Fermionic Renormalization Group Flows: Technique and Theory

Abstract

We give a self-contained derivation of the differential equations for Wilson’s renormalization group for the one-particle irreducible Green functions in fermionic systems. The application of this equation to the (t, t � )-Hubbard model appears in Ref. 9). Here we focus on theoretical aspects. After deriving the equations, we discuss the restrictions imposed by symmetries on the effective action. We discuss scaling properties due to the geometry of the Fermi surface and give precise criteria to determine when they justify the use of one-loop flows. We also discuss the relationship of this approach to other RG treatments, as well as aspects of the practical treatment of truncated equations, such as the projection to the Fermi surface and the calculation of susceptibilities. Renormalization group (RG)studies of fermionic models are important for understanding models of solid-state theory and high-energy physics. In this paper, we set up a system of equations suitable for studying flows for general fermionic systems, in particular those with a Fermi surface at any density, and discuss general, but nontrivial, properties of their solutions. The system of equations applies both to “normal” and to symmetry-broken situations of fermionic systems with short-range interactions for d ≥ 1. By “short-range” we mean that the two-particle interaction decays so fast that its Fourier transform is a regular function. In Ref. 9)we have, together with Furukawa and Rice, applied the RG equations that we derive here to the two-dimensional (t, t � )-Hubbard model in the regime relevant for high-temperature superconductivity. The purpose of the present paper is to give more background on the theoretical aspects of the equations, in particular on a detailed argument which justifies the one-loop flow in a certain scale range, even if the scale-dependent interaction is no longer small. In the following, we give an overview; the details are filled in in the later sections. 1.1. The RG and the three scale regimes The Wilsonian RG organizes the functional integration corresponding to the grand canonical trace as an iterated integral over degrees of freedom with energies in a certain range, i.e. those that belong to a corresponding length scale. As such, it is simply a rewriting of the generating functional of the correlation functions as a function of an energy scale � . In contrast to the situation in other RG schemes, this approach does not require any assumptions about perfect scaling laws as a