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Physical review. B, Condensed matter

Strongly interacting fermions

Khandker Quader, Kevin S. Bedell, G. E. Brown

Published Jul 1, 1987• 46 citations
Publisher

Abstract

It is shown that strongly interacting Fermi systems, such as $^{3}\mathrm{He}$, transition metals, and the heavy-fermion systems can be consistently described within the ``induced''-interaction Fermi-liquid formalism. The general limiting features of the model are considered. In agreement with the Gutzwiller (half-filled case) solution to the Hubbard model, there is a localization of excitations as ${F}_{0}^{s}$ and ${F}_{1}^{s}$\ensuremath{\rightarrow}\ensuremath{\infty}. Some of the large ${m}^{\mathrm{*}}$ originate from a large ``dynamic'' mass, ${m}_{\ensuremath{\lambda}}$ for f electrons, which is then shown to be further increased by many-body effects. In disagreement with the Gutzwiller solution in the half-filled case, but in agreement with the less-than-half-filled case, the compression modulus goes to a finite value as ${F}_{0}^{s}$,${F}_{1}^{s}$\ensuremath{\rightarrow}\ensuremath{\infty}, so that---in a jellium model---the Debye frequency ${\mathrm{FTHETA}}_{D}$ has a finite value. The implication of the spin-dependent parameter ${F}_{0}^{a}$ going to a finite limit are discussed. In particular, it is shown that ferromagnetic ordering cannot occur in the case of a short-ranged potential.

DOI
10.1103/physrevb.36.156

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arXiv:1503.05544•Physical Review B•Jun 8, 2015•18 citations
It is shown that strongly interacting Fermi systems, such as 3He^{3}\mathrm{He}3He, transition metals, and the heavy-fermion systems can be consistently described within the induced''-interaction Fermi-liquid formalism. The general limiting features of the model are considered. In agreement with the Gutzwiller (half-filled case) solution to the Hubbard model, there is a localization of excitations as ${F}_{0}^{s}$ and ${F}_{1}^{s}$\ensuremath{\rightarrow}\ensuremath{\infty}. Some of the large ${m}^{\mathrm{*}}$ originate from a large dynamic'' mass, m\ensuremathλ{m}_{\ensuremath{\lambda}}m\ensuremathλ​ for f electrons, which is then shown to be further increased by many-body effects. In disagreement with the Gutzwiller solution in the half-filled case, but in agreement with the less-than-half-filled case, the compression modulus goes to a finite value as F0s{F}_{0}^{s}F0s​,F1s{F}_{1}^{s}F1s​\ensuremath{\rightarrow}\ensuremath{\infty}, so that---in a jellium model---the Debye frequency FTHETAD{\mathrm{FTHETA}}_{D}FTHETAD​ has a finite value. The implication of the spin-dependent parameter F0a{F}_{0}^{a}F0a​ going to a finite limit are discussed. In particular, it is shown that ferromagnetic ordering cannot occur in the case of a short-ranged potential.
Heavy fermion systems are explained on the basis of the Fermi liquid theory. The specific heat, magnetic susceptibility and electrical resistivity are discussed. Using Anderson's orthogonality theorem, we show that the Fermi liquid is nothing but a local spin singlet state at every site. Heavy fermion systems The Fermi liquid theory is independent of a model Hamiltonian, and can be applied to any system as long as the system remains a Fermi liquid. The theory tells us that even if the electron interaction becomes strong, physical quantities behave as those of the non-interacting Fermi gas. The difference between them with and without interaction is not qualitative but quantitative. As seen at the Mott transition in Hubbard systems, the effective mass of the electron increases near the transition point. As a system realizing such a large effective mass, a Fermi liquid system called the heavy electron system or the heavy fermion system attracts general interest. The heavy fermion systems are composed of the rare earth metals such as Ce and Yb, and actinide atoms such as U. The heavy fermions are nothing but the quasi-particles in the Fermi liquid theory. As a quasi-particle in a strongly correlated electron system, the heavy fermion is an important issue to be studied in the development of the Fermi liquid theory. The heavy fermion realized in f -electron systems is one of the heavy quasi-particles appearing near the Mott transition.
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.
The Hubbard model with strong correlation is investigated in an auxiliary boson-fermion Hamiltonian, where electrons are described by fermionic pair-excitations of auxiliary particles. Auxiliary particles are localized because of a local gauge symmetry, which is required to guarantee the equivalence between the Hubbard model and the auxiliary-particle Hamiltonian. The density of states of electrons has a double-peak structure in nearly-half filled cases. The main band is broad and deep, and it is consistent with the alloy analogy of the Hubbard model. The other is narrow and at the top of the main band, where the chemical potential lies. The narrow peak can be interpreted in terms of the formation of heavy quasiparticles or heavy electrons. The mass enhancement factor of heavy electrons is consistent with the Gutzwiller variational treatment of the Hubbard model.
The microscopic theory of a superfluid Fermi liquid at finite temperature is developed for the case of a pure system with SSS-wave pairing, and applied to the calculation of the static properties. As a function of $\ensuremath{\theta}\ensuremath{\equiv}\frac{T}{{T}_{c}}$ these properties are determined entirely by the Landau parameters F0{F}_{0}F0​, F1{F}_{1}F1​, Z0{Z}_{0}Z0​, etc., characterizing quasiparticle interactions in the normal phase. In particular the spin susceptibility $\ensuremath{\chi}$ and the density of the normal component ${\ensuremath{\rho}}_{n}$ are given by $\frac{\ensuremath{\chi}(\ensuremath{\theta})}{\ensuremath{\chi}(1)}=(1+\frac{1}{4}{Z}_{0})\frac{f(\ensuremath{\theta})}{[1+\frac{1}{4}{Z}_{0}f(\ensuremath{\theta})]},$ $\frac{{\ensuremath{\rho}}_{n}}{\ensuremath{\rho}}=(1+\frac{1}{3}{F}_{1})\frac{f(\ensuremath{\theta})}{[1+\frac{1}{3}{F}_{1}f(\ensuremath{\theta})]},$ where the universal function $f(\ensuremath{\theta})\ensuremath{\equiv}\ensuremath{-}{[\ensuremath{\nu}(0)]}^{\ensuremath{-}1}{\ensuremath{\Sigma}}_{\mathrm{p}}(\frac{\mathrm{dn}}{d{E}_{\mathrm{p}}})$ is the "effective density of states near the Fermi surface" relative to its value $\ensuremath{\nu}(0)$ in the normal phase. Thus the often-quoted expression ${\ensuremath{\rho}}_{n}=\frac{1}{3}{\ensuremath{\Sigma}}_{\mathrm{p}}{\mathrm{p}}^{2}(\frac{\mathrm{dn}}{d{E}_{\mathrm{p}}})$ is valid for an interacting system only in the limit $T\ensuremath{\rightarrow}0$. In the latter part of the paper a simple phenomenological theory of "Fermi-liquid" effects on $\ensuremath{\chi}$ and ${\ensuremath{\rho}}_{n}$ is developed for arbitrary conditions (including the presence of impurities and pairing with $l\ensuremath{\ne}0$); it is found that under most circumstances explicit expressions for $\ensuremath{\chi}$ and ${\ensuremath{\rho}}_{n}$ may be obtained which involve only the Landau parameters and a suitably generalized effective density of states. The theory should apply to the possible superfluid phase of He3{\mathrm{He}}^{3}He3 and to most superconductors. It is suggested that the Knight shift in nontransition-metal superconductors should display some "Fermi-liquid" effects. The weak-field dc penetration depth $\ensuremath{\lambda}(T)$ is shown to be insensitive to such effects both in the Pippard limit and near Tc{T}_{c}Tc​; however, in a London superconductor at lower temperatures the correction to $\ensuremath{\lambda}(T)$ should be observable and yield a direct estimate of F1{F}_{1}F1​.
We propose a novel statistical-mechanical approach to a system of strongly correlated electrons in a partially filled narrow band which reproduces rigorously the situation in the atomic limit. In this approach the doubly occupied configurations are excluded both in the real and in the reciprocal spaces. Employing this principle we calculate the thermodynamic properties of those electrons. The metallic state, corresponding to a partially-filled-band case, exhibits a Fermi-liquid--type behavior at low temperatures and transforms into a spin liquid at high temperatures. This feature of our results is reminiscent of the behavior in both magnetic metals and heavy-fermion systems. Additionally, the approach reproduces correctly the statistical properties of the system of localized moments in the limit of the half-filled band, corresponding to the Mott insulating phase. These results are not obtained if one starts from the Fermi-Dirac distribution for the quasiparticles. The last assertion is reinforced by an explicit derivation of a statistical distribution function for correlated electrons which reduces to the Fermi-Dirac distribution in the limit of weak interactions, as well as to the distribution for the spin liquid derived earlier in the limit of strong correlation between the particles.
We investigate the Landau parameters for the instabilities in spin and charge channels in the nondegenerate extended Hubbard model with intersite Coulomb and exchange interactions. To this aim we use the spin rotationally invariant slave boson approach and we determine the necessary inverse propagator matrix. The analytically derived spin Landau parameter F0a{F}_{0}^{a}F0a​ for the half filled band uncovers the intrinsic instability of the nondegenerate Hubbard model towards ferromagnetism---negative intersite exchange interaction triggers a ferromagnetic instability at half filling before the metal-insulator transition, indicated by the divergence of the magnetic susceptibility at ${F}_{0}^{a}=\ensuremath{-}1$. This result is general and the instability occurs in the strongly correlated metallic regime for any lattice, in three or two dimensions. Next as an illustrative example we present numerical results obtained for the cubic lattice with nearest neighbor exchange JJJ and Coulomb VVV elements and arbitrary electron density. One finds that the range of small doping near half filling is the most unstable one towards spin polarization, but only in the case of ferromagnetic intersite exchange J<0J<0J<0. Charge Landau parameter F0s{F}_{0}^{s}F0s​ is lowered near half filling by increasing UUU when the intersite Coulomb interaction VVV is attractive, but in contrast to F0a{F}_{0}^{a}F0a​ at J<0J<0J<0 it requires an attraction beyond a critical value Vc{V}_{c}Vc​ to generate the divergence of the charge susceptibility at ${F}_{0}^{s}=\ensuremath{-}1$ in the metallic phase. This instability was found for a broad range of electronic filling away from half filling for moderate attraction.