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arXiv:cond-mat/9209018 · Journal of Physics A Mathematical and General

Fermi linearization scheme for itinerant electrons with Clifford variables

Arianna Montorsi, Alessandro Pelizzola

Published Nov 21, 1992• 6 citations
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Abstract

The authors propose an alternative interpretation of the Fermi linearization approach to interacting electron systems, based on the requirement that the coefficients of the linearized operators are Clifford-like variables, whose anticommutator equals an unknown constant c. They apply the approximation to the Falicov-Kimball model (1969), explicitly solving the self-consistency equation for the unknown, which turns out to behave as an order parameter. They discuss its relation with a metal-insulator transition and some thermodynamical quantities. In particular they show that the approximation in the T=0 limit reproduces exactly the Gutzwiller results (1965) for the Hubbard model.

DOI
10.1088/0305-4470/25/22/009

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The authors propose an alternative interpretation of the Fermi linearization approach to interacting electron systems, based on the requirement that the coefficients of the linearized operators are Clifford-like variables, whose anticommutator equals an unknown constant c. They apply the approximation to the Falicov-Kimball model (1969), explicitly solving the self-consistency equation for the unknown, which turns out to behave as an order parameter. They discuss its relation with a metal-insulator transition and some thermodynamical quantities. In particular they show that the approximation in the T=0 limit reproduces exactly the Gutzwiller results (1965) for the Hubbard model.
We propose an alternative interpretation of the fermi linarization approach to interacting electron systems, based on the requirement that the coefficients of the linearized operators are Clifford-like variables, whose anticommutator equals an unknown constant c. We apply the approximation to the Falicov-Kimball model, explicitly solving the self-consistency equation for the unknown, which turns out to behave as an order parameter. We discuss its relation with a metal-insulator transition and some thermodynamical quantities. In particular we show that our approximation in the T = 0 limit reproduces exactly the Gutzwiller results for the Hubbard model
The spin-12\frac{1}{2}21​ Falicov-Kimball model is solved exactly on an infinite-coordination-number Bethe lattice in the thermodynamic limit. This model is a paradigm for a charge-transfer metal-insulator transition, where the occupancy of localized and delocalized electronic orbitals rapidly changes at the metal-insulator transition (rather than the character of the electronic states changing from insulating to metallic as in a Mott-Hubbard transition). The exact solution displays both continuous and discontinuous (first-order) transitions.
After reviewing the fermi-linearization approach to interacting electron systems, we propose here a refinement of the method based on the extension to itinerant ferminon systems of the philosophy of the Bethe-Peierls treatment of classical systems. We focus our attention on a Hubbard-like model, deriving a general self-consistent scheme which we solve explicitly in the simplest case. We obtain a rich phase-space, exhibiting in particular two phase transitions depending on the band filling. Among the results, we discuss in detail the local magnetic moment, by comparing the outcome of our scheme to some known numerical results in one and two dimensions respectively.
We study the metal-insulator transition of the d-dimensional Hubbard model by treating the hopping term between adjacent 1-d chains in the frame of a Clifford linearization scheme, and keeping the full model along the chains. A general equation for the critical point is worked out in terms of the correlation functions of the one-dimensional model, assuming that the transition is of the second order. The equation, which holds at any temperature, is here investigated especially at T=0, where the latter condition holds true. The solution shows the existence of an insulating ground state in any d only at half-filling for U strictly positive, as in the exact 1-d case. The transition is found to be related to a parameter reminiscent of entropy.
The diagrammatic formulation of the single-site approximation to the Hubbard model immediately yields the exact solution to the Falikov-Kimball model in infinite dimensions. By an extension of the alloy analogy, we then derive a two-fluid model, dividing self-consistently the electron system among itinerating and relatively localized particles. Our theory coincides with the CPA only the at half-filling and u=∞. We start by studying the mean field approximation to our theory and identify a phase boundary in the particle density-interaction strength (n−u) plane at which the Fermi liquid model breaks down. Then the effect of fluctuations in the Fermi liquid regime is considered and, finally, we construct, at half-filling, a model which interpolates between the Fermi liquid and the Mott insulating phases.
The Falicov-Kimball model was introduced in 1969 as a statistical model for metal-insulator transitions; it includes itinerant and localized electrons that mutually interact with a local Coulomb interaction and is the simplest model of electron correlations. It can be solved exactly with dynamical mean-field theory in the limit of large spatial dimensions, which provides an interesting benchmark for the physics of locally correlated systems. In this review, the authors develop the formalism for solving the Falicov-Kimball model from a path-integral perspective and provide a number of expressions for single- and two-particle properties. Many important theoretical results are examined that show the absence of Fermi-liquid features and provide a detailed description of the static and dynamic correlation functions and of transport properties. The parameter space is rich and one finds a variety of many-body features like metal-insulator transitions, classical valence fluctuating transitions, metamagnetic transitions, charge-density-wave order-disorder transitions, and phase separation. At the same time, a number of experimental systems have been discovered that show anomalies related to Falicov-Kimball physics [including YbInCu4,{\mathrm{YbInCu}}_{4},YbInCu4​, ${\mathrm{EuNi}}_{2}({\mathrm{Si}}_{1\ensuremath{-}x}{\mathrm{Ge}}_{x}{)}_{2},$ NiI2,{\mathrm{NiI}}_{2},NiI2​, and TaxN].{\mathrm{Ta}}_{x}\mathrm{N}].Tax​N].