arXiv:cond-mat/0205263 · Physical review. B, Condensed matter

Mott transition at large orbital degeneracy: Dynamical mean-field theory

Serge Florens, Antoine Georges, Gabriel Kotliar, Olivier Parcollet

Published Nov 8, 2002• 79 citations

Abstract

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 ${(U}_{c1})$ and the one at which the metal becomes unstable ${(U}_{c2})$ 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 ${U}_{c2}/N$ 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.

DOI: 10.1103/physrevb.66.205102

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arXiv:cond-mat/0205263 · Physical review. B, Condensed matter

Mott transition at large orbital degeneracy: Dynamical mean-field theory

Serge Florens, Antoine Georges, Gabriel Kotliar, Olivier Parcollet

Published Nov 8, 2002• 79 citations

arXiv HTML PDF

Abstract

{(U}_{c1})

and the one at which the metal becomes unstable

{(U}_{c2})

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

{U}_{c2}/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.

DOI: 10.1103/physrevb.66.205102

Related papers

Dynamical slave-boson mean-field study of the Mott transition in the Hubbard model in the large- z limit

Sen Zhou, Long Liang, Ziqiang Wang

The Mott metal-insulator transition in the Hubbard model is studied by constructing a dynamical slave-boson mean-field theory in the limit of large lattice coordination number $z$ that incorporates the binding between doubly occupied (doublon) and empty (holon) sites. On the Mott insulating side where all doublons and holons bond in real space into excitonic pairs leading to the charge gap, the theory simplifies considerably to leading order in $1/\sqrt{z}$ and becomes exact on the infinite- $z$ Bethe lattice. An asymptotic solution is obtained for a continuous Mott transition associated with the closing of the charge gap at a critical value of the Hubbard ${U}_{c}$ and the corresponding doublon density ${n}_{d}^{c}$ , hopping ${\ensuremath{\chi}}_{d}^{c}$ and doublon-holon pairing ${\mathrm{\ensuremath{\Delta}}}_{d}^{c}$ amplitudes. We find ${U}_{c}={U}_{\mathrm{BR}}[1\ensuremath{-}2{n}_{d}^{c}\ensuremath{-}\sqrt{z}({\ensuremath{\chi}}_{d}^{c}+{\mathrm{\ensuremath{\Delta}}}_{d}^{c})]\ensuremath{\simeq}0.8{U}_{\mathrm{BR}}$, where ${U}_{\mathrm{BR}}$ is the critical value for the Brinkman-Rice transition in the Gutzwiller approximation captured in the static mean-field solution of the slave-boson formulation of Kotliar and Ruckenstein. Thus, the Mott transition can be viewed as the quantum correction to the Brinkman-Rice transition due to doublon-holon binding. Quantitative comparisons are made to the results of the dynamical mean-field theory, showing good agreement. In the absence of magnetic order, the Mott insulator is a $U(1)$ quantum spin liquid with nonzero intersite spinon hopping that survives the large- $z$ limit and lifts the ${2}^{N}$ -fold degeneracy of the local moments. We show that the spinons are coupled to the doublons/holons by a dissipative compact $U(1)$ gauge field in the deconfined phase, realizing the spin-charge separated gapless spin liquid Mott insulator.The Mott metal-insulator transition in the Hubbard model is studied by constructing a dynamical slave-boson mean-field theory in the limit of large lattice coordination number $z$ that incorporates the binding between doubly occupied (doublon) and empty (holon) sites. On the Mott insulating side where all doublons and holons bond in real space into excitonic pairs leading to the charge gap, the theory simplifies considerably to leading order in $1/\sqrt{z}$ and becomes exact on the infinite-$z$ Bethe lattice. An asymptotic solution is obtained for a continuous Mott transition associated with the closing of the charge gap at a critical value of the Hubbard ${U}_{c}$ and the corresponding doublon density ${n}_{d}^{c}$, hopping ${\ensuremath{\chi}}_{d}^{c}$ and doublon-holon pairing ${\mathrm{\ensuremath{\Delta}}}_{d}^{c}$ amplitudes. We find ${U}_{c}={U}_{\mathrm{BR}}[1\ensuremath{-}2{n}_{d}^{c}\ensuremath{-}\sqrt{z}({\ensuremath{\chi}}_{d}^{c}+{\mathrm{\ensuremath{\Delta}}}_{d}^{c})]\ensuremath{\simeq}0.8{U}_{\mathrm{BR}}$, where ${U}_{\mathrm{BR}}$ is the critical value for the Brinkman-Rice transition in the Gutzwiller approximation captured in the static mean-field solution of the slave-boson formulation of Kotliar and Ruckenstein. Thus, the Mott transition can be viewed as the quantum correction to the Brinkman-Rice transition due to doublon-holon binding. Quantitative comparisons are made to the results of the dynamical mean-field theory, showing good agreement. In the absence of magnetic order, the Mott insulator is a $U(1)$ quantum spin liquid with nonzero intersite spinon hopping that survives the large-$z$ limit and lifts the ${2}^{N}$-fold degeneracy of the local moments. We show that the spinons are coupled to the doublons/holons by a dissipative compact $U(1)$ gauge field in the deconfined phase, realizing the spin-charge separated gapless spin liquid Mott insulator.

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Luca de’ Medici, Antoine Georges, Silke Biermann

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Mott transitions in correlated electron systems with orbital degrees of freedom

Yūichi Ono, Michael Potthoff, R. Bulla

Mott metal-insulator transitions in an M-fold orbitally degenerate Hubbard model are studied by means of a generalization of the linearized dynamical mean-field theory. The method allows for an efficient and reliable determination of the critical interaction ${U}_{\mathrm{c}}$ for any integer filling n and different M at zero temperature. For half filling a linear dependence of ${U}_{\mathrm{c}}$ on M is found. Inclusion of the (full) Hund's rule exchange J results in a strong reduction of ${U}_{\mathrm{c}}.$ The transition turns out to change qualitatively from continuous for $J=0$ to discontinuous for any finite J. This result is confirmed using the exact-diagonalization method for $M=2.$ Mott metal-insulator transitions in an M-fold orbitally degenerate Hubbard model are studied by means of a generalization of the linearized dynamical mean-field theory. The method allows for an efficient and reliable determination of the critical interaction ${U}_{\mathrm{c}}$ for any integer filling n and different M at zero temperature. For half filling a linear dependence of ${U}_{\mathrm{c}}$ on M is found. Inclusion of the (full) Hund's rule exchange J results in a strong reduction of ${U}_{\mathrm{c}}.$ The transition turns out to change qualitatively from continuous for $J=0$ to discontinuous for any finite J. This result is confirmed using the exact-diagonalization method for $M=2.$

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Mott transition in correlated electron systems with orbital degree of freedom

Yoshiaki Ōno, R. Bulla, Michael Potthoff

Mott metal-insulator transitions in an M-fold orbitally degenerate Hubbard model are studied by means of a generalization of the linearized dynamical mean-field theory. The method allows for an efficient and reliable determination of the critical interaction Uc for any integer filling n and different M at zero temperature. For half-filling a linear dependence of Uc on M is found. Inclusion of the (full) Hund's rule exchange J results in a strong reduction of Uc. The transition turns out to change qualitatively from continuous for J=0 to discontinuous for any finite J. This result is confirmed using the exact-diagonalization method for M=2.Mott metal-insulator transitions in an M-fold orbitally degenerate Hubbard model are studied by means of a generalization of the linearized dynamical mean-field theory. The method allows for an efficient and reliable determination of the critical interaction Uc for any integer filling n and different M at zero temperature. For half-filling a linear dependence of Uc on M is found. Inclusion of the (full) Hund's rule exchange J results in a strong reduction of Uc. The transition turns out to change qualitatively from continuous for J=0 to discontinuous for any finite J. This result is confirmed using the exact-diagonalization method for M=2.

Jan 1, 20030 citations

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

Hideo Hasegawa

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.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.

arXiv:cond-mat/9612142Journal of the Physical Society of JapanMay 15, 199738 citations

{(U}_{c1})

and the one at which the metal becomes unstable

{(U}_{c2})

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

{U}_{c2}/N

The Mott metal-insulator transition in the Hubbard model is studied by constructing a dynamical slave-boson mean-field theory in the limit of large lattice coordination number

z

that incorporates the binding between doubly occupied (doublon) and empty (holon) sites. On the Mott insulating side where all doublons and holons bond in real space into excitonic pairs leading to the charge gap, the theory simplifies considerably to leading order in

1/\sqrt{z}

and becomes exact on the infinite-

z

Bethe lattice. An asymptotic solution is obtained for a continuous Mott transition associated with the closing of the charge gap at a critical value of the Hubbard

{U}_{c}

and the corresponding doublon density

{n}_{d}^{c}

, hopping ${\ensuremath{\chi}}_{d}^{c}$ and doublon-holon pairing ${\mathrm{\ensuremath{\Delta}}}_{d}^{c}$ amplitudes. We find ${U}_{c}={U}_{\mathrm{BR}}[1\ensuremath{-}2{n}_{d}^{c}\ensuremath{-}\sqrt{z}({\ensuremath{\chi}}_{d}^{c}+{\mathrm{\ensuremath{\Delta}}}_{d}^{c})]\ensuremath{\simeq}0.8{U}_{\mathrm{BR}}$, where

{U}_{\mathrm{BR}}

is the critical value for the Brinkman-Rice transition in the Gutzwiller approximation captured in the static mean-field solution of the slave-boson formulation of Kotliar and Ruckenstein. Thus, the Mott transition can be viewed as the quantum correction to the Brinkman-Rice transition due to doublon-holon binding. Quantitative comparisons are made to the results of the dynamical mean-field theory, showing good agreement. In the absence of magnetic order, the Mott insulator is a

U(1)

quantum spin liquid with nonzero intersite spinon hopping that survives the large-

z

limit and lifts the

{2}^{N}

-fold degeneracy of the local moments. We show that the spinons are coupled to the doublons/holons by a dissipative compact

U(1)

gauge field in the deconfined phase, realizing the spin-charge separated gapless spin liquid Mott insulator.

J

when the bandwidth ratio is small and only beyond a critical value of

J

when the bandwidth ratio is larger. Dynamical mean-field theory partially confirms these findings, but the intermediate phase at

J=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

V

and replaced at small

V

by a state with a large effective mass (and a low quasiparticle coherence scale) for the narrower band.

{U}_{\mathrm{c}}

for any integer filling n and different M at zero temperature. For half filling a linear dependence of

{U}_{\mathrm{c}}

on M is found. Inclusion of the (full) Hund's rule exchange J results in a strong reduction of

{U}_{\mathrm{c}}.

The transition turns out to change qualitatively from continuous for

J=0

to discontinuous for any finite J. This result is confirmed using the exact-diagonalization method for

M=2.

1/3

filling, in comparison with the preceding results for different fillings and for two-orbital models. Strong competition between the two insulators results in an intriguing first-order transition to an insulating phase having intermediate characters between Mott and band insulators. It is also found that the two insulators show different behaviors in the phase boundary with the metallic phase in the interaction-temperature plane, which is reflected in a difference in the quasiparticle behavior around the transition. We also discuss the orbital selective Mott transition for larger Hund's coupling, which is compared with previous study at zero temperature.

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.