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International Journal of Modern Physics B

NEW EXACTLY TRACTABLE LIMITS FOR CORRELATED LATTICE FERMIONS: HIGH DIMENSIONALITY AND INFINITE RANGE HOPPING

D. Vollhardt

Published Dec 1, 1989• 18 citations
Publisher

Abstract

We discuss two different limits where correlated lattice fermions are explicitly tractable: that of infinite range hopping and that of high dimensions. While in the first case results are almost trivial, the latter limit turns out to be particularly significant, since in d=∞ theoretical investigations are enormously simplified, while at the same time correlations remain non-trivial. Contact to finite dimensions can be made via systematic 1/d-expansions. We review the state-of-art of this approach.

DOI
10.1142/s0217979289001408

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We discuss two different limits where correlated lattice fermions are explicitly tractable: that of infinite range hopping and that of high dimensions. While in the first case results are almost trivial, the latter limit turns out to be particularly significant, since in d=∞ theoretical investigations are enormously simplified, while at the same time correlations remain non-trivial. Contact to finite dimensions can be made via systematic 1/d-expansions. We review the state-of-art of this approach.
A brief introduction to the bosonization method for interacting one-dimensional fermion systems is given. Using these results, the long-distance decay of correlation functions in the one-dimensional Hubbard model is determined exactly for arbitrary bandfilling and correlation strength, using the exact solution of Lieb and Wu. For infinite U the results are generalized to the case of nonzero nearest-neighbour interaction. The behaviour of thermodynamic quantities, of the frequency-dependent conductivity, and of the thermopower is also discussed, in particular in the proximity of the metal-insulator transitions occurring for half- and quarter-filling. The one-dimensional Luttinger liquid is shown to be unstable in the presence of interchain hopping. The results for the metal-insulator transition are compared with other scenarios developed in higher dimensions.
A brief introduction to the bosonization method for interacting one-dimensional fermion systems is given. Using these results, the long-distance decay of correlation functions in the one-dimensional Hubbard model is determined exactly for arbitrary bandfilling and correlation strength, using the exact solution of Lieb and Wu. For infinite U the results are generalized to the case of nonzero nearest-neighbour interaction. The behaviour of thermodynamic quantities, of the frequency-dependent conductivity, and of the thermopower is also discussed, in particular in the proximity of the metal-insulator transitions occurring for half- and quarter-filling. The one-dimensional Luttinger liquid is shown to be unstable in the presence of interchain hopping. The results for the metal-insulator transition are compared with other scenarios developed in higher dimensions.
We present a self-consistent treatment of Hubbard-like lattice models at large, but finite, spatial dimensions ddd. This involves a systematic expansion in powers of 1/d1/d1/d about the limit $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\infty}$. The first-order corrections can be obtained by self-consistent solution of coupled one- and two-impurity models. We calculate the leading corrections to properties of the Falicov-Kimball model. The infinite-dimensional limit seems to describe this particular model remarkably well even for d=2d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2d=2.