A Geometric Approach to Correlated Systems
Over the past few decades there has been an impressive and a steady progress in computational material science [1, 2, 3]. This development is fueled by the ever growing computational resources and by the demand for yet more precise information on technologically relevant material properties, such as the optical, transport and magnetic characteristics. A microscopic description of these properties entails the knowledge of the quantum spectrum of the system under study. Thus for real materials one has to deal with the notoriously difficult many-body problem in a computationally acceptable manner. For this purpose remarkably successful and efficient conceptual schemes have been developed where the multi-particle system is mapped onto a one body problem for a particle moving in an effective (non local) field created by all the other constituents of the system [1, 2, 4]. Usually, this effective field is further simplified according to certain recipes such as the local density approximation within the density functional theory . These computationally manageable concepts have rendered possible the routine and accurate calculation of a wealth of static material properties, such as the ground state energies. On the other hand, however, it has been observed that the ground-state of certain compounds, e.g. transition metal oxides, is not described adequately within a single particle picture . In addition, for the theoretical description of the excitation spectrum  and for the treatment of dynamical processes, such as many particle reactive scattering, methods have to be envisaged that go beyond the effective single particle model.
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