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Modern Physics Letters B

MODIFIED HUBBARD DECOUPLING FOR THE EMERY MODEL

Lew Gehlhoff

Published Oct 30, 1995• 0 citations
Publisher

Abstract

We consider a version of the Emery model with large spin degeneracy N and use the X-operator formulation and the equation-of-motion method to determine the single-particle Green’s functions. We propose a modified Hubbard decoupling technique for the higher-order Green’s functions appearing in this equation of motion. By applying it to the above model in the limit N→∞ we obtain the exact results.

DOI
10.1142/s0217984995001649

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We consider the Hubbard model on the infinite-dimensional Bethe lattice and construct a systematic series of self-consistent approximations to the one-particle Green's function, G(n){\mathit{G}}^{(\mathit{n})}G(n)(\ensuremath{\omega}), n=2,3,.... The first n-1 equations of motion are exactly fullfilled by G(n){\mathit{G}}^{(\mathit{n})}G(n)(\ensuremath{\omega}) and the nth equation of motion is decoupled following a simple set of decoupling rules. G(2){\mathit{G}}^{(2)}G(2)(\ensuremath{\omega}) corresponds to the Hubbard-III approximation. We present analytic and numerical results for the Mott-Hubbard transition at half filling for n=2,3,4.
We introduce a new approach to strongly correlated systems, based on equations of motion for composite ("quasiparticle") operators. Conserving approximations to the appropriate Green's functions are generated through functional differentiation. For illustration, we apply this scheme to the infinite-UUU, single-band Hubbard model. The analogue of the Hartree-Fock approximation becomes exact in the limit of infinite spin degeneracy (NNN), while the random-phase approximation leads to the corresponding 1N\frac{1}{N}N1​ fluctuations.
In the usual theory of the two-time Green's function, decoupling techniques are introduced in order to obtain a self-closed equation for the lowest order Green's function. These decouplings give neither the correct temperature dependence (the absence of a T3{T}^{3}T3 term) of magnetization nor the correct correlation function $〈{{S}_{j}}^{\ensuremath{-}}{{S}_{j}}^{\ensuremath{-}}{{S}_{k}}^{+}{{S}_{l}}^{+}〉$, which should be zero by definition. In this paper the equation of motion for the higher order Green's function is solved in the approximation which will lead to correct results up to terms of order T3{T}^{3}T3 in these quantities. Employing this higher order Green's function, it is found that the correction to the magnetization consists of a term of order T4{T}^{4}T4 and a divergent term and that the correlation function consists of a term of order T3{T}^{3}T3 and a divergent term. A method extracting a significant part from this divergent term is found by analyzing a system of two interacting \textonehalf{} spins. By this method, it is shown that the T3{T}^{3}T3 term in the magnetization is completely eliminated and that the correlation function $〈{{S}_{j}}^{\ensuremath{-}}{{S}_{j}}^{\ensuremath{-}}{{S}_{k}}^{+}{{S}_{l}}^{+}〉$ vanishes identically.
It is shown that the spin [Formula: see text] Ising model can be formulated as a spinless fermion many-body problem and that the Green's function technique can be applied to it. The hierarchy of Green's function equations of motion terminates at the (q + 1)-particle Green's function, where q is the coordination number. This finite number of equations yields Fisher's transformation of correlations. The technique discussed in this paper can be used to obtain exact results for the one-dimensional Ising model.