Numerical Analytic Continuation Research Articles

Quantum time correlation functions from complex time Monte Carlo simulations: A maximum entropy approach

We present a way of combining real-time path integral Monte Carlo simulations with a maximum entropy numerical analytic continuation scheme in a new approach for calculating time correlation functions for finite temperature many body quantum systems. The real-time dynamics is expressed in the form of the symmetrized time correlation function, which is suitable for Monte Carlo methods, and several simulation techniques are presented for evaluating this function accurately up to moderate values of time. The symmetrized time correlation function is then analytically continued in combination with imaginary time data to obtain the real-time correlation function. We test this approach on several exactly solvable problems, including two one-dimensional systems, as well two cases of vibrational relaxation of a system coupled to a dissipative environment. The computed time correlation functions are in good agreement with exact results over several multiples of the thermal time βℏ, and exhibit a significant improvement over analytic continuation of imaginary time correlation functions. Moreover, we show how the method can be systematically improved.

Quantum mechanical canonical rate theory: A new approach based on the reactive flux and numerical analytic continuation methods

We present the reactive flux analytic continuation (RFAC) method, based on the quantum reactive flux formalism combined with a numerical analytic continuation approach to calculate quantum canonical rates in condensed phase systems. We express the imaginary time reactive-flux correlation function in terms of a frequency dependent rate constant, and use path integral formalism to derive a working expression suitable for Monte Carlo simulation techniques. The imaginary time data obtained by simulation is analytically continued to the real time using the maximum entropy method to obtain the reaction rate. Motivated by the success of the method to predict the rates for a simple one dimensional parabolic barrier model, we assess its accuracy for a condensed phase reaction modeled by a double-well coupled to a harmonic bath. We note that the method is applicable to a more general Hamiltonian as long as the reaction coordinate can be identified. The reaction rates computed in this fashion are in very good agreement with analytic and numerically exact results. We demonstrate the applicability of the method for a wide range of model parameters and temperatures.

Real time quantum correlation functions. II. Maximum entropy numerical analytic continuation of path integral Monte Carlo and centroid molecular dynamics data

We propose a method which uses centroid molecular dynamics (CMD) [J. Cao and G. A. Voth, J. Chem. Phys. 100, 5106 (1994)] real-time data in conjunction with the imaginary-time data generated using path integral Monte Carlo simulations in a numerical analytic continuation scheme based on the maximum entropy approach. We show that significant improvement is achieved by including short-time CMD data with the imaginary-time data. In particular, for a particle bilinearly coupled to a harmonic bath, these methods lead to significant improvements over previous calculations and even allow accurate determination of transport coefficients such as the diffusion coefficient and mobility for this system. In addition we show how maximum entropy method can be used to extract accurate dynamic information from short-time CMD data, and that this approach is superior to the direct Fourier transform of long-time data for systems characterized by broad, featureless spectral distributions.

On the application of numerical analytic continuation methods to the study of quantum mechanical vibrational relaxation processes

A major problem still confronting molecular simulations is how to determine time-correlation functions of many-body quantum systems. In this paper the results of the maximum entropy (ME) and singular value decomposition (SVD) analytic continuation methods for calculating real time quantum dynamics from path integral Monte Carlo calculations of imaginary time time-correlation functions are compared with analytical results for quantum mechanical vibrational relaxation processes. This system studied is an exactly solvable system: a harmonic oscillator bilinearly coupled to a harmonic bath. The ME and SVD methods are applied to exact imaginary-time correlation functions with various level of added random noise, and also to imaginary-time data from path integral Monte Carlo (PIMC) simulations. The information gathered in the present benchmark study is valuable for the application of the analytic continuation of PIMC data to complex systems.

Numerical integration, analytic continuation, and decomposition

One-step formulas are developed using the decomposition method to explore relationships with numerical integration, analytic continuation, and numerical continuation. Analytic continuation in one or multiple dimensions is considered for decomposition solutions and the effects of chaos.

Monte Carlo approach to the dynamical coherent-potential approximation in metallic magnetism.

A Monte Carlo approach to the dynamical coherent-potential approximation (CPA) has been proposed on the basis of the functional-integral method to deal with the dynamical spin fluctuations in the Gutzwiller-Hubbard model. The functional integral on a site in the effective medium is replaced by the ${\mathit{N}}_{\ensuremath{\xi}}$-fold integral, which is evaluated by the Monte Carlo method. Numerical calculations have been performed up to ${\mathit{N}}_{\ensuremath{\xi}}$=64 for intermediate Coulomb interaction strength in the paramagnetic state. It is demonstrated that the present theory recovers the amplitude of local moment, reduces the effective Coulomb interaction, and leads to more Fermi-liquid-like momentum distribution when they are compared with the static approximation. Furthermore, the single-particle excitation spectra calculated by a numerical analytic continuation are shown to have some shoulders due to many-body excitations.

Dynamical Propertiesof the One-Dimensional Hubbard Model–A Numerical Study–

We investigate the dynamical properties of the one-dimensional Hubbard model using the quantum Monte Carlo method and the numerical analytic continuation method based on the maximum entropy principle. Results for the spin, charge and current excitation spectrum are presented for various values of the coupling and the band filling for the system of 24 sites with the periodic boundary condition. We find that the total spectral density of the spin excitation exhibits a gapless structure irrespective of the coupling or the band filling. The charge excitation spectrum, on the other hand, clearly shows the gap formation at the half filled band and shifts its weight toward the higher energy region with increase of the coupling. This charge excitation gap, however, disappears with the hole doping. We also discuss the momentum dependence of the spectral functions.

Maximum-entropy method for analytic continuation of quantum Monte Carlo data.

An outstanding problem in the simulation of condensed-matter phenomena is how to obtain dynamical information. We consider the numerical analytic continuation of imaginary-time quantum Monte Carlo data to obtain real-frequency spectral functions. This is an extremely ill-posed problem similar to the inversion of a Laplace transform. We suggest an image-reconstruction approach, which has been widely applied to data analysis in experimental research. Specifically, we apply the maximum-entropy method (ME) to the analytic continuation of quantum Monte Carlo data. We report encouraging preliminary results for the Fano-Anderson model of an impurity state in a continuum. The incorporation of additional prior information, such as sum rules and asymptotic behavior, can be expected to significantly improve results. We compare (ME) to alternative methods. We also discuss statistical error propagation for the analytic continuation problem via the likelihood function, which is independent of the choice of image-reconstruction method. This includes the sensitivity of the data to structure in the spectral function, the optimization of Monte Carlo simulations, and how to incorporate covariance in the statistical errors of the Monte Carlo method.

A numerical method of analytical continuation of Green-function-like expressions

Green functions, being the basic entities of a great part of condensed matter theory, are 'Herglotz', i.e. they can be analytically continued into the open half-planes forming the physical sheet of complex energy. On the real-energy axis they are usually nonanalytic and may even be distributions, which makes their numerical calculation there hard to perform. On the contrary, being smooth functions in the complex plane, they may easily be computed numerically there, provided closed formal expressions are given. The problem of a numerical analytical continuation towards the singularities (deconvolution) has been treated rather empirically in recent literature. Here a systematic analysis of the extrapolation error propagation is given, and an optimised deconvolution procedure is proposed. The focus is on the spectral function, the density of states and integrated density of states problems. Examples are given demonstrating the power of the procedure.

Pressure Spikes and Stability Considerations in Elastohydrodynamic Lubrication Models

Analytical tools useful in mathematical modeling of the elastohydrodynamic lubrication are developed. Stability maps which delimit the applicability of the two parameter EHL model are computed using linearized stability analysis and numerical analytic continuation. An exact solution is derived (for a region of constant film thickness within the lubricant) which explains the nature of a class of pressure spikes which arise under severe conditions. The relationship between stability and pressure spikes is discussed.

Numerical conformal mapping and analytic continuation

A numerical method for determination of least-square approximations of an arbitrary complex mapping function is derived here and implemented with fast Fourier transforms (FFTs). An essential feature of the method is the factoring of a discrete Hilbert transform in a pair of Fourier transforms in order to reduce the operation count of the longest computation to O ( N log ⁡ N ) O\left ( {N\log N} \right ) . A similar factoring of the discrete Poisson integral formula allows an explicit inversion of it in O ( N log ⁡ N ) O\left ( {N\log N} \right ) operations instead of O ( N 3 ) O\left ( {{N^3}} \right ) (N 3 ^{3} ). The resulting scheme for analytic continuation appears to be considerably more reliable than the evaluation of polynomials. Examples are treated, and APL implementations of algorithms are provided.

On the possibility of analytically continuing stabilization graphs to determine resonance positions and widths accurately

Previous efforts to extract resonance widths by analytic continuation of stabilization graphs were limited by non-analytic behavior of the eigenvalues as functions of the stabilization parameter. We describe a procedure which avoids this difficulty by numerical analytic continuation of the characteristic polynomial, truncated to low order, of the hamiltonian matrix.

Theory of resonance states based on analytical continuation in the coupling constant

A new approach to the resonance state theory based on analytical continuation in the coupling constant is presented. The proposed approach is used to develop an effective method of calculation which makes it possible to find the resonant state characteristics by means of numerical analytical continuation of the respective characteristics of bound states in the coupling constant using the Padé approximant technique.

Numerical analytic continuation of holomorphic functions in $C^n$

Numerical analytic continuation of holomorphic functions in $C^n$

Weyl’s theory applied to predissociation by rotation. I. Mercury hydride

Weyl’s theory for singular ordinary second order differential equations is presented as a numerical method for analyzing the spectral properties of the Schrödinger operator associated with predissociation by rotation in diatomic molecules. It is pointed out that the poles of Weyl–Titchmarsh’s m function characterize both the discrete eigenvalues and the resonances in the continuous spectrum, thus allowing a unified treatment of the entire spectrum. The conventional hard core treatment of the origin is compared to two alternative approaches, both applicable to more general singular behavior of the potential. A detailed discussion of the other singularity, infinity, is given and Kodaira’s theorem is generalized to an expression involving Wronskian’s between the regular and the asymptotic solutions. The calculation of the Weyl–Titchmarsh m function is based on numerical Runge–Kutta integrations of both independent solutions, combined with asymptotic information derived from a Riccati expansion. Im(m) is shown to have its physical significance as flux of predissociating particles. The resonances in the continuous spectrum are obtained by numerical analytical continuation of Im(m) across the real energy axis. The method is applied to Stwalley’s isotopically combined (IC1) potential for HgH. All resonance energies and lifetimes are reported and compared to Stwalley’s phase shift results. Excellent agreement is found for sharp resonances whereas considerable discrepancies occur for the broader ones. This latter fact is attributed to the failure of the Breit–Wigner fit to account for asymmetric non-Lorentzian shapes.

Numerical evaluation of partial-wave born approximations

A method is developed for the numerical evaluation of any term in the Born series for the off shell partial-wave Tmatrix. At negative energies, the integrals may be evaluated by using any appropriate quadrature formula. Numerical analytic continuation of the negative energy results to positive energies is carried out by means of Pad& approximants. The method enables the rate of convergence of eigenfunction expansions for the two-body off-shell amplitude to be accelerated. Numerical results are presented for a Yukawa potential and for a neutron-proton 3S 1 interaction.

Some applications of excited-state-excited-state transition densities

We derive an approximation for transition moments between excited states consistent with the approximations and assumptions normally used to obtain transition moments betwen the ground and excited states in the random-phase approximation and its higher-order approximations. We apply the result to the calculation of the photoionization cross sections of the $2^{3}S$ and $2^{1}S$ metastable states of helium by a numerical analytical continuation of the frequency-dependent polarizability. The procedure completely avoids the need for continuum basis functions. The cross sections agree well with the results of other calculations. We also predict an accurate two-photon decay rate for the $2^{1}S$ metastable state of helium. The entire procedure is immediately applicable to several problems involving photoionization of metastable states of molecules.

On the numerical analytic continuation of power series with application to the two-body and three-body problems

Summation methods can be used for the numerical analytical continuation of power series. In this paper the special class of “general Euler methods” is applied to the power series which result from the classical elliptic two-body problem and the regularized three-body problem. The “region of convergence” and the rate of convergence are considered. The numerical results compare favorably with some other methods which have been tried by other authors.

Phenomenological Dispersion Theory of pp, p̄p Forward Scattering

AbstractA review is given of the present state of experimental and theoretical knowledge of the proton‐proton and antiproton‐proton forward scattering, with special attention to applications of phenomenological dispersion theory in elucidating this interaction. The topics covered include the treatment of unphysical region, low‐energy parameterizations, dispersion relation calculations of the real parts and the meson‐nucleon coupling constants, analyses of the data by recently developed numerical analytic continuation techniques, high‐energy phenomenology, and evaluation of sum rules.

Photoionization cross sections for H 2 in the random phase approximation with a square-integrable basis

Total photoionization cross sections for H 2 are calculated in the random phase approximation (RPA) through a numerical analytic continuation procedure applied to the polarizability for complex values of the frequency. The representation of the polarizability that is required is obtained from a discrete set of excitation energies and oscillator strengths that satisfies the Thomas-Reiche-Kuhn sum rule exactly and other energy-weighted sum rules approximately. The fact that the excitation spectrum is obtained through a solution of the RPA equations with no continuum functions added to the basis makes the method well suited for general molecular photoionization calculations. The results are compared with experiment and good agreement is found.