Numerical Expansions Research Articles

Time-kernel for lattice determinations of NLO hadronic vacuum polarization contributions to the muon g-2

We study the time-momentum representation of the kernel needed to compute the hadronic vacuum polarization contribution to the muon g-2 in the space-like region at next-to-leading order. For small values of the time, we present analytical series expansions; for large values of the time, we present numerical series expansions which overcome the problems showed by naïve asymptotic expansions. These results are to be employed in lattice QCD determination of the hadronic vacuum polarization contribution to the muon g-2 at next-to-leading order.

Numerical linked-cluster expansions for two-dimensional spin models with continuous disorder distributions.

We show that numerical linked cluster expansions (NLCEs) based on sufficiently large building blocks allow one to obtain accurate low-temperature results for the thermodynamic properties of spin lattice models with continuous disorder distributions. Specifically, we show that such results can be obtained computing the disorder averages in the NLCE clusters before calculating their weights. We provide a proof of concept using three different NLCEs based on L, square, and rectangle building blocks. We consider both classical (Ising) and quantum (Heisenberg) spin-1/2 models and show that convergence can be achieved down to temperatures that are up to two orders of magnitude lower than the relevant energy scale in the model. Additionally, we provide evidence that in one dimension one can obtain accurate results for observables such as the energy down to their ground-state values.

The influence of gas heating on the calculated supersonic parameters of gas-powder flow at slag blower in the converter. Message 1

The article shows that one of the priority tasks in steel production technology is to increase the durability of the lining of oxygen converters. It is shown that the slag blower is a radical way to increase the durability of the lining of oxygen converters. When blowing the melt with oxygen, the MgO content in the slag is 6-8%. To increase the chemical affinity of the slag and the lining, it is proposed to blow the converter slag with jets of nitrogen or powder gas. To achieve optimal chemical affinity of the lining and slag, it is recommended to modify the latter by increasing the content of magnesium oxide in it to 12-14%. Increased durability of the lining makes it possible to solve the problem of replacing wasteful wastes of coal slag and changing environmental concerns on the excess media. It is shown that an attempt to increase the efficiency of modeling very complex processes of interaction of supersonic jets with converter slag has not yet been solved a. A critical analysis of existing methods for constructing mathematical models based on the use of known gas-dynamic laws of interaction of free turbulent jets with the melt is carried out. It has been established that only supersonic jets with their characteristic shock-wave structure always enter the slag melt. An integral method for calculating the parameters of a gas-powder flow in a supersonic nozzle taking into account the powder concentration is presented. It is determined how heating of the gas suspension affects the required pressure in front of the nozzle block, the speed and flow density in the outlet section of the supersonic nozzle. The gas dynamics system is connected to the heat exchange of the flowing gas-dispersed mixture in the Laval expansion nozzles to establish the heating effect in front of the nozzle block of the two-phase gas suspension to a temperature that will ensure The increase in the fluidity of the gas flow is doubled and its kinetic energy is increased threefold. The method of expanding the parameters of the gas-powder flow between the Laval expansion nozzle involves the infusion of 10 physical effects into 14 parameters of the gas-powder flow. It is shown that numerical expansions allow the tuyere body to be used as a heat exchanger for heating the gas suspension, as well as for increasing the pressure of the gas-powder jet that flows from the nozzles

Bayesian inversion with α-stable priors

We propose using Lévy α-stable distributions to construct priors for Bayesian inverse problems. The construction is based on Markov fields with stable-distributed increments. Special cases include the Cauchy and Gaussian distributions, with stability indices α = 1, and α = 2, respectively. Our target is to show that these priors provide a rich class of priors for modeling rough features. The main technical issue is that the α-stable probability density functions lack closed-form expressions, and this limits their applicability. For practical purposes, we need to approximate probability density functions through numerical integration or series expansions. For Bayesian inversion, the currently available approximation methods are either too time-consuming or do not function within the range of stability and radius arguments. To address the issue, we propose a new hybrid approximation method for symmetric univariate and bivariate α-stable distributions that is both fast to evaluate and accurate enough from a practical viewpoint. In the numerical implementation of α-stable random field priors, we use the constructed approximation method. We show how the constructed priors can be used to solve specific Bayesian inverse problems, such as the deconvolution problem and the inversion of a function governed by an elliptic partial differential equation. We also demonstrate hierarchical α-stable priors in the one-dimensional deconvolution problem. For all numerical examples, we use maximum a posteriori estimation. To that end, we exploit the limited-memory BFGS and its bounded variant for the estimator.

Chiral discrimination by polarization singularities of a metal sphere

The practical applications of chiral discrimination are usually limited by the weak chiral response of enantiomers and the high complexity of detection methods. Here, we propose to use the C lines (i.e., lines of polarization singularities) emerged in light scattering by a metal sphere to detect small chiral particles. Using full-wave numerical simulations and analytical multipole expansions, we determined the absorption dissymmetry of deep-subwavelength helices (i.e., chiral particles) at different positions on the C lines. We find that the absorption dissymmetry can be much larger than that induced by circularly polarized plane wave excitation, and it can be attributed to the asymmetric absorption of the induced electric and magnetic dipoles. We also show that the absorption dissymmetry strongly depends on the anisotropy of the helices. The results may find applications in optical manipulations, optical sensing, and chiral quantum optics.

Thermodynamics of the disordered Hubbard model studied via numerical linked-cluster expansions

The interplay of disorder and strong correlations in quantum many-body systems remains an open question. That is despite much progress made in recent years with ultracold atoms in optical lattices to better understand phenomena such as many-body localization or the effect of disorder on Mott metal-insulator transitions. Here, we utilize the numerical linked-cluster expansion technique, extended to treat disordered quantum lattice models, and study exact thermodynamic properties of the disordered Fermi-Hubbard model on the square and cubic geometries. We consider box distributions for the disorder in the onsite energy, the interaction strength, as well as the hopping amplitude and explore how energy, double occupancy, entropy, heat capacity and magnetic correlations of the system in the thermodynamic limit evolve as the strength of disorder changes. We compare our findings with those obtained from determinant quantum Monte Carlo simulations and discuss the relevance of our results to experiments with cold fermionic atoms in optical lattices.

Coherent forward scattering peak and multifractality

It has recently been shown that interference effects in disordered systems give rise to two non-trivial structures: the coherent backscattering (CBS) peak, a well-known signature of interference effects in the presence of disorder, and the coherent forward scattering (CFS) peak, which emerges when Anderson localization sets in. We study here the CFS effect in the presence of quantum multifractality, a fundamental property of several systems, such as the Anderson model at the metal-insulator transition. We focus on Floquet systems, and find that the CFS peak shape and its peak height dynamics are generically controlled by the multifractal dimensions $D_1$ and $D_2$, and by the spectral form factor. We check our results using a 1D Floquet system whose states have multifractal properties controlled by a single parameter. Our predictions are fully confirmed by numerical simulations and analytic perturbation expansions on this model. Our results, which we believe to be generic, provide an original and direct way to detect and characterize multifractality in experimental systems.

Polarization singularities in light scattering by small particles

Using full-wave numerical simulations and analytical multipole expansions we investigated the properties of real-space polarization singularities that emerge in light scattering by subwavelength particles. We considered spherical and torus particles made of dielectric or perfect-electric-conductor material. We determined the topological indices and the trajectories of electric-field polarization singularities in both the near-field and far-field regions. In the far-field region, a total of four singularities are identified and the sum of their polarization topological indices is two, independent of the particle's geometric shape. In the near-field region, the polarization singularities strongly depend on the particle's shape and the polarization of incident light, and their index sum is not governed by the Poincar\'e-Hopf theorem anymore due to the non-transverse nature of the fields. From near field to far field, a flipping of sign can happen to the polarization topological indices of the C lines. The far-field properties of the singularities can be well explained by the interference of the excited multipoles, but their near-field properties can be strongly affected by the evanescent fields that are not captured by the multipole expansions. Our work uncovers the important relationship between particles' geometric properties and the polarization singularities of their scattering field. The results can be applied to manipulate polarization singularities in nanophotonic systems and could generate novel applications in optical sensing.

High order numerical schemes for transport equations on bounded domains

This article is an account of the NABUCO project achieved during the summer camp CEMRACS 2019 devoted to geophysical fluids and gravity flows. The goal is to construct finite difference approximations of the transport equation with nonzero incoming boundary data that achieve the best possible convergence rate in the maximum norm. We construct, implement and analyze the so-called inverse Lax-Wendroff procedure at the incoming boundary. Optimal convergence rates are obtained by combining sharp stability estimates for extrapolation boundary conditions with numerical boundary layer expansions. We illustrate the results with the Lax-Wendroff and O3 schemes.

Parameterization Method for State-Dependent Delay Perturbation of an Ordinary Differential Equation

We consider state-dependent delay equations (SDDEs) obtained by adding delays to a planar ODE with a limit cycle. Situations of this type appear in models of several physical processes, where small delay effects are added. Even if the delays are small, they are very singular perturbations since the natural phase space of an SDDE is an infinite-dimensional space. We show that for the SDDE, there are initial values which lead to solutions similar to those of the ODE. That is, there exist a periodic solution and a two parameter family of solutions whose evolution converges to the periodic solution (in the ODE case, these are called the isochrons). The method of proof bypasses the theory of existence, uniqueness, dependence on parameters of SDDE. We consider the class of functions of time that have a well defined behavior (e.g., periodic, or asymptotic to periodic) and derive functional equations which impose that they are solutions of the SDDE. These functional equations are studied using functional analysis methods. We provide a result in “a posteriori” format: given an approximate solution of the functional equation, which has some good condition numbers, we prove that there is a true solution close to the approximate one. Thus, our result can be used to validate the results of numerical computations or formal expansions. The method of proof also leads to practical algorithms. In a companion paper, we present the implementation details and representative results. One feature of the method presented here is that it allows us to obtain smooth dependence on parameters for the periodic solutions and their slow stable manifolds without studying the smoothness of the flow (which seems to be problematic for SDDEs, for now the optimal result on smoothness of the flow is $C^1$).

Lanczos-boosted numerical linked-cluster expansion for quantum lattice models.

Numerical linked-cluster expansions allow one to calculate finite-temperature properties of quantum lattice models directly in the thermodynamic limit through exact solutions of small clusters. However, full diagonalization is often the limiting factor for these calculations. Here we show that a partial diagonalization of the largest clusters in the expansion using the Lanczos algorithm can be as useful as full diagonalization for the method while mitigating some of the time and memory issues. As test cases, we consider the frustrated Heisenberg model on the checkerboard lattice and the Fermi-Hubbard model on the square lattice. We find that our approach can surpass state of the art in performance in a parallel environment.

Numerical linked-cluster expansions for disordered lattice models

Imperfections in correlated materials can alter their ground state as well as finite-temperature properties in significant ways. Here, we develop a method based on numerical linked-cluster expansions for calculating exact finite-temperature properties of disordered lattice models directly in the thermodynamic limit. We show that a continuous distribution for disordered parameters can be achieved using a set of carefully chosen discrete modes in the distribution, which allows for the averaging of properties over all disorder realizations. We benchmark our results for thermodynamic properties of the square lattice Ising and quantum Heisenberg models with bond disorder against Monte Carlo simulations and study them as the strength of disorder changes. We also apply the method to the disordered Heisenberg model on the frustrated checkerboard lattice, which is closely connected to Sr$_2$Cu(Te$_{0.5}$W$_{0.5}$)O$_6$. Our method can be used to study finite-temperature properties of other disordered quantum lattice models including those for interacting lattice fermions.

An A Posteriori KAM Theorem for Whiskered Tori in Hamiltonian Partial Differential Equations with Applications to some Ill-Posed Equations

The goal of this paper is to develop a KAM theory for tori with hyperbolic directions, which applies to Hamiltonian partial differential equations, even to some ill-posed ones. The main result has an a-posteriori format, that is, we show that if there is an approximate solution of an invariance equation which also satisfies some non-degeneracy conditions, then there is a true solution nearby. This allows, besides dealing with the quasi-integrable case, for the validation of numerical computations or formal perturbative expansions as well as for obtaining quasi-periodic solutions in degenerate situations. The a-posteriori format also has other automatic consequences (smooth dependence on parameters, bootstrap of regularity, etc.). We emphasize that the non-degeneracy conditions required are just quantities evaluated on the approximate solution (no global assumptions on the system such as twist). Hence, they are readily verifiable in perturbation expansions. We will pay attention to the quantitative relations between the sizes of the approximation and the non-degeneracy conditions. This will allow us to prove what experts call small twist theorems (the non-degeneracy conditions vanishes as the perturbation goes to zero but much slower than the error of the approximation). The method of proof is based on an iterative method for solving a functional equation for the parameterization of the torus expressing that the range of the parameterization admits an evolution and is invariant. We also solve functional equations for bundles which imply that are invariant under the linearization. The iterative method does not use transformation theory nor action-angle variables. The main result does not assume that the system is close to integrable. More surprisingly, we do not need that the equations we study define an evolution for all initial conditions and are well posed. Even if the systems we study do not admit solutions for all initial conditions, we show that there is a systematic way to choose initial conditions on which one can define an evolution which is quasi-periodic. We first develop an abstract theorem. Then, we show how this abstract result applies to some concrete examples. The examples considered in this paper are the scalar Boussinesq equation and the Boussinesq system (both are PDE models that aim to describe water waves in the long wave limit). For these equations we construct small amplitude time quasi-periodic solutions which are even in the spatial variable. The strategy for the abstract theorem is inspired by that in Fontich et al. (Electron Res Announc Math Sci 16:9–22, 2009; J Differ Equ 246(8):3136–3213, 2009). The main part of the paper is to study infinite dimensional analogues of dichotomies which applies even to ill-posed equations and which is stable under addition of unbounded perturbations. This requires that we assume smoothing properties. We also present very detailed bounds on the change of the splittings under perturbations.

Modeling Walking with an Inverted Pendulum Not Constrained to the Sagittal Plane. Numerical Simulations and Asymptotic Expansions

Inverted pendulum models are commonly used to study the bio-mechanics of biped walkers. In its simplest form, the inverted pendulum consists of a point mass attached to two straight mass-less legs. Most works constrain the motion of the mass to the sagittal plane, i.e. the plane perpendicular to the ground that contains the direction toward the biped is walking. In this article, we remove this constrain to study the oscillations, the mass experiences in the direction perpendicular to the sagittal plane as the biped walks. While small, these lateral oscillations are unavoidable and of importance in the understanding of balance and stability of walkers, as well as walkers induced oscillations in pedestrian bridges.

Density of the p2gg-4c1 packing of ellipses (II)

Abstract As the second paper of the series, the first of which was published in 1997 (M. Tanemura, T. Matsumoto, Density of the p2gg-4c1 packing of ellipses (I). Z. Kristallogr. 1997, 212, 637), the p2gg-4c1(b) packing of identical ellipses where four ellipses are included in the rectangular unit cell and where every ellipse has six contacting neighbors, is discussed. It is shown here that the p2gg-4c1(b) packing of ellipses does not exceed the maximum density ρ = π / 12 $\rho \; = \;\pi /\sqrt {12} $ through numerical computations and series expansions. It is also shown that the p2gg-2a2 packing of ellipses has the similar properties and that it is considered as a special case of p2gg-4c1(b) packing. The method of computing the density for every parameter values of aspect ratio and tilt angle is given.

Modulational instability in the Whitham equation with surface tension and vorticity

We study modulational stability and instability in the Whitham equation, combining the dispersion relation of water waves and a nonlinearity of the shallow water equations, and modified to permit the effects of surface tension and constant vorticity. When the surface tension coefficient is large, we show that a periodic traveling wave of sufficiently small amplitude is unstable to long wavelength perturbations if the wave number is greater than a critical value, and stable otherwise, similarly to the Benjamin–Feir instability of gravity waves. In the case of weak surface tension, we find intervals of stable and unstable wave numbers, whose boundaries are associated with the extremum of the group velocity, the resonance between the first and second harmonics, the resonance between long and short waves, and a resonance between dispersion and the nonlinearity. For each constant vorticity, we show that a periodic traveling wave of sufficiently small amplitude is unstable if the wave number is greater than a critical value, and stable otherwise. Moreover it can be made stable for a sufficiently large vorticity. The results agree with those based upon numerical computations or formal multiple-scale expansions to the physical problem.

Finite-temperature superconducting correlations of the Hubbard model

We utilize numerical linked-cluster expansions (NLCEs) and the determinantal quantum Monte Carlo algorithm to study pairing correlations in the square lattice Hubbard model. To benchmark the NLCE, we first locate the finite-temperature phase transition of the attractive model to a superconducting state away from half filling. We then explore the superconducting properties of the repulsive model for the d-wave and extended s-wave pairing symmetries. The pairing structure factor shows a strong tendency to d-wave pairing and peaks at an interaction strength comparable to the bandwidth. The extended s-wave structure factor and correlation length are larger at higher temperatures but clearly saturate as temperature is lowered, whereas the d-wave counterparts, which start off lower at high temperatures, continue to rise near half filling. This rise is even more dramatic in the d-wave susceptibility. The convergence of NLCEs breaks down as the susceptibilities and correlation lengths become large, so we are unable to determine the onset of long-range order. However, our results extend the conclusion, previously restricted to only magnetic and charge correlations, that NLCEs offer unique window into pairing in the Hubbard model at strong coupling.

A new approach to the epsilon expansion of generalized hypergeometric functions

Assuming that the parameters of a generalized hypergeometric function depend linearly on a small variable ε, the successive derivatives of the function with respect to that small variable are evaluated at ε=0 to obtain the coefficients of the ε-expansion of the function. The procedure, which is quite naive, benefits from simple explicit expressions of the derivatives, to any order, of the Pochhammer and reciprocal Pochhammer symbols with respect to their argument. The algorithm may be used algebraically, irrespective of the values of the parameters. It reproduces the exact results obtained by other authors in cases of especially simple parameters. Implemented numerically, the procedure improves considerably, for higher orders in ε, the numerical expansions given by other methods.

Finite-temperature properties of strongly correlated fermions in the honeycomb lattice

We study finite-temperature properties of strongly interacting fermions in the honeycomb lattice using numerical linked-cluster expansions and determinantal quantum Monte Carlo simulations. We analyze a number of thermodynamic quantities, including the entropy, the specific heat, uniform and staggered spin susceptibilities, short-range spin correlations, and the double occupancy at and away from half filling. We examine the viability of adiabatic cooling by increasing the interaction strength for homogeneous as well as for trapped systems. For the homogeneous case, this process is found to be more efficient at finite doping than at half filling. That, in turn, leads to an efficient adiabatic cooling in the presence of a trap, which, starting with even relatively high entropies, can drive the system to have a Mott insulating phase with substantial antiferromagnetic correlations.

Numerical linked-cluster expansion for the distorted kagome lattice Heisenberg model

Motivated by experimental results for the thermodynamic properties of the Rb2Cu3SnF12 material and the discovery of its valence-bond solid ground state, we utilize the numerical linked-cluster expansions (NLCEs) and devise an expansion tailored to solve the Heisenberg model on a pinwheel-distorted kagome lattice. Using the exchange interactions that are relevant to Rb2Cu3SnF12, we calculate its uniform spin susceptibility and find a very good agreement with experiment. Next, we focus on the ground state of a simplified model of the distorted kagome lattice and take advantage of a zero-temperature Lanczos-based NLCE to study the approach to the ground state of the uniform kagome lattice Heisenberg model using a tuning parameter. Together with results from exact diagonalization of finite clusters, we find evidence that a phase transition occurs before the uniform limit is reached.