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Modern Physics Letters B

EXACTNESS OF FERMIONIC LINEARIZATION SCHEME WITH CLIFFORD VARIABLES IN d=∞

Arianna Montorsi

Published Jul 10, 1995• 1 citation
Publisher

Abstract

We show that the fermionic linearization scheme for dealing with strongly correlated electron systems — when implemented with Clifford variables — becomes exact in the d=∞ limit, at least for Hubbard-like models. In this case, the model is mapped exactly into a single-site problem. The conditions under which such a feature allows to obtain an exact solution are also discussed.

DOI
10.1142/s0217984995000930

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We show that the fermionic linearization scheme for dealing with strongly correlated electron systems — when implemented with Clifford variables — becomes exact in the d=∞ limit, at least for Hubbard-like models. In this case, the model is mapped exactly into a single-site problem. The conditions under which such a feature allows to obtain an exact solution are also discussed.
We propose an alternative interpretation of the fermi linarization approach to interacting electron systems, based on the requirement that the coefficients of the linearized operators are Clifford-like variables, whose anticommutator equals an unknown constant c. We apply the approximation to the Falicov-Kimball model, explicitly solving the self-consistency equation for the unknown, which turns out to behave as an order parameter. We discuss its relation with a metal-insulator transition and some thermodynamical quantities. In particular we show that our approximation in the T = 0 limit reproduces exactly the Gutzwiller results for the Hubbard model
The authors propose an alternative interpretation of the Fermi linearization approach to interacting electron systems, based on the requirement that the coefficients of the linearized operators are Clifford-like variables, whose anticommutator equals an unknown constant c. They apply the approximation to the Falicov-Kimball model (1969), explicitly solving the self-consistency equation for the unknown, which turns out to behave as an order parameter. They discuss its relation with a metal-insulator transition and some thermodynamical quantities. In particular they show that the approximation in the T=0 limit reproduces exactly the Gutzwiller results (1965) for the Hubbard model.
We present an exact mapping of the Hubbard model in infinite dimensions onto a single-impurity Anderson (or Wolff) model supplemented by a self-consistency condition. This provides a mean-field picture of strongly corrrelated systems, which becomes exact as d\ensuremath{\rightarrow}\ensuremath{\infty}. We point out a special integrable case of the mean-field equations, and study the general case using a perturbative renormalization group around the atomic limit. Three distinct Fermi-liquid regimes arise, corresponding to the Kondo, mixed-valence, and empty-orbitals regimes of the single-impurity problem. The Kondo resonance and the satellite peaks of the single-impurity model correspond to the quasiparticle and Hubbard-bands features of the Hubbard model, respectively.
Exact relations are derived for the Fermi Hubbard spectral weight function for infinite U at zero temperature in the thermodynamic limit for any dimension,any lattice structure and general hopping matrix. These relations involve moments of the spectral weight function but differ from similar work by (a) restricting the moments over the interesting low energy (lower Hubbard band) spectrum and (b) without any of the usual approximations (e.g. decoupling) for the requisite higher order correlation functions. The latter are determined exactly as explicit functions of the single particle correlation function in the infinite U limit. The methods behind these results have the additional virtue of simplicity - cumbersome, but entirely straightforward. In a companion paper, the relations offer rich insights into the model behavior and offer severe tests - a computer lab - of any proposed solution to the Hubbard model in the strong correlation (large finite U) region. As the electron density approaches one electron per site, then the correlation functions become local, so are trivial to compute exactly. In that limit and for the paramagnetic case, the second central moment of the spectral weight function is not zero, but quite large.