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International Journal of Modern Physics B

CHARGE AND SPIN FLUCTUATIONS IN THE HUBBARD MODEL: FERMI LIQUID PROPERTIES AT LOW TEMPERATURES

M. Lavagna

Published Apr 1, 1991• 12 citations
Publisher

Abstract

We review the slave-boson representations of the Hubbard model. When both spin and charge fermion degrees of freedom are represented by Bose fields, the approach is equivalent at the "self-consistent" saddle point level to the Gutzwiller approximation (GA). We show how the determination of the Lagrange multipliers, introduced to enforce the constraints, involves a Mott-Hubbard gap near half-filling for U > U c (localization edge). The quantum fluctuations are then considered within a renormalized basis of boson fields (restoring the spin-rotation invariance), which brings out two distinct channels, symmetric and antisymmetric. It is remarkable that both spin and charge fluctuations are obtained at the same level of the gaussian fluctuations, as oppossed to the standard 1/N expansion. This provides the microscopic basis for a Fermi liquid theory. Both dynamic effects and extension to finite temperature which were beyond the scope of the variational procedure can now be described by this approach. The physical implications of the model (dynamic correlation functions, T3Ln T term of the specific heat, superfluid instability) are finally considered, all of which reflect the Fermi liquid nature of the system.

DOI
10.1142/s021797929100047x

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We review the slave-boson representations of the Hubbard model. When both spin and charge fermion degrees of freedom are represented by Bose fields, the approach is equivalent at the "self-consistent" saddle point level to the Gutzwiller approximation (GA). We show how the determination of the Lagrange multipliers, introduced to enforce the constraints, involves a Mott-Hubbard gap near half-filling for U > U c (localization edge). The quantum fluctuations are then considered within a renormalized basis of boson fields (restoring the spin-rotation invariance), which brings out two distinct channels, symmetric and antisymmetric. It is remarkable that both spin and charge fluctuations are obtained at the same level of the gaussian fluctuations, as oppossed to the standard 1/N expansion. This provides the microscopic basis for a Fermi liquid theory. Both dynamic effects and extension to finite temperature which were beyond the scope of the variational procedure can now be described by this approach. The physical implications of the model (dynamic correlation functions, T3Ln T term of the specific heat, superfluid instability) are finally considered, all of which reflect the Fermi liquid nature of the system.
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.
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 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.
We present a theoretical approach to describing the Mott transition of electrons on a two-dimensional lattice that begins with the low-energy effective theory of the Fermi liquid. The approach to the Mott transition must be characterized by the suppression of density and current fluctuations that correspond to specific shape deformations of the Fermi surface. We explore the nature of the Mott insulator and the corresponding Mott transition when these shape fluctuations of the Fermi surface are suppressed without making any a priori assumptions about other Fermi surface shape fluctuations. Building on insights from the theory of the Mott transition of bosons, we implement this suppression by identifying and condensing vortex degrees of freedom in the phase of the low-energy electron operator. We show that the resulting Mott insulator is a quantum spin liquid with a spinon Fermi surface coupled to a U(1)U(1)U(1) gauge field, which is usually described within a slave particle formulation. Our approach thus provides a coarse-grained treatment of the Mott transition and the proximate spin liquid that is nevertheless equivalent to the standard slave particle construction. This alternate point of view provides further insight into the novel physics of the Mott transition and the spin liquid state that is potentially useful. We describe a generalization that suppresses spin antisymmetric fluctuations of the Fermi surface that leads to a spin-gapped charge metal previously also discussed in terms of slave particle constructions.
We study the finite-temperature Fermi-liquid to non-Fermi-liquid crossover in the two-dimensional Hubbard model for a range of dopings using the self-consistent ladder dual fermion method. We consider relatively high temperatures where by comparing the single-particle self-energy and the two-particle spin susceptibility we identify a suppression of the density of states near the Fermi level caused by a quasi-antiferromagnetic behavior that is itself characterized by a long, but finite, correlation length scale which we extract. We compute the single-particle self-energy, and the charge/spin susceptibilities for the entire doping range at relatively high temperature, and also for the half-filled weak-coupling case at relatively low temperatures. We perform fluctuation diagnostics to decompose the single-particle self-energy into scattering qqq-vector and bosonic frequency contributions in the spin channel. Within this framework we find that the key contributions to the single-particle self-energy that give non-Fermi-liquid character show complicated dependence on zero, positive, and negative frequency bosonic modes. We find that for a nominal choice of ${t}^{\ensuremath{'}}$ that electron- and hole-doped systems with similar Fermi-liquid character have distinct spin and charge susceptibility and distinct spin-fluctuation diagnostics spectra.
A concise general expression for the dynamical charge susceptibility of the Hubbard model is presented. The results are obtained from the spin-rotation-invariant slave-boson theory, whose paramagnetic saddle point corresponds to Gutzwiller's variational approximation. The charge structure factor and charge susceptibility are calculated for a wide range of interactions and temperatures and are compared thoroughly with the available quantum Monte Carlo and variational Monte Carlo results for the one-dimensional Hubbard model. The comparison shows reasonable agreement. All the calculations are done in a totally self-consistent way, without adjustable parameters.