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arXiv:2402.05701 · Annalen der Physik

Exact Functional Integration of Radial and Complex Slave‐Boson Fields: Thermodynamics and Dynamics of the Two‐Site Extended Hubbard Model

Vu Hung Dao, Raymond Frésard

Published May 16, 2024• 3 citations
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Abstract

Abstract The functional integral formulation of the Hubbard model when treated in its Kotliar‐Ruckenstein representation in the radial gauge involves fermionic, as well as complex and radial slave boson fields. In order to improve on the understanding of the interplay of the three types of fields, and on the nature of the latter, a comprehensive investigation of an exactly solvable two‐site cluster is performed, as it entails all pitfalls embodied in this approach. It is first shown that the exact partition function is recovered, even when incorporating in the calculation the square root factors that are at the heart of the representation, when suitably regularized. It is shown that using radial slave boson fields allows to overcome all hurdles following from the normal ordering procedure. It is then demonstrated that this applies to the Green's function as well, and to the correlation functions of physical interest, thereby answering the criticisms raised by Schönhammer [K. Schönhammer, Phys. Rev. B 1990 42, 2591]. In addition, the investigation generalizes the calculations to the Hubbard model extended by a non‐local Coulomb interaction.

DOI
10.1002/andp.202400029

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Abstract The functional integral formulation of the Hubbard model when treated in its Kotliar‐Ruckenstein representation in the radial gauge involves fermionic, as well as complex and radial slave boson fields. In order to improve on the understanding of the interplay of the three types of fields, and on the nature of the latter, a comprehensive investigation of an exactly solvable two‐site cluster is performed, as it entails all pitfalls embodied in this approach. It is first shown that the exact partition function is recovered, even when incorporating in the calculation the square root factors that are at the heart of the representation, when suitably regularized. It is shown that using radial slave boson fields allows to overcome all hurdles following from the normal ordering procedure. It is then demonstrated that this applies to the Green's function as well, and to the correlation functions of physical interest, thereby answering the criticisms raised by Schönhammer [K. Schönhammer, Phys. Rev. B 1990 42, 2591]. In addition, the investigation generalizes the calculations to the Hubbard model extended by a non‐local Coulomb interaction.
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 The gauge symmetry group of any slave boson representation allows to gauge away the phase of bosonic fields. One benefit of this radial field formulation is the elimination of spurious Bose condensations when saddle‐point approximation is performed. Within the Kotliar–Ruckenstein representation, three of the four bosonic fields can be radial while the last one has to remain complex. In this work, the procedure to carry out the functional integration involving constrained fermionic fields, complex bosonic fields, and radial bosonic fields is presented. The correctness of the representation is verified by exactly evaluating the partition function and the Green's function of the Hubbard model in the atomic limit.
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 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 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-UUU 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.
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.