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arXiv:1305.5135 · Reviews in Mathematical Physics

Translation-invariant quasi-free states for fermionic systems and the BCS approximation

Gerhard Bräunlich, Christian Hainzl, Robert Seiringer

Published Jul 6, 2014• 17 citations
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Abstract

We study translation-invariant quasi-free states for a system of fermions with two-particle interactions. The associated energy functional is similar to the BCS functional but also includes direct and exchange energies. We show that for suitable short-range interactions, these latter terms only lead to a renormalization of the chemical potential, with the usual properties of the BCS functional left unchanged. Our analysis thus represents a rigorous justification of part of the BCS approximation. We give bounds on the critical temperature below which the system displays superfluidity.

DOI
10.1142/s0129055x14500123

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We study translation-invariant quasi-free states for a system of fermions with two-particle interactions. The associated energy functional is similar to the BCS functional but also includes direct and exchange energies. We show that for suitable short-range interactions, these latter terms only lead to a renormalization of the chemical potential, with the usual properties of the BCS functional left unchanged. Our analysis thus represents a rigorous justification of part of the BCS approximation. We give bounds on the critical temperature below which the system displays superfluidity.
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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 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 consider a many-body system of fermionic atoms interacting via a local pair potential and subject to an external potential within the framework of Bardeen-Cooper-Schrieffer (BCS) theory. We measure the free energy of the whole sample with respect to the free energy of a reference state which allows us to define a BCS functional with boundary conditions at infinity. Our main result is a lower bound for this energy functional in terms of expressions that typically appear in Ginzburg-Landau functionals.
We study a translation-invariant BCS functional of superconductivity for a system of spin-1/21/21/2 fermions with spin-dependent chemical potentials, μ+≠μ−μ_+\neqμ_-μ+​=μ−​. Numerical work of Caldas and Mota and analytical work of Freiji, Hainzl, and Seiringer has suggested an unusual first-order phase transition in spin-imbalanced BCS models, in which the normal state becomes globally unstable while remaining locally stable. Here, we prove that a first-order phase transition occurs in such spin-imbalanced BCS functionals. For each sufficiently small fixed spin imbalance and sufficiently low temperature, there is a coexistence coupling λfoλ_{\mathrm{fo}}λfo​ strictly below the linear-stability threshold λlinλ_{\mathrm{lin}}λlin​ of the normal state at which the Cooper pair density exhibits a nonzero jump and the minimized energy has a kink. The methodological insight is a coercivity estimate obtained by decomposing the BCS energy minimization into two parts, where the ``inner'' problem minimizes over the one-particle density matrices at fixed pairing and can be exactly solved by a Legendre--Fenchel transform.