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

TL;DR

This paper introduces a new functional renormalization group formulation for interacting fermions using the two-particle-irreducible formalism, enabling analysis of phase transitions and broken-symmetry phases.

Contribution

It develops a 2PI RG scheme that captures phase transitions without singularities and reproduces exact results in mean-field theories, extending the analysis of interacting fermion systems.

Findings

The 2PI RG scheme reproduces exact results in BCS theory.

It allows continuation of RG flow into broken-symmetry phases.

Normal phase RG equations can derive Ginzburg-Landau expansion.

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 a fixed point of the RG, 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 $\Phi^{(2)}$. Susceptibilities are calculated from the Bethe-Salpeter equation that relates them to $\Phi^{(2)}$. 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.