by D. Vollhardt

Lecture-Notes for the 9th Jerusalem Winter School for Theoretical Physics, Jerusalem 30. Dec. 1991 - 8. Jan. 1992, "Correlated Electron Systems", ed. V. J. Emery (World Scientific, Singapore), ISBN 98102112321.

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Introductory part:

The High-d Limit: General Properties and Weak-Coupling Perturbation Theory

General Remarks

It is a well-known fact that theoretical investigations of quantum-mechanical many-body systems are faced with severe technical problems - particularly in those dimensions which are most interesting to us, i. e. d=2,3. This is due to the complicated dynamics of such systems and the non-trivial algebra needed to describe the quantum-mechanical world. In the absence of exact methods there is clearly a great need for reliable, controlled approximation schemes. The construction of such schemes is not at all straightforward. In fact, for fermionic lattice models even the familiar concept of a "mean field theory" is considerably more delicate than in the case of most classical systems. There exists a well-established branch of approximation techniques which makes use of the simplifications that occur when some parameter (e.g. the length of the spins S, the spin degeneracy N, etc.) is taken to be large (in fact, infinite). Investigations in this limit, supplemented by an expansion in the inverse of the large parameter, may even provide valuable insight into the fundamental properties of a system where this parameter is not large.

In this set of lectures I will discuss a new approach to fermionic systems, based on the limit of high spatial dimensions d. The limit d→∞, which is well-known from classical physics (e. g. the Ising model), is not as academic as it might seem. In fact, we will find that in many respects the dimension d=3 of our real world is already high. In particular, we will discover that a large number of standard approximations, which are commonly used to explain experimental results, are only correct in d=∞. In this respect the limit d→∞ is useful even for physicists who are mainly interested in the application of theories.

Motivation for the large-dimension limit

In a perfectly crystalline system every lattice site has the same number of nearest neighbors (coordination number Z). In three dimensions (d=3) one has Z=6 for a simple cubic lattice (Z=2d for a hypercubic lattice in general dimensions d), Z=8 for a bcc lattice and Z=12 for an fcc-lattice. The dimensionality of a lattice system is directly described by the number Z, rather than by the somewhat more abstract "number of dimensions d". Since Z~O(10) is already quite large in d=3, such that 1/Z is rather small, it is only natural and in the general spirit of theoretical physics to consider the extreme limit Z→∞ first, and then use 1/Z as a small expansion parameter to reach finite Z.