The periodic Anderson model: Symmetry-based results and some exact solutions

The topic “magnetic impurities in metals” is certainly one of the most studied problems of the solid-state physics in the last years. The interest toward this argument relies on the fact that the interaction between the magnetic moment of the impurities and the conduction electrons of the host metal, is responsible for a large variety of physical phenomena. The simplest model that captures the essential physics of the systems previously mentioned is certainly the periodic Anderson model. This model appeared in the literature for the first time in 1961 in a paper by P.W. Anderson as an attempt to describe in a simplified way the effects of correlations for d-electrons in transition metals. The Hamiltonian of this model cannot be exactly solved in general. Nevertheless, exact results are known in some special cases. The argument of this review is the discussion of some of these exact solutions and the symmetry properties exhibited by the microscopic model Hamiltonian. The review has been organized in such a way that an introductory material is presented to make the main points intelligible to a non-specialist reader even though very recent developments on this topic are also presented. In particular, we will discuss special solutions of the model, holding in any dimension, when one of the interacting couplings of the model vanishes. We want to mention that, in spite of the crudeness of the models so derived, some physical insights can be derived from these simplified versions of the Anderson Hamiltonian. The impossibility of ordering, magnetic or superconducting, will be also discussed. These results hold for any temperature, electron filling and any strength of the parameters of the model, but are confined to low-dimensional cases and are based on the application of the Bogoliubov’s inequality. It is also discussed the T =0 version of the Bogoliubov’s inequality and it is shown that quantum effects disorder the system, at least in one dimension. Recent studies of the Anderson model showing exact solutions holding for specific values of the microscopic parameters and/or for special filling will be also analyzed. These results are based on the application of spin reflection positivity and on symmetry properties exhibited by Anderson Hamiltonian. Some results in the U =∞ limit are also presented; namely, we discuss the conditions under which a ferromagnetic ground state is established in one dimension when the number of electrons exceeds by one the number of sites and then, for decorated lattices, we derive the ground-state energy and we construct the corresponding eigenstate. Finally, a simple theorem on the total momentum of the ground state of the symmetric version of the Hamiltonian is presented.