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Formulation and general results

The formulation of the Composite Operator Method(COM) and some general results are given in Refs. [1,2,3,4,5,6,7,8,9].

The COM formulation is based on the following points:

  1. The old schemes, based on the perturbation expansion, are absolutely inadequate to describe the new physics coming out from the recent experiments. The appropriate description of a strongly correlated system must be obtained in terms of some new particles coming out by the interactions and whose properties should be fixed by a self-consistent procedure [1,4,5].

  2. The microscopic level, where we are concerned with the operators, and the macroscopic level, where we are concerned with the averages, are intimately connected [2,4,5,7].

The first point above is well known and has been crystallized in many formulations (the projection method is one example [10,11,12,13]; the dynamical map and the rearrangement of symmetry, formulated by H. Umezawa and collaborators, is another one [14]). The latter point, instead, is genuine and really new. It appears as an obvious statement, but we think that no much attention has been paid to it in the literature. For instance, when we use the Green's function formalism, we must specify the Hilbert space we are considering. The field equations give relations among the matrix elements, but they are not sufficient to uniquely determine the Green's function. The relations among the operators, coming from the algebra and the boundary conditions, must be implemented to correctly choose the right Hilbert space. This can be done by determining the parameters not bounded by the dynamics in such a way that the Hilbert space has the right properties to conserve the relations among matrix elements imposed by symmetry conditions. In particular, the chosen representation should not violate the symmetry required by the Pauli principle. By Pauli principle we mean all the relations among operators dictated by the algebra.

The issue of ergodicity [15,16,17] is also relevant. In general, in the analysis of the two-point Green's function the so-called ergodicity constants naturally appear. These constant are not present in the spectral functions and cannot be determined by the dynamics. While in the existing literature it is common to assume the ergodicity of the dynamics of all quantities, we have shown [7] that this is not always the case. In a fully self-consistent scheme the ergodicity constants must be calculated case by case. The non-ergodicity of the bosonic dynamics, in the bulk systems too, cannot be ruled out a priori. Again, the proper representation will fix the value of the ergodicity constants.

We have explicitly shown [7] that this formulation gives the exact answer when we consider solvable models like small clusters. In this latter case, the ergodicity issue appears really relevant as the non-ergodicity of the bosonic dynamics is a direct consequence of the discreteness of the energy levels.

The self-consistent scheme of the COM has been solved by using two approximations: the polar approximation [2,4] and the two-site approximation [3,18]. The relation with other approaches has been discussed in Refs. [4,6]. A generalization of the polar approximation has been presented in Ref. [8].

We have also formulated an extension of the COM which is able to resolve coherent low-energy features embedded in a broad high-energy background [9]. This approach, namely the Asymptotic Field Approach(AFA), in a problem with more than one energy scale, which is typical of strongly correlated systems, succeed to capture low-energy features. It extends and improves upon the Roth's method by combining the advantages of the methods based on the equations of motion and the slave boson techniques. It is worth noting that when there is an expansion parameter such as the size of the group, or the size of the representation, this approach can be formulated so as to reduce to the correct solution in the exactly solvable limit.

next up previous
Next: The 1D Hubbard model Up: contents Previous: contents
avella 2002-06-10