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arXiv:1311.4157 · Progress of Theoretical and Experimental Physics

Fermionic functional renormalization group approach to Bose-Einstein condensation of dimers

Yuya Tanizaki

Published Feb 10, 2014• 3 citations
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Abstract

The fermionic functional renormalization group (f-FRG) is applied to describe Bose–Einstein condensation (BEC) of dimers for a two-component fermionic system with attractive contact interaction. In order to describe the system of dimers without introducing auxiliary bosonic fields (bosonization), we propose a new exact evolution equation of the effective action in f-FRG with an infrared regulator for the fermion vertex. Then we analyze its basic properties in detail. We show explicitly that the critical temperature of the free Bose gas is obtained naturally by this method without bosonization. Methods to make systematic improvement from the deep BEC limit are briefly discussed.

DOI
10.1093/ptep/ptu009

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The fermionic functional renormalization group (f-FRG) is applied to describe Bose–Einstein condensation (BEC) of dimers for a two-component fermionic system with attractive contact interaction. In order to describe the system of dimers without introducing auxiliary bosonic fields (bosonization), we propose a new exact evolution equation of the effective action in f-FRG with an infrared regulator for the fermion vertex. Then we analyze its basic properties in detail. We show explicitly that the critical temperature of the free Bose gas is obtained naturally by this method without bosonization. Methods to make systematic improvement from the deep BEC limit are briefly discussed.
Fermionic functional renormalization group is applied to describe Bose-Einstein condensation of composite dimers in a two-component fermionic system with a short-range interaction. Since fermions are bounded in dimers, they do not affect long-range nature of the system. This requires us to invent a renormalization group procedure, which controls the dispersion relation of composite bosons without affecting that of fermionic excitations. In order to realize it without introducing auxiliary bosonic fields, we introduce a vertex infrared regulator so as to compute the critical temperature.
New formulation of fermionic functional renormalization group (f-FRG) with multiple regulators is proposed. It is applied to a two-component fermionic system with an attractive contact interaction in order to study the whole region of the Bardeen-Cooper-Schrieffer to Bose-Einstein condensation crossover. Combining a conventional formalism of FRG with a two-point infrared (IR) regulator and a new formalism with an IR regulator inside the four-fermion vertex, we control both one-particle fermion excitations and collective bosonic excitations. This justifies a simple approximation on the f-FRG method, so that the connection of the f-FRG formalism to the Nozi\`eres-Schmitt-Rink (NSR) theory is made clear. Aspects of f-FRG to go beyond the NSR theory are also discussed.
We propose a method of multi-regulator functional renormalization group (MR-FRG) which is a novel formulation of functional renormalization group with multiple infrared (IR) regulators. It is applied to a two-component fermionic system with an attractive contact interaction to study crossover phenomena between the Bardeen–Cooper–Schrieffer (BCS) phase and the Bose–Einstein condensation (BEC) phase. To control both the fermionic one-particle excitations and the bosonic collective excitations, IR regulators are introduced, one for the fermionic two-point function and another for the four-fermion vertex. It is shown that the Nozières–Schmitt-Rink (NSR) theory, which is successful to capture qualitative features of the BCS–BEC crossover, can be derived from MR–FRG. Some aspects of MR-FRG to go beyond the NSR theory are also discussed.
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.
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