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Physical review. B, Condensed matter

Functional-integral approach to strongly correlated Fermi systems: Quantum fluctuations beyond the Gutzwiller approximation

M. Lavagna

Published Jan 1, 1990• 117 citations
Publisher

Abstract

A systematic extension of the model of almost localized fermions to finite temperatures is presented. It consists in accounting for the quantum fluctuations beyond the Gutzwiller approximation (GA). We use the slave-boson formulation of the Hubbard model which has been proved to be equivalent at the saddle-point level to the GA. We determine the values of the Lagrange multipliers introduced to enforce the constraints, and show how they involve a Mott gap at nearly half filling for U>${U}_{c}$ (localization edge). The results of the GA for the correlation functions are reproduced at the saddle point when taking into account the implicit dependence of the boson fields with the external excitation. The Gaussian fluctuations are then considered in a renormalized basis of the boson fields which brings out two distinct channels (symmetric s and antisymmetric a). We show how the free energy and the correlation functions can be simply expressed as a function of the same Landau parameters ${F}_{0}^{s}$ and ${F}_{0}^{a}$ defined in the GA.

DOI
10.1103/physrevb.41.142

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A systematic extension of the model of almost localized fermions to finite temperatures is presented. It consists in accounting for the quantum fluctuations beyond the Gutzwiller approximation (GA). We use the slave-boson formulation of the Hubbard model which has been proved to be equivalent at the saddle-point level to the GA. We determine the values of the Lagrange multipliers introduced to enforce the constraints, and show how they involve a Mott gap at nearly half filling for U>Uc{U}_{c}Uc​ (localization edge). The results of the GA for the correlation functions are reproduced at the saddle point when taking into account the implicit dependence of the boson fields with the external excitation. The Gaussian fluctuations are then considered in a renormalized basis of the boson fields which brings out two distinct channels (symmetric s and antisymmetric a). We show how the free energy and the correlation functions can be simply expressed as a function of the same Landau parameters F0s{F}_{0}^{s}F0s​ and F0a{F}_{0}^{a}F0a​ defined in the GA.
We review the slave-boson representations of the Hubbard model. When both spin and charge fermion degrees of freedom are represented by Bose fields, the approach is equivalent at the "self-consistent" saddle point level to the Gutzwiller approximation (GA). We show how the determination of the Lagrange multipliers, introduced to enforce the constraints, involves a Mott-Hubbard gap near half-filling for U > U c (localization edge). The quantum fluctuations are then considered within a renormalized basis of boson fields (restoring the spin-rotation invariance), which brings out two distinct channels, symmetric and antisymmetric. It is remarkable that both spin and charge fluctuations are obtained at the same level of the gaussian fluctuations, as oppossed to the standard 1/N expansion. This provides the microscopic basis for a Fermi liquid theory. Both dynamic effects and extension to finite temperature which were beyond the scope of the variational procedure can now be described by this approach. The physical implications of the model (dynamic correlation functions, T3Ln T term of the specific heat, superfluid instability) are finally considered, all of which reflect the Fermi liquid nature of the system.
We propose a new functional integral representation of the Hubbarda and Anderson models of lattice fermions. The simplest saddle-point approximation leads, at zero temperature, to the results derived from the Gutzwiller variational wave function. This approach uncovers the limitations of the Gutzwiller approximation and clarifies its connection to the "auxiliary-boson" mean-field theory of the Anderson model. This formulation leads to a novel strong-coupling mean-field theory which allows for a unified treatment of antiferromagnetism and ferromagnetism, metal-to-insulator transition, and Kondo compensation effects.
We consider slave boson formulations of the Hubbard model, the t–J model and the Anderson model, in which spin and charge fermion degrees of freedom are represented by Bose fields. The approach has been applied to the Hubbard model generalizing the formulation of Kotliar and Ruckenstein. There the Gutzwiller solution is recovered as a saddle point. The method allows to find other types of mean field solutions in a systematic way. Dynamical correlation functions in mean field approximation as well as fluctuation contributions are also calculated.
We study the Hubbard model through a functional-integral formulation proposed by Kotliar and Ruckenstein. A careful use of the U(1${)}^{\ensuremath{\bigotimes}3}$ gauge symmetry of the bosonized action allows us to compute the complete one-loop correction to the free energy in the case of half-filled band. We extract the effective mass of the fermion in the metallic phase and show that the noninteracting limit is not recovered for vanishing on-site repulsion. The bosonic Hamiltonian must be adjusted order by order in perturbation theory.
We supplement (and critically overview) the existing extensive analysis of antiferromagnetic solution for the Hubbard model with a detailed discussion of two specific features, namely (i) the evolution of the magnetic (Slater) gap (here renormalized by the electronic correlations) into the Mott-Hubbard or atomic gap, and (ii) a rather weak renormalization of the effective mass by the correlations in the half-filled-band case, which contrasts with that for the paramagnetic case. The mass remains strongly enhanced in the non-half-filled-band case. We also stress the difference between magnetic and non-magnetic contributions to the gap. These results are discussed within the slave boson approach in the saddle-point approximation, in which there appears a non-linear staggered molecular field due to the electronic correlations that leads to the appearance of the magnetic gap. They reproduce correctly the ground-state energy in the limit of strong correlations. A brief comparison with the solution in the limit of infinite dimensions and the corresponding situation in the doubly-degenerate-band case with one electron per atom is also made. 1 1
Many experimental results seem to relate the superconducting oxides to the phenomenon of Mott localization in strongly correlated systems. We will give a brief review of the different theories developed until then to treat the Mott localization in the Hubbard model: variational Ansatz of Gutzwiller or RVB Ansatz, leading to fundamentally different ground states. In this context, we propose a new approach which consists to take account of the quantum fluctuations around the mean-field solution provided by the Gutzwiller approximation. We show how to understand the ground state in terms of a resonance (split by the Mott gap in the almost-localized regime) in complete analogy with the Abrikosov-Suhl resonance of the Kondo problem — heavy fermions. Superconductivity appears as induced by the constrained intersite correlations developed by quantum fluctuations around this resonant state.