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Physical review. B, Condensed matter

Phase transition as a function of orbital degeneracy in a generalized Hubbard model

L. Zhang, Jainendra K. Jain, Victor J. Emery

Published Sep 1, 1992• 15 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.46.5599

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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}=\ensuremathΔc{\mathrm{\ensuremath{\Delta}}}_{\mathit{c}}\ensuremathΔc​, 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}>\ensuremathΔc{\mathrm{\ensuremath{\Delta}}}_{\mathit{c}}\ensuremathΔc​, i.e., when there is no Bose condensation in the N\ensuremath{\rightarrow}\ensuremath{\infty} limit.
In this short communication we discuss results obtained for the single and multi-orbital Hubbard models within the mean-field-slave-boson description of correlated fermions on a square lattice. For appropriate parameter values the results can be used to mimic some of the normal state properties of the copper oxide systems. Results for the single and multi-orbital models differ in a very characteristic manner. In the single band case the phase diagram consists of metallic (U<Uc) and strongly correlated (U>Uc) regions, which are clearly seperated at half filling (Uc being the critical value of the Brinkman-Rice transition1)). Upon doping the transition between these two phases smoothes out and for large enough doping (δ>0.2, δ=1−n) the system behaves as a weakly correlated metal. In the multi-oibital case depending on the input energies for the orbitals the transition to the insulating state can be either of the Brinkman-Rice or the charge-transfer type. Similar to the single band case doping causes rapid loss of correlation effects. We tentatively conclude that the Kondo-type of effect is absent in the multi-orbital model.
The three-band Hubbard model is a fundamental model for understanding properties of the copper-oxygen planes in cuprate superconductors. We use cutting-edge auxiliary-field quantum Monte Carlo (AFQMC) methods to investigate ground state properties of the model in the parent compound. Large supercells combined with twist averaged boundary conditions are employed to reliably reach the thermodynamic limit. Benchmark quality results are obtained on the magnetic correlations and charge gap. A key parameter of this model is the charge-transfer energy $\mathrm{\ensuremath{\Delta}}$ between the oxygen ppp and the copper ddd orbitals, which appears to vary significantly across different families of cuprates and whose ab initio determination is subtle. We show that the system undergoes a quantum phase transition from an antiferromagnetic insulator to a paramagnetic metal as $\mathrm{\ensuremath{\Delta}}$ is lowered to 3eV3\phantom{\rule{0.28em}{0ex}}\mathrm{eV}3eV.
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 examine whether the Mott transition of a half-filled, two-orbital Hubbard model with unequal bandwidths occurs simultaneously for both bands or whether it is a two-stage process in which the orbital with narrower bandwith localizes first (giving rise to an intermediate ``orbital-selective'' Mott phase). This question is addressed using both dynamical mean-field theory and a representation of fermion operators in terms of slave quantum spins, followed by a mean-field approximation (similar in spirit to a Gutzwiller approximation). In the latter approach, the Mott transition is found to be orbital selective for all values of the Coulomb exchange (Hund) coupling JJJ when the bandwidth ratio is small and only beyond a critical value of JJJ when the bandwidth ratio is larger. Dynamical mean-field theory partially confirms these findings, but the intermediate phase at J=0J=0J=0 is found to differ from a conventional Mott insulator, with spectral weight extending down to arbitrary low energy. Finally, the orbital-selective Mott phase is found, at zero temperature, to be unstable with respect to an interorbital hybridization VVV and replaced at small VVV by a state with a large effective mass (and a low quasiparticle coherence scale) for the narrower band.