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Despite the advances in the development of numerical methods analyticalapproaches play a key role on the way towards a deeper understanding ofstrongly interacting systems. In this regards, renormalization schemes forHamiltonians represent an important new direction in the field. Among theserenormalization schemes the projector-based renormalization method (PRM)reviewed here might be the approach with the widest range of possibleapplications: As demonstrated in this review, continuous unitarytransformations, perturbation theory, non-perturbative phenomena, andquantum-phase transitions can be understood within the same theoreticalframework. This review starts from the definition of an effective Hamiltonianby means of projection operators that allows the evaluation within perturbationtheory as well as the formulation of a renormalization scheme. The developedapproach is then applied to three different many-particle systems: At first, westudy the electron-phonon problem to discuss several modifications of themethod and to demonstrate how phase transitions can be described within thePRM. Secondly, to show that non-perturbative phenomena are accessible by thePRM, the periodic Anderson model is investigated to describe heavy-fermionbehavior. Finally, we discuss the quantum-phase transition in theone-dimensional Holstein model of spinless fermions where both metallic andinsulating phase are described within the same theoretical framework.

Projector-based renormalization method (PRM) and its application to many-particle systems