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Physical review. B, Condensed matter

Mixed boson-fermion description of correlated electrons: Fluctuation corrections in the symmetric treatment

J. J. Vicente Alvarez, Hermenegildo Alejandro Ceccatto, C. A. Balseiro

Published Nov 15, 1995• 2 citations
Publisher

Abstract

We consider the introduction of fluctuation corrections to saddle-point results in the symmetric treatment of a mixed pseudofermion-boson representation of correlated electrons. In our calculations we avoid the complications of working in the discrete imaginary-time formulation of the functional integral, a procedure recently advocated in the literature as mandatory for this problem. For a simple two-site model our approach leads to approximate results in remarkable agreement with the exact results, and without the spurious nonanalyticities of other similar treatments.

DOI
10.1103/physrevb.52.14511

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We consider the introduction of fluctuation corrections to saddle-point results in the symmetric treatment of a mixed pseudofermion-boson representation of correlated electrons. In our calculations we avoid the complications of working in the discrete imaginary-time formulation of the functional integral, a procedure recently advocated in the literature as mandatory for this problem. For a simple two-site model our approach leads to approximate results in remarkable agreement with the exact results, and without the spurious nonanalyticities of other similar treatments.
A systematic extension of the model of almost localized fermions to finite temperatures is presented. It consists in accounting for the quantum fluctuations beyond the Gutzwiller approximation (GA). We use the slave-boson formulation of the Hubbard model which has been proved to be equivalent at the saddle-point level to the GA. We determine the values of the Lagrange multipliers introduced to enforce the constraints, and show how they involve a Mott gap at nearly half filling for U>Uc{U}_{c}Uc​ (localization edge). The results of the GA for the correlation functions are reproduced at the saddle point when taking into account the implicit dependence of the boson fields with the external excitation. The Gaussian fluctuations are then considered in a renormalized basis of the boson fields which brings out two distinct channels (symmetric s and antisymmetric a). We show how the free energy and the correlation functions can be simply expressed as a function of the same Landau parameters F0s{F}_{0}^{s}F0s​ and F0a{F}_{0}^{a}F0a​ defined in the GA.
Functional-integral techniques are used to study correlated fermion models of popular interest: the prototypical Hubbard models. Recent progress has been made in the functional-integral approach to such correlated Fermi systems by introducing a spin-space reference frame that :fluctuates in space and time, Introducing fermions with varying quantization a.xis on the lattice allows the functional-integral representation to assume a uniform approximation over the entire range of the interaction strength U, reproducing the expected mean fields at the saddlepoints along with gapless excitation spectrums. First, the framework necessary for this study is developed: second quantization and the second quantized models of correlated fermions that are to be examined are briefly reviewed. Next, an introduction to first and second quantized functional integrals is followed by basic functional-integral techniques employed to study quantum many-particle systems. Once the background necessary to investigate this approach has been given, the :fluctuating spin-space reference frame is described and is examined in the two-site problem, where an exact diagonalization may be performed. A functional-integral over one-vertex polygonal paths is used to approximate the partition function in a fixed reference frame orientation. This functional-integral representation is studied in the low-to-intermediate and high temperature regimes. A classification of the quantization a.xis configurations, based on classical spin waves, is motivated and presented.
We apply the recently proposed supersymmetric Hubbard operators [P. Coleman, C. P\'epin, and J. Hopkinson, Phys. Rev. B 63, 140411(R) (2001)] to an atomic model. In the limiting case of free spins, we derive exact results for the entropy which are compared with a mean-field + Gaussian corrections description. We show how these results can be extended to the case of charge fluctuations and calculate exact results for the partition function, free energy, and heat capacity of an atomic model for some simple examples. Wave-functions of possible states are listed. We compare the accuracy of large N expansions of the susy spin operators [P. Coleman, C. P\'epin, and A. M. Tsvelik, Phys. Rev. B 62, 3852 (2000); Nucl. Phys. B 586, 641 (2000)] with those obtained using ``Schwinger bosons'' and ``Abrikosov pseudofermions.'' For the atomic model, we compare results of slave boson, slave fermion, and susy Hubbard operator approximations in the physically interesting but uncontrolled limiting case of $\stackrel{\ensuremath{\rightarrow}}{N}2.$ For a mixed representation of spins, we estimate the accuracy of large N expansions of the atomic model. In the single box limit, we find that the lowest-energy susy saddle point reduces to simply either slave bosons or slave fermions, while for higher boxes this is not the case. The highest energy saddle point solution has the interesting feature that it admits a small region of a mixed representation, which bears a superficial resemblance to that observed experimentally close to an antiferromagnetic quantum critical point.
We propose a new functional integral representation of the Hubbarda and Anderson models of lattice fermions. The simplest saddle-point approximation leads, at zero temperature, to the results derived from the Gutzwiller variational wave function. This approach uncovers the limitations of the Gutzwiller approximation and clarifies its connection to the "auxiliary-boson" mean-field theory of the Anderson model. This formulation leads to a novel strong-coupling mean-field theory which allows for a unified treatment of antiferromagnetism and ferromagnetism, metal-to-insulator transition, and Kondo compensation effects.
Electronic states in the two-dimensional Hubbard model are studied in the doped paramagnetic states by use of the auxiliary boson approach. Four auxiliary bosons are introduced by means of the Hubbard-Stratonovich transformation within the functional integral treatment. These bosons correspond to one charge and three spin fluctuations. An effective model is formulated. Going beyond the boson fluctuations around the saddle point, the single-electron Green's function including one-loop self-energy effects of fermion-boson interaction is derived. The behavior of the fermion self-energy, spectral functions, and density of states are investigated for several values of the Coulomb interaction UUU within the limit imposed by the Stoner criterion and also for small and moderate doping concentrations $\ensuremath{\delta}$. For small doping $(\ensuremath{\delta}=0.2)$, as UUU approaches a value determined by the Stoner criterion, the spin-boson spectrum has a striking low-energy enhancement around $\mathbf{q}\ensuremath{\sim}2{\mathbf{k}}_{F}.$ This enhancement of the spin fluctuation causes a non-Fermi-liquid-like low-energy $\ensuremath{\omega}$ linear dependence in the imaginary part of the self-energy. Due to the self-energy effects, band splittings and a band narrowing are produced in the spectral functions. As a consequence of this, the density of states has pseudogap structures around the quasiparticle band with a narrow bandwidth on the Fermi energy. These features seem to be precursors of the metal-insulator (Mott-Hubbard) transition and they might be related to the spin-gap phenomena observed in high-Tc{T}_{c}Tc​ materials. Our results for the density of states, the spectral function, and the band dispersions show qualitative agreement with the data of finite-size cluster simulations.
The one-band Hubbard model is considered in the slave-boson formulation first introduced by Kotliar and Ruckenstein. It is shown that a mean-field approximation, where broken-symmetry states appropriate for a bipartite lattice are allowed, leads to a quantitative agreement with quantum Monte Carlo results for local observables over a wide range of interactions (0U\ensuremath{\le}20t) and doping (0n〉\ensuremath{\le}1). Thus, our saddle-point solutions constitute an excellent starting point for a systematic treatment of fluctuations.