Fermionic Functional Renormalization Group

Abstract

In this chapter we shall derive the fundamental FRG flow equations for one-band models of nonrelativistic fermions. Although these flow equations follow as a special case of the general flow equations derived in Chap. 7, it is still useful to write down these equations explicitly in order to identify the various terms in the vertex expansion with familiar types of scattering processes (Kopietz and Busche 2001, Salmhofer and Honerkamp 2001). Moreover, for fermions with SU(2) spin rotational symmetry it is useful to take the constraints imposed by this symmetry via a proper parameterization of the vertices into account. We shall derive the corresponding exact FRG flow equations in Sect. 10.2. We then show in Sect. 10.3 how to recover standard single-channel approximations such as the random phase approximation or the ladder approximation from our FRG equations. We proceed in Sect. 10.4 with a discussion of the rescaling problem for normal fermions, which leads to a nonperturbative definition of the Fermi surface. Finally, we discuss in Sect. 10.5 the one-loop patching approximation and present numerical results for the square-lattice Hubbard model. © 2010 Springer-Verlag Berlin Heidelberg.