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Journal of Physics C Solid State Physics

A justification of Hubbard's decoupling scheme

D K Ghosh

Published Jun 28, 1971• 2 citations
Publisher

Abstract

Using Sawada's Green function technique, which separates various orders of correlations, the author solves the Hubbard model and shows that the single particle Green function has the same structure as Hubbard's solution using the double time Green function technique. The demonstration helps to clarify the uncertainty with regard to the decoupling procedure.

DOI
10.1088/0022-3719/4/9/002

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Physical review. B, Condensed matter•Dec 15, 1999•1 citation
Using Sawada's Green function technique, which separates various orders of correlations, the author solves the Hubbard model and shows that the single particle Green function has the same structure as Hubbard's solution using the double time Green function technique. The demonstration helps to clarify the uncertainty with regard to the decoupling procedure.
The Hubbard I decoupling approximation [J. Hubbard, Proc. R. Soc. London Ser. A 276, 238 (1963)] is extended by introducing a k-dependent self-energy which describes a mobile correlation hole of opposite spin propagating along with the electron in its way around the lattice. The theory, despite its simplicity, (1) becomes exact in the strong-coupling limit; and (2) reproduces with great accuracy the ground-state energy, the double occupancy, the on-site correlation functions, and the effective hopping for all the values of the Hubbard Coulomb repulsion U. It also gives reasonable spectral functions leading to (3) a single-particle Green function showing the well-known spectral-weight transfer between the Hubbard bands as soon as the electron density deviates from half-filling, and (4) a momentum distribution as well as magnetic and charge structure factors in qualitative agreement with quantum Monte Carlo simulations, slave-boson calculations, and conserving approximations like fluctuation exchange and parquet. On the other hand, as in most decoupling schemes, liftime effects and the near-Fermi-level additional structure characteristic of strongly correlated electron systems are missing.
A nonperturbative, controllable, and moment-conserving decoupling approach is proposed. Our assumption is that a higher-order Green function (GF) essentially contains two-part information. One part relates to the single-particle GF, which is involved in the corresponding moment; the other part is a higher-order quantity, which is independent of the single-particle GF. Thus the higher-order GF may be separated into two corresponding terms. We derive the chain of equations of motion (EOM) of the latter term from the usual EOM for GF's. To solve the equations a truncation approximation is made at the level desired. The single-band Hubbard model is studied using this approach. Moment-conserving solutions up to the order m=2m=2m=2 are obtained. The properties of a solution corresponding to the Hubbard-I approximation level are discussed in detail. The quasiparticle bands and the momentum distribution are in excellent agreement with the results from the quantum Monte Carlo simulation.