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

Solid and Condensed State Physics


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 Φ(2). Susceptibilities are calculated from the Bethe-Salpeter equation that relates them to Φ(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.


Phase Transition Renormalization Group Normal Phase Thermodynamic Potential Group Approach 
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Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2005

Authors and Affiliations

  1. 1.Department of Mathematics, Imperial College, 180 Queen's Gate, London SW7 2AZUK and Laboratoire de Physique des Solides, CNRS UMR 8502, Université Paris-SudOrsayFrance

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