paper, necessary and sufficient conditions for the boundedness and compactness of one class of integral operators with power and logarithmic singularities in weighted Lebesgue spaces are obtained.

Approaching logarithmic singularities in quasilinear chemotaxis-consumption systems with signal-dependent sensitivities

<p style='text-indent:20px;'>The chemotaxis system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{array}{l}\left\{ \begin{array}{l} u_t = \nabla \cdot \big( D(u) \nabla u \big) - \nabla \cdot \big( uS(x, u, v)\cdot \nabla v\big), \\ v_t = \Delta v -uv, \end{array} \right. \end{array} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>is considered in a bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ n\ge 2 $\end{document}</tex-math></inline-formula>, with smooth boundary.</p><p style='text-indent:20px;'>It is shown that if <inline-formula><tex-math id="M3">\begin{document}$ D: [0, \infty) \to [0, \infty) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ S: \overline{\Omega}\times [0, \infty)\times (0, \infty)\to \mathbb{R}^{n\times n} $\end{document}</tex-math></inline-formula> are suitably smooth functions satisfying</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{array}{l}D(u) \ge k_D u^{m-1} \qquad {\rm{for\; all}}\; u\ge 0 \end{array} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ \begin{array}{l}|S(x, u, v)| \le \frac{S_0(v)}{v^\alpha} \qquad {\rm{for\; all}}\; (x, u, v)\; \in \Omega\times (0, \infty)^2 \end{array} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with some</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE4"> \begin{document}$ \begin{array}{l}m>\frac{3n-2}{2n} \qquad {\rm{and}}\;\alpha\in [0, 1), \end{array} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and with some <inline-formula><tex-math id="M5">\begin{document}$ k_D>0 $\end{document}</tex-math></inline-formula> and nondecreasing <inline-formula><tex-math id="M6">\begin{document}$ S_0: (0, \infty)\to (0, \infty) $\end{document}</tex-math></inline-formula>, then for all suitably regular initial data a corresponding no-flux type initial-boundary value problem admits a global bounded weak solution which actually is smooth and classical if <inline-formula><tex-math id="M7">\begin{document}$ D(0)>0 $\end{document}</tex-math></inline-formula>.</p>

Asymptotic stability of spiky steady states for a singular chemotaxis model with signal-suppressed motility.

We study the nonlinear stability of spiky solutions to a chemotaxis model of consumption type with singular signal-suppressed motility in the half space. We show that, when the no-flux boundary condition for the bacteria density and the nonhomogeneous Dirichlet boundary condition for the nutrient are prescribed, this chemotaxis model admits a unique smooth spiky steady state, and it is nonlinearly stable under appropriate perturbations. The challenge of the problem is that there are two types of singularities involved in the model: one is the logarithmic singularity of the sensitive function; and the other is the inverse square singularity of the motility. We employ a Cole-Hopf transformation to relegate the former singularity to a nonlocality that can be resolved by the method of anti-derivative. To deal with the latter singularity, we construct an approximate system that retains a key structure of the original singular system in the local theory, and develop a new strategy, which combines a weighted elliptic estimate and the weighted energy estimate, to establish a priori estimate in the global theory.

Exact solution and the multidimensional Godunov scheme for the acoustic equations

The acoustic equations derived as a linearization of the Euler equations are a valuable system for studies of multi-dimensional solutions. Additionally they possess a low Mach number limit analogous to that of the Euler equations. Aiming at understanding the behaviour of the multi-dimensional Godunov scheme in this limit, first the exact solution of the corresponding Cauchy problem in three spatial dimensions is derived. The appearance of logarithmic singularities in the exact solution of the 4-quadrant Riemann Problem in two dimensions is discussed. The solution formulae are then used to obtain the multidimensional Godunov finite volume scheme in two dimensions. It is shown to be superior to the dimensionally split upwind/Roe scheme concerning its domain of stability and ability to resolve multi-dimensional Riemann problems. It is shown experimentally and theoretically that despite taking into account multi-dimensional information it is, however, not able to resolve the low Mach number limit.

On the absence of logarithmic singularities in the solutions of Lamé-type equations

The object of this research is linear differential equations of the second order with regular singularities. We extend the concept of a regular singularity to linear partial differential equations. The general solution of a linear differential equation with a regular singularity is a linear combination of two linearly independent solutions, one of which in the general case contains a logarithmic singularity. The well-known Lamé equation, where the Weierstrass elliptic function is one of the coefficients, has only meromorphic solutions. We consider such linear differential equations of the second order with regular singularities, for which as a coefficient instead of the Weierstrass elliptic function we use functions that are the solutions to the first Painlevé or Korteweg – de Vries equations. These equations will be called Lamé-type equations. The question arises under what conditions the general solution of Lamé-type equations contains no logarithms. For this purpose, in the present paper, the solutions of Lamé-type equations are investigated and the conditions are found that make it possible to judge the presence or absence of logarithmic singularities in the solutions of the equations under study. An example of an equation with an irregular singularity having a solution with an logarithmic singularity is given, since the equation, defining it, has a multiple root.

Comparing the Performance of Four Sinc Methods for Numerical Indefinite Integration

Aims/ Objectives: To compare the performance of four Sinc methods for the numerical approximation of indefinite integrals with algebraic or logarithmic end-point singularities.
 Methodology: The first two quadrature formulas were proposed by Haber based on the sinc method, the third is Stengers Single Exponential (SE) formula and Tanaka et al.s Double Exponential (DE) sinc method completes the number. Furthermore, an application of the four quadrature formulas on numerical examples, reveals convergence to the exact solution by Tanaka et al.s DE sinc method than by the other three formulae. In addition, we compared the CPU time of the four quadrature methods which was not done in an earlier work by the same author.
 Conclusion: Haber formula A is the fastest as revealed by the CPU time.

Staticity and regularity for zero rest-mass fields near spatial infinity on flat spacetime

Linear zero-rest-mass fields generically develop logarithmic singularities at the critical sets where spatial infinity meets null infinity. Friedrich’s representation of spatial infinity is ideally suited to study this phenomenon. These logarithmic singularities are an obstruction to the smoothness of the zero-rest-mass field at null infinity and, in particular, to peeling. In the case of the spin-2 field it has been shown that these logarithmic singularities can be precluded if the initial data for the field satisfies a certain regularity condition involving the vanishing, at spatial infinity, of a certain spinor (the linearised cotton spinor) and its totally symmetrised derivatives. In this article we investigate the relation between this regularity condition and the staticity of the spin-2 field. It is shown that while any static spin-2 field satisfies the regularity condition, not every solution satisfying the regularity condition is static. This result is in contrast with what happens in the case of general relativity where staticity in a neighbourhood of spatial infinity and the smoothness of the field at future and past null infinities are much more closely related.

Lie polynomials and a twistorial correspondence for amplitudes

We review Lie polynomials as a mathematical framework that underpins the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of mathcal {M}_{0,n}, the moduli space of n points on the Riemann sphere up to Mobiüs transformation. We introduce a twistorial correspondence between the cotangent bundle T^*_Dmathcal {M}_{0,n}, the bundle of forms with logarithmic singularities on the divisor D as the twistor space, and mathcal {K}_n the space of momentum invariants of n massless particles subject to momentum conservation as the analogue of space–time. This gives a natural framework for Cachazo He and Yuan (CHY) and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories as being the corresponding Penrose transform. In particular, we show that it gives a natural correspondence between CHY half-integrands and scattering forms, certain n-3-forms on mathcal {K}_n, introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral n-3-planes in mathcal {K}_n introduced by ABHY.

Spin-valley locked instabilities in moiré transition metal dichalcogenides with conventional and higher-order Van Hove singularities

Recent experiments have observed correlated insulating and possible superconducting phases in twisted homobilayer transition metal dichalcogenides (TMDs). Besides the spin-valley locked moir\'e bands due to the intrinsic Ising spin-orbit coupling, homobilayer moir\'e TMDs also possess either logarithmic or power-law divergent Van Hove singularities (VHSs) near the Fermi surface, controllable by an external displacement field. The former and the latter are dubbed conventional and higher-order VHSs, respectively. Here, we perform a perturbative renormalization group (RG) analysis to unbiasedly study the dominant instabilities in homobilayer TMDs for both the conventional and higher-order VHS cases. We find that the spin-valley locking largely alters the RG flows and leads to instabilities unexpected in the corresponding extensively studied graphene-based moir\'e systems, such as spin- and valley-polarized ferromagnetism and topological superconductivity with mixed parity. In particular, for the case with two higher-order VHSs, we find a spin-valley-locking-driven metallic state with no symmetry breaking in the TMDs despite the diverging bare susceptibility. Our results show how the spin-valley locking significantly affects the RG analysis and demonstrate that moir\'e TMDs are suitable platforms to realize various interaction-induced spin-valley locked phases, highlighting physics fundamentally different from the well-studied graphene-based moir\'e systems.

High-order series expansion of non-Hermitian quantum spin models

We investigate the low-energy physics of non-Hermitian quantum spin models with $PT$-symmetry. To this end we consider the one-dimensional Ising chain and the two-dimensional toric code in a non-Hermitian staggered field. For both systems dual descriptions in terms of non-Hermitian staggered Ising interactions in a conventional transverse field exist. We perform high-order series expansions about the high- and low-field limit for both systems to determine the ground-state energy per site and the one-particle gap. The one-dimensional non-Hermitian Ising chain is known to be exactly solvable. Its ground-state phase diagram consists of second-order quantum phase transitions, which can be characterized by logarithmic singularities of the second derivative of the ground-state energy and, in the symmetry-broken phase, the gap closing of the low-field gap. In contrast, the gap closing from the high-field phase is not accessible perturbatively due to the complex energy and the occurrence of exceptional lines in the high-field gap expression. For the two-dimensional toric code in a non-Hermitian staggered field we study the quantum robustness of the topologically ordered phase by the gap closing of the low-field gap. We find that the well-known second-order quantum phase transition of the toric code in a uniform field extends into a large portion of the non-Hermitian parameter space. However, the series expansions become unreliable for a dominant anti-Hermitian field. Interestingly, the analysis of the high-field gap reveals the potential presence of an intermediate region.