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arXiv:cond-mat/0501674 · Progress of Theoretical Physics

Eliashberg Equations Derived from the Functional Renormalization Group

Carsten Honerkamp A, Manfred Salmhofer B

Published Jun 1, 2005• 13 citations
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Abstract

We describe how the fermionic functional renormalization group (fRG) flow of a Cooper+forward scattering problem can be continued into the superconducting state. This allows us to reproduce from the fRG flow the fundamental equations of the Eliashberg theory for superconductivity at all temperatures including the symmetry-broken phase. We discuss possible extensions of this approach like the inclusion of vertex corrections.

DOI
10.1143/ptp.113.1145

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Fermionic functional renormalization group (FRG) is applied to describe the superfluid phase transition of the two-component fermionic system with attractive contact interaction. Connection between the fermionic FRG approach and the Bardeen-Cooper- Schrieffer (BCS) theory with its Gorkov and Melik-Barkhudarov (GMB) correction is made clear, and the FRG flow of the fermion self-energy is also studied to go beyond the BCS+GMB theory. The superfluid transition temperature and the associated chemical potential are calculated in the region of the negative scattering length using fermionic FRG.
A fermionic functional renormalization group (FRG) is applied to describe the super-fluid phase transition of the two-component fermionic system with attractive contact interaction. The connection between the fermionic FRG approach and the conventional Bardeen-Cooper-Schrieffer (BCS) theory with Gorkov and Melik-Barkhudarov (GMB) correction are clarified in detail in the weak coupling region by using the renormalization group flow of the fermionic four-point vertex with particle-particle and particle-hole scat-tering contributions. To go beyond the BCS+GMB theory, coupled FRG flow equations of the fermion self-energy and the four-point vertex are studied under an Ansatz con-cerning their frequency/momentum dependence. We found that the fermion self-energy turns out to be substantial even in the weak couping region, and the frequency depen-dence of the four-point vertex is essential to obtain the correct asymptotic-ultraviolet behavior of the flow for the self-energy. The superfluid transition temperature and the associated chemical potential are calculated in the region of negative scattering lengths. Subject Index A40, A63, B32, I22
Fermionic functional renormalization group (FRG) is applied to describe the superfluid phase transition of the two-component fermionic system with attractive contact interaction. Connection between the fermionic FRG approach and the Bardeen-Cooper- Schrieffer (BCS) theory with its Gorkov and Melik-Barkhudarov (GMB) correction is made clear, and the FRG flow of the fermion self-energy is also studied to go beyond the BCS+GMB theory. The superfluid transition temperature and the associated chemical potential are calculated in the region of the negative scattering length using fermionic FRG. © Published under licence by IOP Publishing Ltd.
We present a functional renormalization group (fRG) flow for many-fermion lattice models into phases with broken spin-rotational symmetry. The flow is expressed purely in terms of fermionic vertex functions. The symmetry breaking is seeded by a small initial anomalous self-energy which grows at the transition scale and which prevents a runaway flow at nonzero scales. Focusing on the case of commensurate antiferromagnetism, we discuss how the interaction vertex can be parametrized efficiently. For reduced models with long-range bare interactions, we show the results of standard mean-field theory are reproduced by the fRG and how anisotropies in the spin sector change the flows. We then describe a more efficient decomposition of the interaction vertex that should allow for the treatment of more general models.
We present a general frame to extend functional renormalization group (fRG) based computational schemes by using an exactly solvable interacting reference problem as starting point for the RG flow. The systematic expansion around this solution accounts for a nonperturbative inclusion of correlations. Introducing auxiliary fermionic fields by means of a Hubbard-Stratonovich transformation, we derive the flow equations for the auxiliary fields and determine the relation to the conventional weak-coupling truncation of the hierarchy of flow equations. As a specific example we consider the dynamical mean field theory (DMFT) solution as reference system, and discuss the relation to the recently introduced DMF2RG{\mathrm{DMF}}^{2}\mathrm{RG}DMF2RG and the dual-fermion formalism.