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Efficient Computational Procedure for the Analytic Continuation of Eliashberg Equations

J. Johansson, Fredrik Laurén

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

The superconducting order parameter and the mass renormalization function can be solved either at discrete frequencies along the imaginary axis, or as a function of continuous real frequencies. The latter is done with a method called analytic continuation. The analytic continuation can conveniently be done by approximating a power series to the functions, the Padè approximation. Studied in this project is the difference between the Padè approximation, and a formally exact analytic continuation of the functions. As it turns out, the Padè approximant is applicable to calculate the superconducting order parameter at temperatures sufficiently below the critical temperature. However close to the critical temperature the approximation fails, while the solution presented in this report remains reliable.

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The superconducting order parameter and the mass renormalization function can be solved either at discrete frequencies along the imaginary axis, or as a function of continuous real frequencies. The latter is done with a method called analytic continuation. The analytic continuation can conveniently be done by approximating a power series to the functions, the Padè approximation. Studied in this project is the difference between the Padè approximation, and a formally exact analytic continuation of the functions. As it turns out, the Padè approximant is applicable to calculate the superconducting order parameter at temperatures sufficiently below the critical temperature. However close to the critical temperature the approximation fails, while the solution presented in this report remains reliable.
Analytic continuation is a central step in the simulation of finite-temperature field theories in which numerically obtained Matsubara data are continued to the real frequency axis for a physical interpretation. Numerical analytic continuation is considered to be an ill-posed problem where uncertainties on the Matsubara axis are amplified exponentially. Here, we present a systematic and controlled procedure that approximates any Matsubara function by a minimal pole representation to within a predefined precision. We then show a systematic convergence to the exact spectral function on the real axis as a function of our control parameter for a range of physically relevant setups. Our methodology is robust to noise and paves the way towards reliable analytic continuation in many-body theory and, by providing access to the analytic structure of the functions, a direct theoretical interpretation of physical properties.
We explore in detail how analytic continuation of divergent perturbation series by generalized hypergeometric functions is achieved in practice. Using the example of strong-coupling perturbation series provided by the two-dimensional Bose–Hubbard model, we compare hypergeometric continuation to Shanks and Padé techniques, and demonstrate that the former yields a powerful, efficient and reliable alternative for computing the phase diagram of the Mott insulator-to-superfluid transition. In contrast to Shanks transformations and Padé approximations, hypergeometric continuation also allows us to determine the exponents which characterize the divergence of correlation functions at the transition points. Therefore, hypergeometric continuation constitutes a promising tool for the study of quantum phase transitions.
In this paper we have presented a general scheme whereby the method of Padé approximants may be applied to the power series expansion of time dependent correlation functions. It has been demonstrated by actual numerical calculations, using the computer simulated results of Berne and Harp [5], that even a few terms in the power series expansion are capable of reproducing the correlation curves to a fairly large extent. It is pointed out that the method can be extended to higher orders without the introduction of fitting parameters, and without violating the time parity requirements of the Liouville equation. In the final analysis the method is a truly microscopic method since it relies on the microscopic hamiltonian, and the underlying principles are very straightforward. The basic requirements are the spectral moments, but we do not need many of these as the Padé approximants are highly convergent.
A simple method for numerical analytic continuation is developed. It is designed to analytically continue the imaginary time (Matsubara frequency) quantum Monte Carlo simulation results to the real time (real frequency) domain. Such a method is based on the Padé approximation. We modify it to be a linear regression problem, and then use bootstrapping statistics to get the averaged result and estimate the error. Unlike maximum entropy method, no prior information is needed. Test-cases have shown that the spectrum is recovered for inputs with relative error as high as 1%.
We develop a reliable parameter-free analytic continuation method for quantum many-body calculations. Our method is based on a kernel grid, a causal spline, a regularization using the second-derivative roughness penalty, and the L-curve criterion. We also develop the L-curve averaged deviation to estimate the precision of our analytic continuation. To deal with statistically obtained data more efficiently, we further develop a bootstrap-averaged analytic continuation method. In the test using the exact imaginary-frequency Green's function with added statistical error, our method produces the spectral function that converges systematically to the exact one as the statistical error decreases. As an application, we simulate the two-orbital Hubbard model for various electron numbers with the dynamical-mean field theory in the imaginary time and obtain the real-frequency self-energy with our analytic continuation method, clearly identifying a non-Fermi-liquid behavior as the electron number approaches the half filling from the quarter filling. Our analytic continuation can be used widely, and it will facilitate drawing clear conclusions from imaginary-time quantum many-body calculations.
Analytic continuation (AC) from the imaginary-time Green's function to the spectral function is a crucial process for numerical studies of the dynamical properties of quantum many-body systems. This process, however, is an ill-posed problem; that is, the obtained spectrum is unstable against the noise of the Green's function. Though several numerical methods have been developed, each of them has its own advantages and disadvantages. The sparse modeling (SpM) AC method, for example, is robust against the noise of the Green's function but suffers from unphysical oscillations in the low-energy region. We propose a method that combines the SpM AC with the Pad\'e approximation. This combination, called SpM-Pad\'e, inherits robustness against noise from SpM and low-energy accuracy from Pad\'e, compensating for the disadvantages of each. We demonstrate that the SpM-Pad\'e method yields low-variance and low-biased results with almost the same computational cost as that of the SpM method.