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arXiv:1304.3286 · Progress of Theoretical and Experimental Physics

Flow equation of functional renormalization group for three-body scattering problems

Yuya Tanizaki

Published Nov 1, 2013• 4 citations
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Abstract

Functional renormalization group (FRG) is applied to the three-body scattering problem in the two-component fermionic system with an attractive contact interaction. We establish an exact flow equation on the basis of FRG and show that our flow equation is consistent with integral equations obtained from the Dyson–Schwinger equation. In particular, the relation of our flow equation and the Skornyakov and Ter-Martirosyan equation for the atom-dimer scattering is made clear.

DOI
10.1093/ptep/ptt092

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Functional renormalization group (FRG) is applied to the three-body scattering problem in the two-component fermionic system with an attractive contact interaction. We establish an exact flow equation on the basis of FRG and show that our flow equation is consistent with integral equations obtained from the Dyson–Schwinger equation. In particular, the relation of our flow equation and the Skornyakov and Ter-Martirosyan equation for the atom-dimer scattering is made clear.
Functional renormalization group (FRG) is applied to the three-body scattering prob-lem in the two-component fermionic system with an attractive contact interaction. We establish an exact flow equation on the basis of FRG and show that our flow equation is consistent with integral equations obtained from the Dyson-Schwinger equation. In particular, the relation of our flow equation and the Skornyakov and Ter-Martirosyan equation for the atom-dimer scattering is made clear. Subject Index A12,B32,D05
We consider fermion-dimer scattering in the presence of a large positive scattering length in the frame of functional renormalization group equations. A flow equation for the momentum dependent fermion-dimer scattering amplitude is derived from first principles in a systematic vertex expansion of the exact flow equation for the effective action. The resummation obtained from the nonperturbative flow is shown to be equivalent to the one performed by the integral equation by Skorniakov and Ter-Martirosian (STM). The flow equation approach allows to integrate out fermions and bosons simultaneously, in line with the fact that the bosons are not fundamental but build up gradually as fluctuation induced bound states of fermions. In particular, the STM result for atom-dimer scattering is obtained by choosing the relative cutoff scales of fermions and bosons such that the fermion fluctuations are integrated out already at the initial stage of the RG evolution.
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.
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.
A fermionic functional renormalization group (FRG) is applied to describe the superfluid 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 scattering 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 concerning their frequency/momentum dependence. We found that the fermion self-energy turns out to be substantial even in the weak coupling region, and the frequency dependence 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.
We derive functional renormalization group (fRG) schemes for Fermi systems which are based on the two-particle irreducible approach to the quantum many-body problem. In a first step, the cutoff is introduced in the non-interacting propagator as it is commonly done in fRG based on one-particle irreducible vertex functions. The most natural truncation of the resulting infinite hierarchy of flow equations is shown to be fully equivalent to self-consistent perturbation theory. An earlier suggested alternative truncation strategy is considered as well. In a second step, the cutoff is introduced in the two-particle interaction. Again two truncation procedures are investigated, one of which was derived before. In the latter, the mean-field solution of the many-body problem is considered as the starting point of the renormalization group flow. We compare the performance and the required numerical resources for solving the coupled flow equations for all the approximate schemes by applying them to the problem of the quantum anharmonic oscillator. In a functional integral representation, this model has a formal similarity to the quantum many-body problem. The perspectives for applying the derived two-particle irreducible fRG approaches to zero- and one-dimensional systems of correlated fermions are discussed.