Many Particle Strongly Interacting Systems.pdf


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Introduction

The basic problems of field theory and statistical mechanics are much similar
in many aspects, especially, when we use the method of second quantization and Green functions[1]. In both the cases, we are dealing with systems
possessing a large number of degrees of freedom (the energy spectrum is
practically a continuous one) and with averages of quantum mechanical operators [2]. In quantum field theory, we mostly consider averages over the
ground state, while in statistical mechanics, we consider finite temperatures
(ensemble averages) as well as ground-state averages. Great advances have
been made during the last decades in statistical physics and condensed matter theory through the use of methods of quantum field theory [3] - [5].
It was widely recognized that a successful approximation for determining
exited states is based on the quasi-particle concept and the Green function method. For example, the study of highly correlated electron systems
has attracted much attention recently [6] - [9], especially after discovery of
copper oxide superconductors, a new class of heavy fermions [7], and lowdimensional compounds [3], [8]. Although much work for strongly correlated
systems has been performed during the last years, it is worthy to remind
that the investigation of excitations in many-body systems has been one of
the most important and interesting subjects for the last few decades.
The quantum field theoretical techniques have been widely applied to statistical treatment of a large number of interacting particles. Many-body
calculations are often done for model many-particle systems by using a perturbation expansion. The basic procedure in many-body theory [10] is to
find a suitable unperturbed Hamiltonian and then to take into account a
small perturbation operator. This procedure that works well for weakly
interacting systems needs a special reformulation for many-body systems
with complex spectra and strong interaction. For many practically interesting cases (e.g. in quantum chemistry problems ), the standard schemes of
perturbation expansion must be reformulated greatly [11] - [15]. Moreover,
many-body systems on a lattice have their own specific features and in some
important aspects differ greatly from continuous systems.
In this review that is largely pedagogical we are primarily dealing with the
spectra of elementary excitations to learn about quasi-particle many-body
dynamics of interacting systems on a lattice. Our analysis is based on the
equation-of-motion approach, the derivation of the exact representation of
the Dyson equation and construction of an approximate scheme of calculations in a self-consistent way. In this review only some topics in the field
are discussed. The emphasis is on the methods rather than on a detailed
comparison with the experimental results. We attempt to prove that the approach we suggest produces a more advanced physical picture of the problem
of the quasi-particle many-body dynamics.
The most characteristic feature of the recent advancement in the basic re3