by D. Vollhardt

Proceedings of the International School of Physics "Enrico Fermi", Course CXXI, eds. R. A. Broglia and J. R. Schrieffer (North Holland, Amsterdam, 1994), p. 31.

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Introductory part:

Strongly correlated lattice eletrons


Condensed matter physics involves the study of interacting many-particle systems. Due to the structure of matter it is therefore to a large extent the physics of many-electron systems. Their theoretical investigations are notoriously difficult - particularly in the dimensions most interesting to us, i. e. d=2 and 3. Hence many fundamental questions are still open. These difficulties are well-known for a long time already, and exactly five years ago, in the introduction of his lectures here in Varenna, Anderson presented an authoritative and fascinating account of the development of the theory of strongly correlated electron systems during 1937-87.

Due to the discovery of high-Tc superconductivity and the unprecedented activity triggered by it, the entire community suddenly realized that, in spite of many years of research, some of the most basic problems in condensed matter physics were not yet understood. In particular, it soon become clear that high-Tc superconductivity had to be the result of an interplay between various different phenomena which were not yet sufficiently understood even by themselves. These phenomena include (i) the generation of magnetic correlations as a consequence of strong interactions between electrons and the influence of mobile vacancies on such magnetic states, (ii) the particular behavior of electrons in the vicinity of a Mott-Hubbard metal-insulator transition, (iii) the effect of (static) disorder on the correlated behavior of electrons, (iv) the peculiarities of two-dimensional Fermi systems, and of course, (v) the preconditions for superconductivity in a strongly correlated system. In this respect the discovery of high_Tc superconductivity had a sobering effect. Suddenly the new motto was "back to the roots", and there was general consensus about the need for controlled approximations, etc.. The ensueing concerted research efforts during the last five years has provided significant new insight into some of these fundamental questions, although we are still far away from a satisfactory situation.

Most of the questions listed above involve the presence of intermediate or strong interactions between the electrons. Consequently, a large part of recent theoretical investigations of correlated electron systems - perhaps the largest part - dealt with strong-coupling approaches. In view of the plethora of available material the title of my lectures is bound to lead to false expectations. It is clearly impossible to present in four lectures a half-way adequate account of all the concepts, ideas, techniques and physical results developed in the course of even the most recent investigations - in particular since these lectures are supposed to have pedagogical value. I am therefore bound to limit myself to a presentation of only a few strong-coupling approaches. By this selection I do not wish to imply that these approaches are necessarily the most important, reliable, potent or promising ones - it is simply a limitation by necessity. The two recent books on correlated electron systems by Fradkin and Fulde provide extensive discussions of several strong-coupling approaches from quite different perspectives, and I refer the reader to these books for some approaches not discussed by me, e. g. effective fieldtheoretical models and gauge field theories, or Liouville projection techniques.

To elucidate the typical problem involved in strong-coupling approaches I will concentrate on the Hubbard model and its generalization, since it is the generic model for correlated lattice fermions. I will try to point out and clarify, using different perspectives and methods, why it is precisely the Hubbard interaction which makes the strong-coupling approach so difficult. For pedagogical reasons I will first discuss the problem of a single vacancy in a quantum-mechanical spin background and will then present a discussion of variational wave functions, especially Gutzwiller-type wave function, because they are explicit, conceptually simple and physically intuitive and provide immediate insight into almost all fundamental questions of the strong-coupling problem, without demanding a solid background of rather complicated techniques. These two topics have been addressed already five years ago by Rice; however, since then considerable new insight has been gained which I wish to present, too. I will then discuss a projection method which employs auxiliary ("slave") particles to enforce the local constraints in the strong-coupling limit. We will find intimate connections between the mean-field results for slave bosons and Gutzwiller-type wave functions. Finally, I will discuss the concept of a mean-field theory for strongly correlated electrons, i.e. a non-perturbative approach, as obtained by an exact solution in the limit of high spatial dimensions (Z→∞). This will lead us to an effective, dynamical single-site problem of considerable complexity. - Although the topics of the four lectures are all different, the emerging physics is closely related. It is my intention to stress these connections, i. e. the common features, as much as possible.