Method Of Analytic Continuation Research Articles

All-electron self-consistentGWin the Matsubara-time domain: Implementation and benchmarks of semiconductors and insulators

The $GW$ approximation is a well-known method to improve electronic structure predictions calculated within density functional theory. In this work, we have implemented a computationally efficient $GW$ approach that calculates central properties within the Matsubara-time domain using the modified version of elk, the full-potential linearized augmented plane wave (FP-LAPW) package. Continuous-pole expansion (CPE), a recently proposed analytic continuation method, has been incorporated and compared to the widely used Pad\'e approximation. Full crystal symmetry has been employed for computational speedup. We have applied our approach to 18 well-studied semiconductors/insulators that cover a wide range of band gaps computed at the levels of single-shot ${G}_{0}{W}_{0}$, partially self-consistent $G{W}_{0}$, and fully self-consistent $GW$ (full-$GW$), in conjunction with the diagonal approximation. Our calculations show that ${G}_{0}{W}_{0}$ leads to band gaps that agree well with experiment for the case of simple $s\ensuremath{-}p$ electron systems, whereas full-$GW$ is required for improving the band gaps in $3d$ electron systems. In addition, $G{W}_{0}$ almost always predicts larger band gap values compared to full-$GW$, likely due to the substantial underestimation of screening effects as well as the diagonal approximation. Both the CPE method and Pad\'e approximation lead to similar band gaps for most systems except strontium titantate, suggesting that further investigation into the latter approximation is necessary for strongly correlated systems. Moreover, the calculated cation $d$ band energies suggest that both full-$GW$ and $G{W}_{0}$ lead to results in good agreement with experiment. Our computed band gaps serve as important benchmarks for the accuracy of the Matsubara-time $GW$ approach.

A procedure for combining rotating-coil measurements of large-aperture accelerator magnets

The rotating search coil is a precise and widely used tool for measuring the magnetic field harmonics of accelerator magnets. This paper deals with combining several such multipole measurements, in order to cover magnet apertures largely exceeding the diameter of the available search coil.The method relies on the scaling laws for multipole coefficients and on the method of analytic continuation along zero-homotopic paths. By acquiring several measurements of the integrated magnetic flux density at different transverse positions within the bore of the accelerator magnet, the uncertainty on the field harmonics can be reduced at the expense of tight tolerances on the positioning. These positioning tolerances can be kept under control by mounting the rotating coil and its motor-drive unit on precision alignment stages. Therefore, the proposed technique is able to yield even more precise results for the higher-order field components than a dedicated rotating search coil of larger diameter. Moreover, the versatility of the measurement bench is enhanced by avoiding the construction of rotating search coils of different measurement radii.

Rules of correspondence in atomic physics

The Smirnov method of analytic continuation (B.M. Smirnov, Sov. Phys. JETP 20, 345 (1964)) has been justified and developed for atomic physics. It has been shown that the polarizability of alkali atoms α, their van der Waals interaction constant C 6, and the oscillator strength of the transition to the first P state f 01 are related to the parameter 〈r 2〉 and gap in the spectrum \( \frac{3}{2}\frac{f}{\Delta } \approx \frac{3}{2}\alpha \Delta \approx {\left( {3{C_6}\Delta } \right)^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}}} \approx \left\langle {{r^2}} \right\rangle \). The average square of the coordinate of the valence electron 〈r 2〉 in the first approximation has a hydrogen dependence \( {J_1} = \frac{1}{{2{v^2}}}.\) on the filling factor ν, which is defined in terms of the first ionization potential: xxxxxxxxx

Solutions of a crack interacting with a three-phase composite in plane elasticity

The interaction between a crack and a circularly cylindrical layered media under a remote uniform load for plane elasticity is investigated. Based on the method of analytical continuation associated with the alternation technique, the solutions to the crack problem for a three-phase composite are derived. A rapidly convergent series solution for the stress field, expressed in terms of an explicit general term of the corresponding homogeneous potential, is obtained in an elegant form. The solution procedures for solving this problem consist of two parts. In the first part, the complex potential functions of dislocation interacting with a three-phase composite are obtained. In the second part, the derivation of logarithmic singular integral equations by introducing the complex potential functions of dislocation along the crack border is made. The stress intensity factors (SIFs) are then obtained numerically in terms of the dislocation density functions of the logarithmic singular integral equations. The stress intensity factors (SIFs) as a function of the dimensionless crack length for various material properties and geometric parameters are shown in graphic form. The obtained results may provide some guidance for material and geometry selections by minimizing the SIF.

Bounds on the complex permittivity of polycrystalline materials by analytic continuation

An analytic continuation method for obtaining rigorous bounds on the effective complex permittivity ε * of polycrystalline composite materials is developed. It is assumed that the composite consists of many identical anisotropic crystals, each with a unique orientation. The key step in obtaining the bounds involves deriving an integral representation for ε *, which separates parameter information from geometrical information. Forward bounds are then found using knowledge of the single crystal permittivity tensor and mean crystal orientation. Inverse bounds are also developed, which recover information about the mean crystal orientation from ε *. We apply the polycrystalline bounds to sea ice, a critical component of the climate system. Different ice types, which result from different growth conditions, have different crystal orientation and size statistics. These characteristics significantly influence the fluid transport properties of sea ice, which control many geophysical and biogeochemical processes important to the climate and polar ecosystems. Using a two-scale homogenization scheme, where the single crystal tensor is numerically computed, forward bounds for sea ice are obtained and are in excellent agreement with columnar sea ice data. Furthermore, the inverse bounds are also applied to sea ice, helping to lay the groundwork for determining ice type using remote sensing techniques.

The contact problem of the mathematical theory of elasticity with stick and slip areas. The theory of rolling and tribology

The contact problem of the mathematical theory of elasticity, taking adhesion of the contact into account, is considered as a topic in fracture mechanics. An exact solution of the general contact problem of fracture mechanics under plane deformation conditions with stick and slip areas of two contacting half-spaces with dissimilar elastic properties is given. In fact, this problem is the basis of theoretical tribology. The solution is obtained in closed form for a class of inhomogeneous materials. The problem of the pressure of an absolutely rigid punch on an elastic body under plane deformation conditions, taking adhesion on the stick and slip areas into account, is also solved in closed form, when Poisson's ratio is 1/2. The initial mathematical problem also covers problems in fracture mechanics of composites concerning the propagation of cracks along the interfaces of two different elastic materials, taking into account superposition/slip areas of the faces of the cracks. The analytic continuation method is used to reduce the problems to a single generalized Riemann boundary value problem, the solution of which is obtained in closed form. Using the example of the solution of typical contact problems of fracture mechanics, a rigorous quantitative theory of the fundamental rolling modes and the stick-slip phenomenon is given and analysed. It is shown that when there is no slip and adhesion the rolling friction coefficient in Coulomb's law is directly proportional to (NRP)1/2 for wheels and cylinders, and (NRP)1/3 for spheres, where N is the normal force (the weight of the sphere or the weight per unit length of the cylinder), R is the radius of the wheel or sphere and P is the elastic compliance of the system. The effect of adhesion and roughness of the materials on the rolling, and also wear of the materials during rolling are characterized by two material constants of fracture mechanics. By a decision of the editorial board of the journal, the last section is added as an answer to critical comments on the paper, published after this paper.

Spectral measure computations for composite materials

The analytic continuation method of homogenization theory provides Stieltjes integral representations for the eective parameters of composite media. These representations involve the spectral measures of self-adjoint random operators which depend only on the composite geometry. On nite bond lattices, these random operators are represented by random matrices and the spectral measures are given explicitly in terms of their eigenvalues and eigenvectors. Here we provide the mathematical foundation for rigorous computation of spectral measures for such composite media, and develop a numerically ecient projection method to enable such computations. This is accomplished by providing a unied formulation of the analytic continuation method which is equivalent to the original formulation and holds for nite and innite lattices, as well as in continuum settings. We also introduce a family of bond lattices and directly compute the associated spectral measures and eective parameters. The computed spectral measures are in excellent agreement with known theoretical results. The behavior of the associated eective parameters is consistent with the symmetries and theoretical predictions of models, and the computed values fall within rigorous bounds. Some previous calculations of spectral measures have relied on nding the boundary values of the imaginary part of the eective parameter in the complex plane. Our method instead relies on direct computation of the eigenvalues and eigenvectors which enables, for example, statistical analysis of the spectral data.

Screw Dislocation Interacting with a Wedge Crack Penetrating a Fibrous Three-phase Magnetoelectroelastic Composite

AbstractOn the basis of the linear magnetoelectroelasticity, the interaction problem between a generalized screw dislocation and a fibrous three-phase magnetoelectroelastic composite penetrated by a semi-infinite wedge crack is investigated in this paper. The fibrous magnetoelectroelastic composite is composed of three dissimilar materials bonded along two concentric circular interfaces. The magnetoelectroelastic materials are assumed to be transversely isotropic and have the same poling direction. The analytical derivations are based on the complex variable, conformal mapping, analytical continuation and image singularity methods. Numerical calculations are given graphically for studying the effects of material combinations, geometric models, wedge angles and the load type on the generalized stress fields, the generalized stress intensity factors, and the forces on the dislocation.

Higgs mode near superfluid-to-Mott-insulatortransition studied by the quantum Monte Carlo method

In additional to the phonon (massless Goldstone mode) in Galilean invariant superfluid, there is another type of mode known as the Higgs amplitude mode in superfluid with emergent Lorentz invariance. In two dimensions, due to the strong decay into phonons, whether this Higgs mode is a detectable excitation with sharp linear response has been controversial for decades. Recent progress gives a positive answer to this question. Here, we review a series of numerical studies of the linear response of a two-dimensional Lorentz invariant superfluid near the superfluid-Mott insulator quantum critical point (SF-MI QCP). Particularly, we introduce a numerical procedure to unbiasedly calculate the linear response properties of strongly correlated systems. The numerical procedure contains two crucial steps, i.e., one is to use a highly efficient quantum Monte Carlo method, the worm algorithm in the imaginary-time path-integral representation, to calculate the imaginary time correlation functions for the system in equilibrium; and then, the other is, based on the imaginary time correlation functions, to use the numerical analytical continuation method for obtaining the real-time (real-frequency) linear response function. Applying this numerical procedure to the two-dimensional Bose Hubbard model near SF-MI QCP, it is found that despite strong damping, the Higgs boson survives as a prominent resonance peak in the kinetic energy response function. Further investigations also suggest a similar but less prominent resonance peak near SF-MI QCP on the MI side, and even on the normal liquid side. Since SF-MI quantum criticality can be realized by ultracold aotms in optical lattice, the Higgs resonance peak can be directly observed in experiment. In addition, we point out that the same Higgs resonance peak exists in all quantum critical systems with the same universality, namely (2 + 1)-dimensional relativistic U(1) criticality, as SF-MI QCP.

The two orbital Hubbard model in a square lattice: a DMFT + DMRG approach

We develop a precise and reliable numerical method for the calculation of electronic properties of the two orbital Hubbard model in a square lattice at half filling, based on the Dynamical Mean Field Theory (DMFT). We use the Density Matrix Renormalization Group (DMRG) as the impurity solver for the DMFT's selfconsistent equations to obtain accurate values for the Green's functions on the real axis. This way, reliable densities of states are obtained that do not need to resort to analytical continuation methods as those using quantum Monte Carlo techniques. Large system sizes can be achieved with increasing accuracy.

Periodic discrete energy for long-range potentials

We consider periodic energy problems in Euclidean space with a special emphasis on long-range potentials that cannot be defined through the usual infinite sum. One of our main results builds on more recent developments of Ewald summation to define the periodic energy corresponding to a large class of long-range potentials. Two particularly interesting examples are the logarithmic potential and the Riesz potential when the Riesz parameter is smaller than the dimension of the space. For these examples, we use analytic continuation methods to provide concise formulas for the periodic kernel in terms of the Epstein Hurwitz Zeta function. We apply our energy definition to deduce several properties of the minimal energy including the asymptotic order of growth and the distribution of points in energy minimizing configurations as the number of points becomes large. We conclude with some detailed calculations in the case of one dimension, which shows the utility of this approach.

Eshelby’s problem for infinite, semi-infinite and two bonded semi-infinite laminated anisotropic thin plates

We consider an Eshelby’s inclusion of arbitrary shape with prescribed uniform mid-plane eigenstrains and eigencurvatures in an infinite, semi-infinite and in one of two bonded dissimilar semi-infinite Kirchhoff laminated anisotropic thin plates. The inclusion has the same extensional, coupling and bending stiffnesses as the surrounding material. The boundary of the semi-infinite plate can be described by free, rigidly clamped and simply supported edges. We derive solutions of simple form by using the new Stroh octet formalism for the coupled stretching and bending deformations of anisotropic thin plates and the method of analytic continuation. In particular, real solutions of the far-field elastic fields induced by an inclusion of arbitrary shape are obtained. Specific examples of an elliptical inclusion in an infinite, semi-infinite and in one of two bonded dissimilar semi-infinite anisotropic plates are presented to demonstrate the obtained general solutions.

Determination of the resonance energy and width of the (2)B(2g) shape resonance of ethylene with the method of analytical continuation in the coupling constant.

The method of analytical continuation in the coupling constant, which allows us to determine the energy and width of a shape resonance, has been applied to the study of the (2)B2g shape resonance of ethylene. The procedure was done in two steps. In the first step, we used commercially available quantum-chemistry programs to calculate the electronic energy of a neutral molecule and of a negative ion. In both calculations, the Hamiltonian was altered by the inclusion of an additional attractive potential that helps to keep the negative ion bound. In the second step, the energy difference between the neutral molecule and its negative ion was analytically continued by the use of the statistical Padé approximation.

A cluster calculation for 6He spectrum

The 6He nucleus is considered as the cluster α + n + n system. The excitation energies of the low-lying levels are calculated using the configuration-space Faddeev equations. The analytical continuation method in a coupling constant is applied for calculation of resonance parameters. The αn interaction is constructed to reproduce the results of R-matrix analysis for αn-scattering data. A realistic AV14 potential describes the nn interaction. Additional three-body potential adjusted by the ground state energy of 6He is used. The energies of the low-lying resonances of 6He (01+, 02+, 21+) are reasonably reproduced by the calculations.

An iteration method for stable analytic continuation

In the present paper we consider the problem of numerical analytic continuation for an analytic function f(z)=f(x+iy) on a strip domain Ω+={z=x+iy∈C|x∈R,0<y<y0}. The key difference with the known methods is that a novel iteration regularization method with a priori and a posteriori parameter choice rule is presented. The comparison in numerical aspect with other methods is also discussed.

Steady stratified periodic gravity waves with surface tension II: Global bifurcation

In this paper we consider two-dimensional, stratified, steady water waves propagating over an impermeable flat bed and with a free surface. The motion is assumed to be driven by capillarity (that is, surface tension) on the surface and a gravitational force acting on the body of the fluid. We prove the existence of global continua of classical solutions that are periodic and traveling. This is accomplished by globally continuing the curves of small-amplitude solutions obtained by the author in [25]. We do this in two ways: first, by means of a degree theoretic theorem in the spirit of Rabinowitz, and second via the analytic continuation method of Dancer.

Acceleration of the Stochastic Analytic Continuation Method via an Orthogonal Polynomial Representation of the Spectral Function

Stochastic analytic continuation is an excellent numerical method for analytically continuing Green's functions from imaginary frequencies to real frequencies, although it requires significantly more computational time than the traditional MaxEnt method. We develop an alternate implementation of stochastic analytic continuation which expands the dimensionless field n(x) introduced by Beach using orthogonal polynomials. We use the kernel polynomial method (KPM) to control the Gibbs oscillations associated with truncation of the expansion in orthogonal polynomials. Our KPM variant of stochastic analytic continuation delivers improved precision at a significantly reduced computational cost.

Mode-III Stress Intensity Factors of an Arbitrarily Oriented Crack Crossing Interface in a Layered Structure

ABSTRACTThe problem of a layered structure containing an arbitrarily oriented crack crossing the interface in anti-plane elasticity is considered in this paper. The fundamental solution of displacements and stresses is obtained in a series form via the method of analytical continuation in conjunction with the alternating technique. A dislocation distribution along the prospective site of a crack is used to model a crack crossing the interface and the singular integral equations with logarithmic singular kernels for a line crack are then established. The crack is approximated by several line segments and the linear interpolation equation with undetermined coefficients was applied for the dislocation function along line segments. Once the undetermined dislocation coefficients are solved, the mode-III stress intensity factors KIII at two crack tips can be obtained for various crack inclinations with different material property combinations. All the numerical results are checked to achieve a good approximation that demonstrates the accuracy and the efficiency of the proposed method.

Interaction between a point heat source and a nonuniformly coated circular inclusion

This paper investigates the interaction between a point heat source and a nonuniformly coated circular inclusion under plane deformation. Based on the complex variable theory and conformal mapping in conjunction with the analytical continuation method, the general expressions of the temperature, displacements, and stresses for three dissimilar media are derived explicitly in a series form. Numerical results are provided for some particular examples to investigate the effect of material combinations on the interfacial stresses. The derived analytical solution can also be used as a fundamental solution for the boundary element method as well as for other physical models, including welding and cutting.

Dynamic structure factor for3He in two dimensions

Recent neutron scattering experiments on 3He films have observed a zero-sound mode, its dispersion relation and its merging with -and possibly emerging from- the particle-hole continuum. Here we address the study of the excitations in the system via quantum Monte Carlo methods: we suggest a practical scheme to calculate imaginary time correlation functions for moderate-size fermionic systems. Combined with an efficient method for analytic continuation, this scheme affords an extremely convincing description of the experimental findings.