arXiv:0806.1101 · Physical Review B

Exact results in a slave boson saddle point approach for a strongly correlated electron model

Raymond Frésard, Thilo Kopp

Published Aug 26, 2008• 6 citations

Abstract

We revisit the Kotliar-Ruckenstein (KR) slave boson saddle point evaluation for a two-site correlated electron model. As the model can be solved analytically, it is possible to compare the KR saddle point results with the exact many-particle levels. The considered two-site cluster mimics an infinite-$U$ single-impurity Anderson model with a nearest-neighbor Coulomb interaction: one site is strongly correlated with an infinite local Coulomb repulsion, which hybridizes with the second site, on which the local Coulomb repulsion vanishes. Making use of the flexibility of the representation, we introduce appropriate weight factors in the KR saddle point scheme. Ground-state and all excitation levels agree with the exact diagonalization results. Thermodynamics and correlation functions may be recovered in a suitably renormalized saddle point evaluation.

DOI: 10.1103/physrevb.78.073108

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arXiv:0806.1101 · Physical Review B

Exact results in a slave boson saddle point approach for a strongly correlated electron model

Raymond Frésard, Thilo Kopp

Published Aug 26, 2008• 6 citations

arXiv HTML PDF

Abstract

U

single-impurity Anderson model with a nearest-neighbor Coulomb interaction: one site is strongly correlated with an infinite local Coulomb repulsion, which hybridizes with the second site, on which the local Coulomb repulsion vanishes. Making use of the flexibility of the representation, we introduce appropriate weight factors in the KR saddle point scheme. Ground-state and all excitation levels agree with the exact diagonalization results. Thermodynamics and correlation functions may be recovered in a suitably renormalized saddle point evaluation.

DOI: 10.1103/physrevb.78.073108

Related papers

Barnes slave-boson approach to the two-site single-impurity Anderson model with non-local interaction

Raymond Frésard, Henni Ouerdane, Thilo Kopp

The Barnes slave-boson approach to the U =∞ single-impurity Anderson model extended by a non-local Coulomb interaction is revisited. We demonstrate first that the radial gauge representation facilitates the treatment of such a non-local interaction by performing the exact evaluation of the path integrals representing the partition function, the impurity hole density and the impurity hole density autocorrelation function for a two-site cluster. The free energy is also obtained on the same footing. Next, the exact results are compared to their approximations at saddle-point level, and it is shown that the saddle point evaluation recovers the exact answer in the limit of strong non-local Coulomb interaction, while the agreement between both schemes remains satisfactory in a large parameter range.The Barnes slave-boson approach to the U =∞ single-impurity Anderson model extended by a non-local Coulomb interaction is revisited. We demonstrate first that the radial gauge representation facilitates the treatment of such a non-local interaction by performing the exact evaluation of the path integrals representing the partition function, the impurity hole density and the impurity hole density autocorrelation function for a two-site cluster. The free energy is also obtained on the same footing. Next, the exact results are compared to their approximations at saddle-point level, and it is shown that the saddle point evaluation recovers the exact answer in the limit of strong non-local Coulomb interaction, while the agreement between both schemes remains satisfactory in a large parameter range.

arXiv:0801.4496Europhysics Letters (EPL)Apr 17, 20087 citations

Exact results with the Kotliar‐Ruckenstein slave‐boson representation

Raymond Frésard, Thilo Kopp

Abstract Radial slave boson representations have the particular advantage that the expectation values of their respective fields are finite even without the formal introduction of spurious Bose condensates for each of the bosonic fields. The expectation values of the radial (real) fields are in fact to be interpreted as the density of empty or singly occupied sites. Whereas the radial representation of the Barnes slave bosons has been investigated before, a setup for the functional integral of radial bosonic fields in the more physical Kotliar‐Ruckenstein representation has not been accomplished to date. We implement a path integral procedure with suitable renormalization factors for a strongly correlated two‐site model which allows to control the formal steps in the intricate evaluation, as the results for the partition function and the expectation values are known from exact diagonalization for such a minimal single impurity Anderson model. The partition function is shown to be a trace over a product of matrices local in time and therefore can be calculated analytically. Eventually, we establish the scheme for the evaluation of correlation functions and thermodynamic properties.Abstract Radial slave boson representations have the particular advantage that the expectation values of their respective fields are finite even without the formal introduction of spurious Bose condensates for each of the bosonic fields. The expectation values of the radial (real) fields are in fact to be interpreted as the density of empty or singly occupied sites. Whereas the radial representation of the Barnes slave bosons has been investigated before, a setup for the functional integral of radial bosonic fields in the more physical Kotliar‐Ruckenstein representation has not been accomplished to date. We implement a path integral procedure with suitable renormalization factors for a strongly correlated two‐site model which allows to control the formal steps in the intricate evaluation, as the results for the partition function and the expectation values are known from exact diagonalization for such a minimal single impurity Anderson model. The partition function is shown to be a trace over a product of matrices local in time and therefore can be calculated analytically. Eventually, we establish the scheme for the evaluation of correlation functions and thermodynamic properties.

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Finite-Uslave-boson methods and the intersite magnetic correlations in the Hubbard models

Hiroyuki Kaga

We examine the two existing slave-boson methods for the finite-U Hubbard model by Kotliar and Ruckenstein (KR) and by Newns et al. and Zou and Anderson (NZA) in the light of the Fermi-liquid quasiparticle picture. It is shown that to treat doubly occupied states the KR slave-boson scheme is more consistent with this picture and that the well-known erroneous gauge-symmetry breaking encountered in the mean-field treatment of the NZA double-occupancy boson is due to the presence of the hidden double-fermion operators. Then, the effective intersite antiferromagnetic interaction between quasiparticles is derived in the KR slave-boson approach by taking into account empty and double-occupancy slave-boson fluctuations to O(1/ ${\mathit{N}}^{2}$ ) in the 1/N expansion of the functional-integral formalism. The obtained effective exchange coupling constant ${\mathit{J}}_{\mathrm{eff}}$ (\ensuremath{\delta})=J(2 ${\mathit{p}}_{0}^{2}$ -\ensuremath{\delta}/2) with J\ensuremath{\equiv} ${\mathit{t}}_{\mathit{i}\mathit{j}}^{2}$ /U is strongly dependent upon doping \ensuremath{\delta} and is scaled with the reduced Coulomb repulsion u\ensuremath{\equiv}U/ ${\mathit{U}}_{\mathit{c}}$ in the two-dimensional (2D) and 3D Hubbard models, where ${\mathit{U}}_{\mathit{c}}$ is the critical value for the metal-to-insulator transition. The roles of this intersite correlation and the spin-dependent band-renormalization correlation included in the mean-field theory for the paramagnetic-to-antiferromagnetic instability are compared in the two Hubbard models. The intersite contribution is more significant in the 2D case than in the 3D case, but is generally small compared with the band effect except for small dopings \ensuremath{\delta} and smaller U values. The maximum doping ${\mathrm{\ensuremath{\delta}}}_{\mathrm{cr}}$ for the paramagnetic-to-ferromagnetic instability in the strong-coupling limit (U\ensuremath{\rightarrow}\ensuremath{\infty}) of the 2D and 3D Hubbard models is found to be ${\mathrm{\ensuremath{\delta}}}_{\mathrm{cr}}$\ensuremath{\sim}0.28 and ${\mathrm{\ensuremath{\delta}}}_{\mathrm{cr}}$\ensuremath{\sim}0.38, respectively.We examine the two existing slave-boson methods for the finite-U Hubbard model by Kotliar and Ruckenstein (KR) and by Newns et al. and Zou and Anderson (NZA) in the light of the Fermi-liquid quasiparticle picture. It is shown that to treat doubly occupied states the KR slave-boson scheme is more consistent with this picture and that the well-known erroneous gauge-symmetry breaking encountered in the mean-field treatment of the NZA double-occupancy boson is due to the presence of the hidden double-fermion operators. Then, the effective intersite antiferromagnetic interaction between quasiparticles is derived in the KR slave-boson approach by taking into account empty and double-occupancy slave-boson fluctuations to O(1/${\mathit{N}}^{2}$) in the 1/N expansion of the functional-integral formalism. The obtained effective exchange coupling constant ${\mathit{J}}_{\mathrm{eff}}$(\ensuremath{\delta})=J(2${\mathit{p}}_{0}^{2}$-\ensuremath{\delta}/2) with J\ensuremath{\equiv}${\mathit{t}}_{\mathit{i}\mathit{j}}^{2}$/U is strongly dependent upon doping \ensuremath{\delta} and is scaled with the reduced Coulomb repulsion u\ensuremath{\equiv}U/${\mathit{U}}_{\mathit{c}}$ in the two-dimensional (2D) and 3D Hubbard models, where ${\mathit{U}}_{\mathit{c}}$ is the critical value for the metal-to-insulator transition. The roles of this intersite correlation and the spin-dependent band-renormalization correlation included in the mean-field theory for the paramagnetic-to-antiferromagnetic instability are compared in the two Hubbard models. The intersite contribution is more significant in the 2D case than in the 3D case, but is generally small compared with the band effect except for small dopings \ensuremath{\delta} and smaller U values. The maximum doping ${\mathrm{\ensuremath{\delta}}}_{\mathrm{cr}}$ for the paramagnetic-to-ferromagnetic instability in the strong-coupling limit (U\ensuremath{\rightarrow}\ensuremath{\infty}) of the 2D and 3D Hubbard models is found to be ${\mathrm{\ensuremath{\delta}}}_{\mathrm{cr}}$\ensuremath{\sim}0.28 and ${\mathrm{\ensuremath{\delta}}}_{\mathrm{cr}}$\ensuremath{\sim}0.38, respectively.

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Slave-boson theory of the extended Falicov-Kimball model

P. M. R. Brydon

The extended Falicov-Kimball model, with both an on-site hybridization potential and dispersive narrow band, is examined within the saddle-point approximation to the Kotliar-Ruckenstein [Phys. Rev. Lett. 57, 1362 (1986)] slave-boson theory. We first set the hybridization potential to zero and find that the phase diagram depends strongly on the orbital structure: for degenerate orbitals, a correlated-insulating state is found at sufficiently strong interaction strengths, whereas a finite orbital energy difference can lead to discontinuous valence transitions. The obtained phase diagram is very sensitive to the presence of a finite hybridization potential. As in Hartree-Fock theory, we find an enhancement of the hybridization by the interorbital Coulomb repulsion. The more precise treatment of correlation effects, however, leads to large deviations from the Hartree-Fock results. In the limit of vanishing hybridization, an excitonic insulator state is only found when the orbitals are degenerate, which restricts this phase to a much smaller parameter space than in other available mean-field theories.The extended Falicov-Kimball model, with both an on-site hybridization potential and dispersive narrow band, is examined within the saddle-point approximation to the Kotliar-Ruckenstein [Phys. Rev. Lett. 57, 1362 (1986)] slave-boson theory. We first set the hybridization potential to zero and find that the phase diagram depends strongly on the orbital structure: for degenerate orbitals, a correlated-insulating state is found at sufficiently strong interaction strengths, whereas a finite orbital energy difference can lead to discontinuous valence transitions. The obtained phase diagram is very sensitive to the presence of a finite hybridization potential. As in Hartree-Fock theory, we find an enhancement of the hybridization by the interorbital Coulomb repulsion. The more precise treatment of correlation effects, however, leads to large deviations from the Hartree-Fock results. In the limit of vanishing hybridization, an excitonic insulator state is only found when the orbitals are degenerate, which restricts this phase to a much smaller parameter space than in other available mean-field theories.

arXiv:0706.1883Physical Review BJan 9, 200817 citations

Variational results as saddle-point approximations: The Anderson impurity model

Kurt Schönhammer

Physical review. B, Condensed matterAug 1, 199017 citations

Slave-boson based configuration-interaction approach for the Hubbard model

G. Seibold

Based on the Kotliar--Ruckenstein slave-boson scheme, we develop a configuration-interaction (CI) approach that is suitable to evaluate energy corrections for symmetry-broken saddle-point solutions. The theory is applied to spin-polaron states in the Hubbard model and compared with analogous results obtained within the Hartree--Fock approximation. In addition, we show that within the infinite $\mathcal{D}$ prescription of the Gutzwiller method a CI approach does not correct the variational result since in the thermodynamic limit matrix elements between different inhomogeneous states vanish due to an ``orthogonality catastrophe.''Based on the Kotliar--Ruckenstein slave-boson scheme, we develop a configuration-interaction (CI) approach that is suitable to evaluate energy corrections for symmetry-broken saddle-point solutions. The theory is applied to spin-polaron states in the Hubbard model and compared with analogous results obtained within the Hartree--Fock approximation. In addition, we show that within the infinite $\mathcal{D}$ prescription of the Gutzwiller method a CI approach does not correct the variational result since in the thermodynamic limit matrix elements between different inhomogeneous states vanish due to an ``orthogonality catastrophe.''

arXiv:0801.4528Physical Review BJun 12, 20083 citations

U

{\mathit{N}}^{2}

) in the 1/N expansion of the functional-integral formalism. The obtained effective exchange coupling constant

{\mathit{J}}_{\mathrm{eff}}

(\ensuremath{\delta})=J(2

{\mathit{p}}_{0}^{2}

-\ensuremath{\delta}/2) with J\ensuremath{\equiv}

{\mathit{t}}_{\mathit{i}\mathit{j}}^{2}

/U is strongly dependent upon doping \ensuremath{\delta} and is scaled with the reduced Coulomb repulsion u\ensuremath{\equiv}U/

{\mathit{U}}_{\mathit{c}}

in the two-dimensional (2D) and 3D Hubbard models, where

{\mathit{U}}_{\mathit{c}}

is the critical value for the metal-to-insulator transition. The roles of this intersite correlation and the spin-dependent band-renormalization correlation included in the mean-field theory for the paramagnetic-to-antiferromagnetic instability are compared in the two Hubbard models. The intersite contribution is more significant in the 2D case than in the 3D case, but is generally small compared with the band effect except for small dopings \ensuremath{\delta} and smaller U values. The maximum doping ${\mathrm{\ensuremath{\delta}}}_{\mathrm{cr}}$ for the paramagnetic-to-ferromagnetic instability in the strong-coupling limit (U\ensuremath{\rightarrow}\ensuremath{\infty}) of the 2D and 3D Hubbard models is found to be ${\mathrm{\ensuremath{\delta}}}_{\mathrm{cr}}$\ensuremath{\sim}0.28 and ${\mathrm{\ensuremath{\delta}}}_{\mathrm{cr}}$\ensuremath{\sim}0.38, respectively.

\mathcal{D}

prescription of the Gutzwiller method a CI approach does not correct the variational result since in the thermodynamic limit matrix elements between different inhomogeneous states vanish due to an ``orthogonality catastrophe.''