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arXiv:1002.3228 · physica status solidi (b)

A slave‐boson mean‐field theory for general multi‐band Hubbard models

Jörg Bünemann

Published Jun 18, 2010• 9 citations
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Abstract

Abstract We introduce a new slave‐boson mean‐field theory which allows the investigation of general multi‐band Hubbard models. Unlike earlier attempts of such a generalization, in our approach the quantum‐mechanical problem is exactly reformulated in an extended Hilbert space of Fermions and Bosons before a mean‐field approximation is applied. Systems with superconducting order parameters are naturally included in our formalism. Our ground‐state energy functional agrees with the corresponding quantity derived within the Gutzwiller theory.

DOI
10.1002/pssb.201046081

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Abstract We introduce a new slave‐boson mean‐field theory which allows the investigation of general multi‐band Hubbard models. Unlike earlier attempts of such a generalization, in our approach the quantum‐mechanical problem is exactly reformulated in an extended Hilbert space of Fermions and Bosons before a mean‐field approximation is applied. Systems with superconducting order parameters are naturally included in our formalism. Our ground‐state energy functional agrees with the corresponding quantity derived within the Gutzwiller theory.
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 propose a new method of solving a class of mean-field (MF) models, which is based on the Maximum Entropy (MaxEnt) principle with additional constraints included. Next, we show equivalence of our method when applied to the Gutzwiller approximation (GA), with the mean-field slave-boson (SB) formalism (on the example of the single-band Hubbard model). This equivalence provides thus an alternative justification of the results obtained within the SB approach which, however, contains ad hoc assumptions to position it in agreement with GA. Our approach implies that all predictions of the MF SB method can be obtained in a simpler, transparent, and controllable manner within GA when supplemented with the statistical-consistency conditions. We call the method as the Statistically-consistent Gutzwiller Approximation (SGA). Explicitly, the present formulation does not require introducing the condensed amplitudes of auxiliary Bose fields, which do not have a direct physical meaning and do not appear in the present formulation. Although the results of SGA are in the present case equivalent to SB, one can improve them further by utilizing more advanced schemes of calculating averages beyond the standard GA. To illustrate our approach, as well as to outline its advantages over alternative treatments of GA, we select the case of almost localized Fermi liquid (ALFL) in two dimensions and analyze it in detail within the tight-binding approximation. We also comment on significance of our method for describing correlated fermions. Namely, the reasoning used here can be applied to the corresponding MF treatment of the multiband Hubbard, the periodic Anderson, the t−Jt-Jt−J, and the t−J−Ut-J-Ut−J−U models. In this manner our method may be applied for strongly correlated electron systems, optical lattices, and other related situations.
A slave-boson functional-integral method has been developed for the Hubbard model with arbitrary, orbital degeneracy DDD. Its saddle-point mean-field theory is equivalent to the Gutzwiller approximation, as in the case of single-band Hubbard model. Our theory is applied to the doubly degenerate (D=2D = 2D=2) model, and numerical calculations have been performed for this model in the paramagnetic states. The effect of the exchange interaction on the metal-insulator (MI) transition is discussed. The critical interaction for the MI transition is analytically calculated as functions of orbital degeneracy and electron occupancy.
We study the negative-U Hubbard model for arbitrary values of the attractive interaction U and band filling by means of the slave-boson mean-field theory. We analyze both the normal and the superconducting states. We show that for the superconducting state our approach gives results similar to those obtained in the usual BCS mean field; only small renormalization of the bare parameters are obtained for U of the order of the bandwidth W. The energy difference between the normal and the superconducting states is maximum for U\ensuremath{\simeq}W. This energy difference is a measure of the superconducting critical temperature, which increases exponentially for small U and decreases as 1/U for large U. We discuss briefly the pair-breaking excitations.
We introduce Gutzwiller-correlated wave functions for the variational investigation of general multi-band Hubbard models. We set up a diagrammatic formalism which allows us to evaluate analytically ground-state properties in the limit of infinite spatial dimensions. In this limit recent results obtained within the Gutzwiller approximation are seen to become exact for these wave functions. We further show that the slave-boson mean-field theory for degenerate bands becomes variationally controlled at zero temperature in infinite dimensions. Lastly, we briefly comment on the variational approach to the Anderson transition in strongly correlated electron systems.