Access by Xinjiang University

Diagonal Padé approximant of the one-body Green's function: A study on Hubbard rings

Walter Tarantino^*

Stefano Di Sabatino

Materials Theory, ETH Zürich, Wolfgang-Pauli-Strasse 27, CH-8093 Zürich, Switzerland and Laboratoire des Solides Irradiés, Ecole Polytechnique, CNRS, CEA, Université Paris-Saclay, F-91128 Palaiseau, France

Laboratoire des Solides Irradiés, Ecole Polytechnique, CNRS, CEA, Université Paris-Saclay, F-91128 Palaiseau, France

^*walter.tarantino@mat.ethz.ch

PDF

Phys. Rev. B 99, 075149 – Published 25 February, 2019

DOI: https://doi.org/10.1103/PhysRevB.99.075149

Abstract

Padé approximants to the many-body Green's function can be built by rearranging terms of its perturbative expansion. The hypothesis that the best use of a finite number of terms of such an expansion is given by the subclass of diagonal Padé approximants is here tested, and largely confirmed, on a solvable model system, namely the Hubbard ring for a variety of site numbers, fillings, and interaction strengths.

Physics Subject Headings (PhySH)

Article Text

References (25)

R. Martin, L. Reining, and D. Ceperley, Interacting Electrons (Cambridge University Press, Cambridge, 2016).
G. Baker and P. Graves-Morris, Padé Approximants, Encyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge, 1996).
C. Bender and S. Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory (Springer, New York, 2013).
S. Hirata, A. E. Doran, P. J. Knowles, and J. Ortiz, J. Chem. Phys. 147, 044108 (2017).
J. Hubbard, Proc. Math. Phys. Eng. Sci. 276, 238 (1963).
G. Alvarez, Comput. Phys. Commun. 180, 1572 (2009).
G. Alvarez, “Dmrg++ website” (2009).
C. Lanczos, J. Res. Natl. Bur. Stand. B 45, 255 (1950).
The underlying mechanism can be illustrated by: $\begin{matrix} \frac{1}{ω - ω_{0} - i η - U} & = & \frac{1}{ω - ω_{0} - i η} + \frac{U}{{(ω - ω_{0} - i η)}^{2}} \\ + \frac{U^{2}}{{(ω - ω_{0} - i η)}^{3}} + ..., \end{matrix}$ which, despite not being a sum of poles, still converges to the correct result for sufficiently small values of $U$ .
In the pole representation, the correct solution to the Dyson Eq. (2) can be achieved by considering the function $\tilde{G} : = {(G_{0}^{- 1} - Σ)}^{- 1}$ and reducing it to a sum of poles by exploiting the fact that $η$ is not a finite number but rather an infinitesimal. For instance, given $G_{0} = 1 / (ω - i η)$ and $Σ = 1 / (ω + i η)$ , one has that $\tilde{G} = 1 / (ω - i η - 1 / (ω + i η))$ which upon reduction to poles becomes $G = \frac{1 / 2}{ω - 1 + i η} + \frac{1 / 2}{ω + 1 + i η}$ . From the computation side, going from $\tilde{G}$ to $G$ involves finding the roots of the denominator of $\tilde{G}$ which can be numerically challenging. A pragmatic way to proceed is to consider $\tilde{G}$ as an approximation to $G$ and carry out all calculations with $\tilde{G}$ rather than $G$ with a sufficiently small value for $η$ . This is indeed what we have done in our study, where (32) has been evaluated with $\tilde{G}$ , rather than $G$ , with $η = 0.1$ .
Unravelling the intricacies of perturbation theory on degenerate states lies beyond the purposes of this work, the interested reader being referred to [25].
R. T. Scalettar, in Quantum Materials: Experiments and Theory, edited by E. Pavarini, E. Koch, J. van den Brink, and G. Sawatzky, Modeling and Simulation (Forschungszentrum Jülich, Jülich, 2016), Chap. 4, pp. 117–147.
R. Žitko, Comput. Phys. Commun. 182, 2259 (2011).
R. Žitko, “Sneg website” (2014).
Wolfram Research, Inc., Mathematica, Version 11.3, Champaign, IL (2018), http://support.wolfram.com/kb/472.
To mimic Coulomb's law, $α$ must be greater than 0 but less than 1. At around $α = 1 / 2$ we noticed what seemed to be a crossing of the two lowest eigenvalues of the Hamiltonian, signaling a possible phase transition. To avoid any possible spurious effects, we focused on the range $0 < α < 1 / 2$ , for which the choice $α = 1 / 10$ seemed quite representative.
G. A. Baker Jr. and J. L. Gammel, The Padé Approximant in Theoretical Physics, Mathematics in Science and Engineering, Vol. 71 (Elsevier Science, New York, 1971).
D. Z. Goodson, Wiley Interdiscip. Rev. Comput. Mol. Sci. 2, 743 (2012).
J. Schött, I. L. M. Locht, E. Lundin, O. Grånäs, O. Eriksson, and I. Di Marco, Phys. Rev. B 93, 075104 (2016).
Y. Pavlyukh, Sci. Rep. 7, 504 (2017).
V. Somà, T. Duguet, and C. Barbieri, Phys. Rev. C 84, 064317 (2011).
J. J. Phillips and D. Zgid, J. Chem. Phys. 140, 241101 (2014).
A. A. Rusakov and D. Zgid, J. Chem. Phys. 144, 054106 (2016).
It must be noticed that current approaches based on the second-order expansion of the self-energy are generally formulated within a self-consistent formalism, according to which diagrams of the expansion of the self-energy are not written in terms of $G_{0}$ but of the fully interacting $G$ itself. While this alters the diagrammatic content of the generic term $Σ_{n}$ , formula (16) remains valid and the right-hand side can be read as a functional of $G$ if $Σ_{1}$ and $Σ_{2}$ are too.
C. Brouder, G. Panati, and G. Stoltz, Phys. Rev. Lett. 103, 230401 (2009).

Physical Review B

Export citation

Export citation

Diagonal Padé approximant of the one-body Green's function: A study on Hubbard rings

Abstract

Physics Subject Headings (PhySH)

Article Text

References (25)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation