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Journal of the Physical Society of Japan

Fast and Accurate Scheme for Green Functions and Eigenvectors: Regulated Polynomial Expansion without Gibbs Oscillation

Shigetoshi Sota, Masaki Itoh

Published May 11, 2007• 15 citations
Publisher

Abstract

An oscillation-free Fourier expansion is proposed and applied to the polynomial expansion of resolvent operator, as an alternative treatment for Gibbs phenomenon to the kernel polynomial method (KPM) by Voter et al. [Phys. Rev. B 53 (1996) 12733]. Adopting regulated Legendre polynomial as basis functions, the recurrence algorithm underlying the KPM is practically preserved, while it eliminates unphysical oscillations entirely. The resolution is uniform in the frequency domain, and improved without limitation by extending the truncation range of the expansion. The method can also be used to calculate the eigenvector with remarkably high precision. It is further extended to the time domain, being able to deal with the quantum evolution in the same algorithm. A numerical test was performed against a lattice dynamics of 19 3 atoms, as well as of a computer generated Lennard–Jones glass of 1000 atoms. Not only it serves as a most efficient simulation scheme for a bulk, confirmed was also that the calculation converges to exact results within realistic computation time.

DOI
10.1143/jpsj.76.054004

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An oscillation-free Fourier expansion is proposed and applied to the polynomial expansion of resolvent operator, as an alternative treatment for Gibbs phenomenon to the kernel polynomial method (KPM) by Voter et al. [Phys. Rev. B 53 (1996) 12733]. Adopting regulated Legendre polynomial as basis functions, the recurrence algorithm underlying the KPM is practically preserved, while it eliminates unphysical oscillations entirely. The resolution is uniform in the frequency domain, and improved without limitation by extending the truncation range of the expansion. The method can also be used to calculate the eigenvector with remarkably high precision. It is further extended to the time domain, being able to deal with the quantum evolution in the same algorithm. A numerical test was performed against a lattice dynamics of 19 3 atoms, as well as of a computer generated Lennard–Jones glass of 1000 atoms. Not only it serves as a most efficient simulation scheme for a bulk, confirmed was also that the calculation converges to exact results within realistic computation time.
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.
Application of orthogonal polynomial expansion to quantum simulations is formulated on a general footing, implementing the regulation technique by Sota and Itoh for treating for the Gibbs oscillation.It is an alternative to the kernel polynomial method using Tchebyshev polynomial, but is simpler to handle and makes it possible to use all the popular orthogonal polynomials, covering finite, semi-infinite and infinite intervals of the eigenvalue spectrum.The accuracy can be made equivalent to direct diagonalization, with the resolution being homogeneous in the whole range of the spectrum.The target quantities can be as diverse as including eigenvectors, as well as all sorts of one-particle properties and correlation functions, involving thermal average and quantum time evolution.It can also be used as a handy tool for solving linear algebraic equations.
Inspired by the recent proposed Legendre orthogonal polynomial representation of imaginary-time Green's functions, we develop an alternate representation for the Green's functions of quantum impurity models and combine it with the hybridization expansion continuous-time quantum Monte Carlo impurity solver. This representation is based on the kernel polynomial method, which introduces various integral kernels to filter fluctuations caused by the explicit truncations of polynomial expansion series and improve the computational precision significantly. As an illustration of the new representation, we reexamine the imaginary-time Green's functions of single-band Hubbard model in the framework of dynamical mean-field theory. The calculated results suggest that with carefully chosen integral kernels the Gibbs oscillations found in previous orthogonal polynomial representation have been suppressed vastly and remarkable corrections to the measured Green's functions have been obtained.
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Quantum many-body systems in thermal equilibrium can be described by the imaginary time Green’s function formalism. However, the treatment of large molecular or solid ab initio problems with a fully realistic Hamiltonian in large basis sets is hampered by the storage of the Green’s function and the precision of the solution of the Dyson equation. We present a Legendre-spectral algorithm for solving the Dyson equation that addresses both of these issues. By formulating the algorithm in Legendre coefficient space, our method inherits the known faster-than-exponential convergence of the Green’s function’s Legendre series expansion. In this basis, the fast recursive method for Legendre polynomial convolution enables us to develop a Dyson equation solver with quadratic scaling. We present benchmarks of the algorithm by computing the dissociation energy of the helium dimer He2 within dressed second-order perturbation theory. For this system, the application of the Legendre spectral algorithm allows us to achieve an energy accuracy of 10−9Eh with only a few hundred expansion coefficients.
The Matsubara Green's function that is used to describe temperature-dependent behavior is expressed on a numerical grid. While such a grid usually has a couple of hundred points for low-energy model systems, for realistic systems with large basis sets the size of an accurate grid can be tens of thousands of points, constituting a severe computational and memory bottleneck. In this paper, we determine efficient imaginary time grids for the temperature-dependent Matsubara Green's function formalism that can be used for calculations on realistic systems. We show that, because of the use of an orthogonal polynomial transform, we can restrict the imaginary time grid to a few hundred points and reach micro-Hartree accuracy in the electronic energy evaluation. Moreover, we show that only a limited number of orthogonal polynomial expansion coefficients are necessary to preserve accuracy when working with a dual representation of the Green's function or self-energy and transforming between the imaginary time and frequency domain.