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Physical review. B, Condensed matter

Spin- and charge-fluctuation contribution to mass enhancement

T. Li

Published Aug 15, 1993• 4 citations
Publisher

Abstract

We study the one-loop contribution of the spin and charge fluctuation to the boson hopping factor in the slave-boson representation of the Hubbard model introduced by Kotliar and Ruckenstein. Whether one uses a low-lying cutoff or sums over the whole bosonic spectrum, the noninteracting limit for the fluctuation correction of the mass enhancement is not recovered for vanishing Coulomb interaction. We argue that one should be cautious when attempting to extend the loop-expansion calculation, in the functional-integral scheme of Kotliar and Ruckenstein, to the observables in which the high-energy excitations may play an important role.

DOI
10.1103/physrevb.48.4991

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We study the one-loop contribution of the spin and charge fluctuation to the boson hopping factor in the slave-boson representation of the Hubbard model introduced by Kotliar and Ruckenstein. Whether one uses a low-lying cutoff or sums over the whole bosonic spectrum, the noninteracting limit for the fluctuation correction of the mass enhancement is not recovered for vanishing Coulomb interaction. We argue that one should be cautious when attempting to extend the loop-expansion calculation, in the functional-integral scheme of Kotliar and Ruckenstein, to the observables in which the high-energy excitations may play an important role.
We study the Hubbard model through a functional-integral formulation proposed by Kotliar and Ruckenstein. A careful use of the U(1${)}^{\ensuremath{\bigotimes}3}$ gauge symmetry of the bosonized action allows us to compute the complete one-loop correction to the free energy in the case of half-filled band. We extract the effective mass of the fermion in the metallic phase and show that the noninteracting limit is not recovered for vanishing on-site repulsion. The bosonic Hamiltonian must be adjusted order by order in perturbation theory.
We determine the free energy of strongly correlated electron systems in the example of the Hubbard model by calculating the contribution of spin and charge fluctuations to the Gutzwiller approximation mean-field result. We employ the slave-boson formulation of Kotliar and Ruckenstein in its spin-rotation-invariant form in the usual continuous-time approximation of the functional integral representation, corrected by “high-frequency contributions” ( SRIKR + ). In contrast to all previous attempts to calculate the fluctuation contribution to the free energy within the spin-rotation-invariant Kotliar-Ruckenstein slave-boson representation (SRIKR) scheme our result preserves the noninteracting limit. We show that the correct form of the kinetic energy renormalization is essential. The results for the ground-state energy in the paramagnetic phase are in very good agreement with state-of-the-art results obtained by methods such as density matrix embedding theory, quantum Monte Carlo, and others. The leading low-temperature behavior of the free energy allows us to extract the quasiparticle effective mass, in particular its enhancement near a continuous phase transition into an ordered state. Our work demonstrates that the SRIKR + method is competitive with the best available alternative methods and equips the slave-boson approach with an improved synoptic power to explore strongly correlated electron systems.
The Hubbard model with finite on-site repulsion U is studied via the functional-integral formulation of the four-slave-boson approach by Kotliar and Ruckenstein. It is shown that a correct treatment of the continuum imaginary time limit (which is required by the very definition of the functional integral) modifies the free energy when fluctuation (1/N) corrections beyond mean field are considered, thus removing the inconsistencies originating from the incorrect handling of this pathologic limit so far performed in the literature. In particular, our treatment correctly restores the decrease of the average number of doubly occupied sites for increasing U. Our analysis requires us to suitably interpret the Kotliar and Ruckenstein choice for the bosonic hopping operator and to abandon the commonly used normal-ordering prescription, in order to obtain meaningful fluctuation corrections. In this way we recover the exact solution at U=0 not only at the mean-field level but also at the next order in 1/N. In addition, we consider alternative choices for the bosonic hopping operator and test them numerically for a simple two-site model for which the exact solution is readily available for any U. We also discuss how the 1/N expansion can be formally generalized to the four-slave-boson approach, and provide a simplified prescription to obtain the additional terms in the free energy which result at the order 1/N from the correct continuum limit.
We study the single-band Hubbard model via the functional-integral four-slave-boson formulation introduced by Kotliar and Ruckenstein, and show that a proper treatment of the continuum limit of the functional integral removes the inconsistencies due to an incorrect handling of this limit so far performed in the literature. It follows from our analysis that the Kotliar and Ruckenstein mean-field solution has to be abandoned in order to include fluctuation corrections in a systematic way. We show explicitly how to construct a suitable alternative solution which admits the inclusion of fluctuations.
We consider the N-component generalization of the four-slave-boson approach to the Hubbard model, where 1/N acts as the small parameter that controls the fluctuations about the saddle point, and address the problem of the appropriate choice of the bosonic hopping operator ${\mathit{z}}_{\mathit{i}\mathrm{\ensuremath{\sigma}}}$. By suitably reorganizing the Fock space, we show that the square-root form for ${\mathit{z}}_{\mathit{i},\mathrm{\ensuremath{\sigma}}}$ (originally introduced by Kotliar and Ruckenstein) reproduces the exact independent-fermion (U=0) results not only at the mean-field (N=\ensuremath{\infty}) level but also to all orders in the 1/N expansion, provided one relaxes the usually adopted normal-ordered prescription for ${\mathit{z}}_{\mathit{i},\mathrm{\ensuremath{\sigma}}}$. This ensures that ${\mathit{z}}_{\mathit{i},\mathrm{\ensuremath{\sigma}}}$ need not be modified at successive orders in the fluctuation expansion, and implies that all correlation functions are correctly recovered in the U=0 limit, a nontrivial result for the slave-boson approach. In addition, it provides a stringent requirement on the form of ${\mathit{z}}_{\mathit{i},\mathrm{\ensuremath{\sigma}}}$, which might be also generalized to alternative slave-boson formalisms (like the spin-rotation-invariant formulation).
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.