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arXiv:cond-mat/9501132 · Physical review. B, Condensed matter

Correct continuum limit of the functional-integral representation for the four-slave-boson approach to the Hubbard model: Paramagnetic phase

Enrico Arrigoni, Giancarlo Calvanese Strinati

Published Jul 15, 1995• 14 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.52.2428

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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 slave-boson method within the coherent-state functional integral representation of the partition function, and show how to deal with the continuum imaginary-time limit required by the very definition of the functional integral. We find that a correct treatment of the continuum limit modifies the free energy when fluctuation (1/N) corrections beyond the mean-field solution are considered. Numerical results are presented for a two-level single-site model system (with an infinite Hubbard repulsion), for which the additional terms in the free energy introduced by the correct continuum limit act to validate the 1/N expansion. Our analysis calls for a revision of several outcomes of the slave-boson method with the inclusion of fluctuation corrections.
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 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 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 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.