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arXiv:cond-mat/9612142 · Journal of the Physical Society of Japan

Slave-Boson Functional-Integral Approach to the Hubbard Model with Orbital Degeneracy

Hideo Hasegawa

Published May 15, 1997• 38 citations
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Abstract

A slave-boson functional-integral method has been developed for the Hubbard model with arbitrary, orbital degeneracy $D$. 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 ($D = 2$) 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.

DOI
10.1143/jpsj.66.1391

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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 analytically the Mott transition of the N-orbital Hubbard model using dynamical mean-field theory and a low-energy projection onto an effective Kondo model. It is demonstrated that the critical interaction at which the insulator appears (Uc1){(U}_{c1})(Uc1​) and the one at which the metal becomes unstable (Uc2){(U}_{c2})(Uc2​) have a different dependence on the number of orbitals as the latter becomes large: ${U}_{c1}\ensuremath{\propto}\sqrt{N}$ while ${U}_{c2}\ensuremath{\propto}N.$ An exact analytical determination of the critical coupling Uc2/N{U}_{c2}/NUc2​/N is obtained in the large-N limit. The metallic solution close to this critical coupling has many similarities at low-energy with the results of slave boson approximations, to which a comparison is made. We also discuss how the critical temperature associated with the Mott critical endpoint depends on the number of orbitals.
We present a slave-boson representation for the Hubbard model, introducing Bose fields for the empty, singly, and doubly occupied sites and show that the boson for the singly occupied site must transform as a tensor of rank two under spin rotations. This generalizes the formulation of Kotliar and Ruckenstein, which is not manifestly spin-rotation invariant. The paramagnetic saddle point of the corresponding functional integral is identical to Gutzwiller's solution. As an illustration of the method, we calculate the T3{T}^{3}T3lnT spin-fluctuation contribution to the specific heat and find it to be fully consistent with Fermi-liquid theory. We discuss further applications of this approach.
We present a slave-boson formulation for the Hubbard model that preserves the spin-rotation and particle-hole symmetries of the paramagnetic phase at half filling. Fermion and boson fields are treated on an equal footing in a functional integral formulation. We show that, while in two and three dimensions there is a metal-insulator transition as the interaction U increases, in one dimension our approach always gives an insulating state. Short-range spin-spin correlations are included at the saddle-point level for all values of U, giving a good description of the insulating state. We show that the energy gap in the one-particle spectrum increases from zero as (U-Uc{\mathit{U}}_{\mathit{c}}Uc​${)}^{\mathrm{\ensuremath{\alpha}}}$, where Uc{\mathit{U}}_{\mathit{c}}Uc​ is the critical value of the on-site interaction and the exponent \ensuremath{\alpha} is 1 and 1/2 for 2 and 3 dimensions, respectively. \textcopyright{} 1996 The American Physical Society.
We reformulate the time-dependent Gutzwiller approximation by M. Schir\'o and M. Fabrizio, [Phys. Rev. Lett. 105, 076401 (2010)] in the framework of slave-boson mean-field theory, which is used to investigate the dynamical Mott transition of the generic two-band Hubbard model at half filling upon an interaction quench. Interorbital fluctuations lead to notable changes with respect to the single-band case. The singular dynamical transition is replaced by a broad regime of long-lived fluctuations between metallic and insulating states, accompanied by intriguing precursor behavior. A mapping to a spin model proves helpful to analyze the different regions in terms of the evolution of an Ising-like order parameter. Contrary to the static case, singlet occupations remain vital in the Mott-insulating regime with finite Hund's exchange.
We study a two-band Hubbard model with one copper and one oxygen orbital in each cell. These orbitals are hybridized within a cell, and different cells are coupled via oxygen-oxygen hopping. The Hubbard interaction is set equal to zero on oxygen and infinity on copper. We show that, for sufficiently small oxygen bandwidth, the system is insulating at half filling whatever the value of \ensuremath{\Delta} (the energy difference between the oxygen and copper orbitals). This contradicts the prediction of the slave-boson mean-field theory that there is a metal-insulator transition at a fixed value of \ensuremath{\Delta}=${\mathrm{\ensuremath{\Delta}}}_{\mathit{c}}$, independent of the oxygen bandwidth in the narrow-bandwidth limit. In order to understand this discrepancy, we study a generalized model in which the degeneracy of the copper orbitals in a given cell is N. Slave-boson mean-field theory is exact in the N\ensuremath{\rightarrow}\ensuremath{\infty} limit, indicating that there is a phase transition at some value of N. This question is studied further by a combination of 1/N expansion around the N=\ensuremath{\infty} point, and numerical diagonalization up to N=16. It is found that for the physical case of N=2, the results obtained from the large-N theory are qualitatively and quantitatively reliable only for \ensuremath{\Delta}>${\mathrm{\ensuremath{\Delta}}}_{\mathit{c}}$, i.e., when there is no Bose condensation in the N\ensuremath{\rightarrow}\ensuremath{\infty} limit.