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Exactness of fermionic linearization scheme in infinite dimension

Arianna Montorsi

Published Jan 1, 1995• 0 citations
OpenAlex

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 = infinity 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

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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.

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International Journal of Modern Physics B•Mar 1, 1992•67 citations
We show that the fermionic linearization scheme for dealing with strongly correlated electron systems - when implemented with Clifford variables - becomes exact in the d = infinity 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 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.
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.
An interacting spin-fermion model is exactly solved on an open chain. In a certain representation, it is the nearest-neighbor Hubbard model in the limit of infinite UUU (local interaction). Exact solution of its complete energy eigenspectrum is accomplished by introducing a unitary transformation which maps the original problem to a tight-binding model of the fermions only. Physically, the exact solution implies the absence of Nagaoka ferromagnetism in the ground state for arbitrary electron densities. The present method solves a class of very general models exactly. Few more problems are discussed as an application of this unitary transform method.