Eur. Phys. J. B 48, 319-338 (2005)
https://doi.org/10.1140/epjb/e2005-00409-7

Renormalization group approach to interacting fermion systems in the two-particle-irreducible formalism

Department of Mathematics, Imperial College, 180 Queen's Gate, London SW7 2AZ, UK and Laboratoire de Physique des Solides, CNRS UMR 8502, Université Paris-Sud, 91405 Orsay, France

Corresponding author: ^a n.dupuis@imperial.ac.uk

Received: 21 June 2005
Revised: 10 October 2005
Published online: 23 December 2005

Abstract

We describe a new formulation of the functional renormalization group (RG) for interacting fermions within a Wilsonian momentum-shell approach. We show that the Luttinger-Ward functional is invariant under the RG transformation, and derive the infinite hierarchy of flow equations satisfied by the two-particle-irreducible (2PI) vertices. In the one-loop approximation, this hierarchy reduces to two equations that determine the self-energy and the 2PI two-particle vertex . Susceptibilities are calculated from the Bethe-Salpeter equation that relates them to . While the one-loop approximation breaks down at low energy in one-dimensional systems (for reasons that we discuss), it reproduces the exact results both in the normal and ordered phases in single-channel (i.e. mean-field) theories, as shown on the example of BCS theory. The possibility to continue the RG flow into broken-symmetry phases is an essential feature of the 2PI RG scheme and is due to the fact that the 2PI two-particle vertex, contrary to its 1PI counterpart, is not singular at a phase transition. Moreover, the normal phase RG equations can be directly used to derive the Ginzburg-Landau expansion of the thermodynamic potential near a phase transition. We discuss the implementation of the 2PI RG scheme to interacting fermion systems beyond the examples (one-dimensional systems and BCS superconductors) considered in this paper.