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International Journal of Modern Physics B

QUANTUM FLUCTUATIONS ON THE GUTZWILLER APPROXIMATION OF THE HUBBARD MODEL

M. Lavagna

Published Oct 1, 1988• 1 citation
Publisher

Abstract

Many experimental results seem to relate the superconducting oxides to the phenomenon of Mott localization in strongly correlated systems. We will give a brief review of the different theories developed until then to treat the Mott localization in the Hubbard model: variational Ansatz of Gutzwiller or RVB Ansatz, leading to fundamentally different ground states. In this context, we propose a new approach which consists to take account of the quantum fluctuations around the mean-field solution provided by the Gutzwiller approximation. We show how to understand the ground state in terms of a resonance (split by the Mott gap in the almost-localized regime) in complete analogy with the Abrikosov-Suhl resonance of the Kondo problem — heavy fermions. Superconductivity appears as induced by the constrained intersite correlations developed by quantum fluctuations around this resonant state.

DOI
10.1142/s0217979288000421

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Many experimental results seem to relate the superconducting oxides to the phenomenon of Mott localization in strongly correlated systems. We will give a brief review of the different theories developed until then to treat the Mott localization in the Hubbard model: variational Ansatz of Gutzwiller or RVB Ansatz, leading to fundamentally different ground states. In this context, we propose a new approach which consists to take account of the quantum fluctuations around the mean-field solution provided by the Gutzwiller approximation. We show how to understand the ground state in terms of a resonance (split by the Mott gap in the almost-localized regime) in complete analogy with the Abrikosov-Suhl resonance of the Kondo problem — heavy fermions. Superconductivity appears as induced by the constrained intersite correlations developed by quantum fluctuations around this resonant state.
A new approach is developed to treat the Mott localization in strongly-correlated systems. It is based on a constrained perturbation theory of the Hubbard model as a function of the superexchange coupling J. The Hartree term which is proved to correspond to the Gutzwiller approximation constitutes the natural ground state around which the system fluctuates. We interpret it in terms of a resonance in complete analogy with the Abriskov-Suhl resonance of the Kondo problem. It is shown how superconductivity is induced by quantum fluctuations around this resonant state.
The principal purpose of this topical review is to single out some of the universal features of high-temperature (high-Tc) and related strongly-correlated systems, which can be compared with experiment in a quantitative manner. The description starts with the concept of exchange-interaction-mediated (real-space) pairing, combined with strong correlations among narrow band electrons, for which the reference state is that of the Mott–Hubbard insulator. The physical discussion of concrete properties relies on variational approach, starting from generalized renormalized mean-field theory (RMFT) in the form of statistically-consistent Gutzwiller approximation (SGA), and its subsequent generalization, i.e., a systematic Diagrammatic Expansion of the Variational (Gutzwiller-type) Wave Function (DE-GWF). The solution leads to the two energy scales, one involving unconventional quasiparticles close to the Fermi energy, and the other reflecting the fully correlated state, involving electrons deeper below the Fermi surface. Those two regimes are separated by a kink in the dispersion relation, which is observed in photoemission. As a result, one obtains both the doping dependent properties in the correlated state, and effective renormalized quasiparticles. The reviewed ground-state characteristics for high-Tc systems encompass high-Tc superconductivity, nematicity, charge- (and pair-) density-wave effects, as well as non-BCS kinetic energy gain in the paired state. The calculated dynamic properties are: the universal Fermi velocity, the Fermi wave-vector, the effective mass enhancement, the pseudogap; and the d-wave gap magnitude. We discuss that, within the variational approach, the minimal realistic model is represented by the so-called t-J-U Hamiltonian. Inadequacy of the t-J and Hubbard models, particularly of their RMFT versions, is discussed explicitly. For heavy fermion systems, modeled by the Anderson lattice model and with the DE-GWF approach, we discuss the phase diagram, encompassing superconducting, Kondo insulating, ferro- and anti-ferromagnetic states. The superconducting state is then a two-d-wave gap system. If the orbital degeneracy of f-electrons is included, the coexistent ferromagnetic- (spin–triplet) superconducting phases appear and match those observed for UGe2 in a semiquantitative manner. Finally, in the second part, we generalize our approach to the collective spin and charge fluctuations in high-Tc systems, starting from variational approach, defining the saddle-point state, and combining it with 1/N expansion. The present scheme differs essentially from that starting from the saddle-point Hartree–Fock approximation, and incorporating the fluctuations in the random phase approximation (RPA). The spectrum of collective spin and charge excitations is determined for the Hubbard and t-J-U models, and subsequently compared quantitatively with recent experiments. The Appendices provide formal details to make this review self-contained.
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 use an effective Hamiltonian for two-dimensional Hubbard model including an antiferromagnetic spin-spin coupling term to study recently proposed Gossamer superconductivity. We apply a renormalized mean field theory to approximately take into account the strong correlation effect in partially projected Gutzwiller wave functions. At the half-filling, there is a first order phase transition to separate a Mott insulator at large Coulomb repulsion UUU from a Gossamer superconductor at small UUU. At the critical value U=UcU={U}_{c}U=Uc​, the charge carrier density and the superconducting (SC) order parameter change discontinuously from zero in the Mott insulating phase to finite values in the Gossamer SC phase. The first order transition is due to the interplay of the kinetic and spin exchange energies. As the electron density changes away from half-filling, the Gossamer SC state changes smoothly, while the Mott insulator is evolved into a resonating valence bond (RVB) SC state. The Gossamer and RVB SC states have the same pairing symmetry. The SC order parameter changes smoothly from a RVB SC state at U>UcU>{U}_{c}U>Uc​ to a Gossamer SC state at U<UcU<{U}_{c}U<Uc​ at a fixed nonhalf-filled electron density. We argue that the RVB SC state is smoothly connected to the Gossamer SC state, hence to the BCS state.
We develop a new mean field-theory for systems with on-site correlations, like the Hubbard and Anderson models. It consists in a generalization of the saddle-point approximation to the functional integral representation proposed by Kotliar and Ruckenstein to more than one saddle point. It contains also the alloy analog approximation proposed by Hubbard as a particular case. For a half-filled Hubbard model and a model density of states appropriate for three dimensions, we obtain a metal to insulator transition as U increases. The effective Hamiltonian for the insulating phase contains two split bands (rather than only one with an extremely heavy mass) which reproduce correctly the Green’s functions of the atomic limit.
The Hubbard model is studied in the vicinity of the Mott transition. Local and quantal spin fluctuations are completely included through the mapping to the Anderson model; a local Kondo temperature TK{T}_{K}TK​ is defined as their characteristic temperature or energy scale. Antiferromagnetic spin fluctuations are included perturbatively in terms of 1d\frac{1}{d}d1​ and to leading order in kBTKU\frac{{k}_{B}{T}_{K}}{U}UkB​TK​​ with ddd being the spatial dimensionality, kB{k}_{B}kB​ the Boltzmann constant, and UUU the intrasite repulsion. The two different kinds of spin fluctuations are responsible for the formation of heavy quasiparticles. Two intersite exchange interactions are responsible for both the development of the antiferromagnetic spin fluctuations and $d\ensuremath{\gamma}$-wave Cooper pairing between the heavy quasiparticles: the superexchange interaction and an exchange interaction due to the virtual exchange of spin excitations within the heavy quasiparticle band. An experimental specific-heat coefficient of about 14 mJ/K2{\mathrm{K}}^{2}K2 CuO2{\mathrm{O}}_{2}O2​ mol and a Wilson ratio of about 0.4 imply that the interplay between the local and the antiferromagnetic spin fluctuations plays a crucial role in both the normal and the superconducting states of $\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ with critical temperatures of ${T}_{c}\ensuremath{\simeq}90$ K. A pairing interaction deduced from the normal-state properties is strong enough to give critical temperatures as high as ${T}_{c0}\ensuremath{\simeq}200$ K in the absence of any pair breaking; there is experimental evidence that the pair-breaking effect of the antiferromagnetic spin fluctuations reduces ${T}_{c0}\ensuremath{\simeq}200$ K down to ${T}_{c0}\ensuremath{\simeq}90$ K.