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Introductory part:

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-T_{c} 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-T_{c} 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_T_{c} 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.

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- 30.12.2016