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
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.
PACS: 05.10.Cc – Renormalization group methods / 05.30.Fk – Fermion systems and electron gas / 71.10.-w – Theories and models of many-electron systems
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2005