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Renormalization Group and Fermi Liquid Theory for Many-Nucleon Systems

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Renormalization Group and Effective Field Theory Approaches to Many-Body Systems

Part of the book series: Lecture Notes in Physics ((LNP,volume 852))

Abstract

We discuss renormalization group approaches to strongly interacting Fermi systems, in the context of Landau’s theory of Fermi liquids and functional methods, and their application to neutron matter.

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Notes

  1. 1.

    At non-zero temperature, the imaginary part of the self-energy never vanishes and the quasiparticle life-time is finite. However, for \(T \ll \mu \) and \(\omega \approx \mu \), the life-time is large and the quasiparticle concept is useful.

  2. 2.

    This is readily seen by evaluating \(G(z=\omega \pm i\delta ,\mathbf p )\) for a spectral function of the quasiparticle form,

    $$\begin{aligned} \rho (\omega ,\mathbf p ) = \frac{1}{\pi } \frac{\varGamma _{\mathbf p }/2}{(\omega - \varepsilon _{\mathbf p })^2 + \varGamma _{\mathbf p }^2/4} , \end{aligned}$$
    (5.42)

    using the Källén-Lehmann representation for the Green’s function at a complex argument,

    $$\begin{aligned} G(z,\mathbf p ) = \int _{-\infty }^{\infty } d\omega ^{\prime } \, \frac{\rho (\omega ^{\prime },\mathbf p )}{z - \omega ^{\prime }} . \end{aligned}$$
    (5.43)

    One then finds that \(G(\omega \pm i\delta ,\mathbf p ) = (\omega - \varepsilon _{\mathbf p } \pm i \varGamma _{\mathbf p }/2)^{-1}\). This implies that for \(\omega \) in the upper half plane the quasiparticle pole is in the lower half plane and vice versa.

  3. 3.

    The BCS singularity for back-to-back scattering, \(P=0\), is discussed in Sect. 5.3.

  4. 4.

    The 1PI effective action is obtained by constraining the Green’s function in the 2PI effective action to the solution of the Dyson equation, Eq. (5.99).

  5. 5.

    The resulting flow equation, Eq. (5.119), is consistent with Eq. (5.5), if we make the natural identification \(\delta n_{\mathbf p } = -n_{\mathbf p }^0 \, \delta (|\mathbf p | -(k_F-\varLambda )) \, \text{ d} \varLambda \).

  6. 6.

    For vanishing external sources, all vertices with an odd number of external fermion lines vanish.

  7. 7.

    In finite systems, the spin and density response differs. In nuclei with cores, the low-lying response is due to surface vibrations. Consequently, induced interactions may be attractive, because the spin response is weaker [57].

  8. 8.

    For \(k_{\text{ F}}\approx 0. 4 \, \text{ fm}^{-1}\), neutron matter is close to the universal regime, but theoretically simpler due to an appreciable effective range \(k_{\text{ F}}r_\mathrm{e } \approx 1\) [58].

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Acknowledgments

This work was supported in part by NSERC and by the Alliance Program of the Helmholtz Association (HA216/EMMI).

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Authors and Affiliations

  1. Theory Division, GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291, Darmstadt, Germany

    Bengt Friman

  2. TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, V6T 2A3, Canada

    Kai Hebeler & Achim Schwenk

  3. Department of Physics, The Ohio State University, Columbus, OH, 43210, USA

    Kai Hebeler

  4. ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291, Darmstadt, Germany

    Achim Schwenk

  5. Institut für Kernphysik, Technische Universität Darmstadt, 64289, Darmstadt, Germany

    Achim Schwenk

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  1. Bengt Friman
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Correspondence to Bengt Friman .

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Editors and Affiliations

  1. Institut für Kernphysik, und Extreme Matter Institute EMMI, Technische Universität Darmstadt, Schlossgarten 2, Darmstadt, 64289, Germany

    Achim Schwenk

  2. Lab. de Physique Therique, Université Louis Pasteur, rue de l'Université 3, Strasbourg, 67084, France

    Janos Polonyi

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Friman, B., Hebeler, K., Schwenk, A. (2012). Renormalization Group and Fermi Liquid Theory for Many-Nucleon Systems. In: Schwenk, A., Polonyi, J. (eds) Renormalization Group and Effective Field Theory Approaches to Many-Body Systems. Lecture Notes in Physics, vol 852. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27320-9_5

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