In Sect. 4.7, we have studied in some detail the correlation effects for the homogeneous free electron system and figured out their dependence on the electron density. Correlation has been considered in the effective single-particle potential for crystal electrons within the density-functional theory (DFT), which has led to the independent particle description of the electronic band structure in Chap. 5. In Chap. 6, we have stressed the importance of the exchange interaction and of the electron spin for magnetic properties. The aim of this chapter is to introduce concepts of treating correlation effects for electrons in a crystalline surrounding beyond the independent particle picture in a more general sense. One aspect will be to describe model systems, which allow to demonstrate correlation effects [122, 193–196]. The motivation comes from the observation that some group of solids, namely those with the Fermi energy within narrow bands deriving from d or f electrons, exhibit properties which cannot be understood within the single-particle band structure. Among those are besides the magnetic properties (Chap. 6), the insulating behavior of some transition metal oxides, and the heavy fermion effects in compounds of lanthanides and actinides. Another aspect is the quasi-particle concept in the context of the Fermi liquid theory and the deviations from Fermi liquid behavior in systems with reduced dimension [197–201]. Finally, we would like to address also correlation in two-dimensional electron systems in high magnetic fields, known as the fractional quantum Hall regime [202–204].
KeywordsGreen Function Hubbard Model Landau Level Fermi Liquid Luttinger Liquid
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