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International Journal of Modern Physics B

FERMIONS ON A LATTICE IN HIGH DIMENSIONS

E. Müller‐Hartmann

Published Dec 1, 1989• 68 citations
Publisher

Abstract

The limit of high dimensions of lattice fermion models has soon met great interest after its recent introduction by Metzner and Vollhardt. Fermion models are greatly simplified in high dimensions. The simplifications achieved are, however, much more subtle and diverse than those known for spin models. In the present contribution I present some of my own work as well as a short review of some other work on this new promising subject.

DOI
10.1142/s0217979289001391

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The limit of high dimensions of lattice fermion models has soon met great interest after its recent introduction by Metzner and Vollhardt. Fermion models are greatly simplified in high dimensions. The simplifications achieved are, however, much more subtle and diverse than those known for spin models. In the present contribution I present some of my own work as well as a short review of some other work on this new promising subject.
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.
We discuss interacting fermion models in two dimensions, and, in particular, such that can be solved exactly by bosonization. One solvable model of this kind was proposed by Mattis as an effective description of fermions on a square lattice. We review recent work on a specific relation between a variant of Mattis' model and such a lattice fermion system, as well as the exact solution of this model. The background for this work includes well-established results for one-dimensional systems and the high-Tc problem. We also mention exactly solvable extensions of Mattis' model.
We discuss interacting fermion models in two dimensions, and, in particular, such that can be solved exactly by bosonization. One solvable model of this kind was proposed by Mattis as an effective description of fermions on a square lattice. We review recent work on a specific relation between a variant of Mattis' model and such a lattice fermion system, as well as the exact solution of this model. The background for this work includes well-established results for one-dimensional systems and the high-Tc problem. We also mention exactly solvable extensions of Mattis' model.
The author explores some of the inherent simplifications of "quantum lattice physics." He distinguishes between fermions and bosons and analyzes the nnn-body problem for each, with n=1,2,3…n=1,2,3\dots{}n=1,2,3… typically a small number. With delta-function (zero-range) interactions, the three-body problem on a lattice is manageable, and some results can even be extrapolated to $n\ensuremath{\ge}4$. Such calculations are not limited to one dimension (where the well-known Bethe ansatz solves a number of nnn-body problems). On the contrary, studies cited are mainly in three dimensions and actually simplify with increasing dimensionality. For example, it is found that bound states of $n\ensuremath{\ge}3$ particles in $d\ensuremath{\ge}3$ dimensions are formed discontinuously as the strength of two-body attractive forces is increased, and are therefore always in the easily analyzed "strong coupling limit." In the Appendix, an exactly solved example from the theory of itinerant-electron magnetism illustrates how a rigorous solution to the few-body problem is capable of yielding information concerning the NNN-body problem.