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Introduction to the Renormalization Group with Applications to Non-relativistic Quantum Electron Gases

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Quantum Many Body Systems

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2051))

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Abstract

In these lectures we review the rigorous work on many Fermions models which led to the first constructions of interacting Fermi liquids in two dimensions, and to the proof that they obey different scaling regimes depending on the shape of the Fermi surface. We also review progress on the three dimensional case.

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Notes

  1. 1.

    If we order the pair (b(ℓ), e(ℓ)), bcan be considered the “beginning” and ethe “end” of the line, which is equivalent to orient the line. However the ordinary real-variable Wick’s theorem does not require such an orientation, which becomes necessary only for complex or Grassmann Gaussian measures.

  2. 2.

    Correlation functions play this fundamental role in statistical mechanics.

  3. 3.

    However the functional space that supports this measure is not in general a space of smooth functions, but rather of distributions. This was already true for functional integrals such as those of Brownian motion, which are supported by continuous but not differentiable paths. Therefore “functional integrals” in quantum field theory should more appropriately be called “distributional integrals”.

  4. 4.

    Strictly speaking this is true only for semi-regular graphs, i.e. graphs without tadpoles, i.e. without lines which start and end at the same vertex, see [21].

  5. 5.

    Because the graphs with N = 2 are quadratically divergent we must Taylor expand the quasi local fat dots until we get convergent effects. Using parity and rotational symmetry, this generates only a logarithmically divergent ∫( ∇ ϕ). ( ∇ ϕ) term beyond the quadratically divergent ∫ϕ2. Furthermore this term starts only at n = 2 or two loops, because the first tadpole graph at N = 2, n = 1 is exactlylocal.

  6. 6.

    And also over vertex joints of graphs, just as in the universality theorem for the Tutte polynomial.

  7. 7.

    It is enough in fact to compute such weights for one-particle irreducible and one-vertex-irreducible graphs, then multiply them in the appropriate way for the general case.

  8. 8.

    This is usually easily done by taking some kind of “square roots” in momentum space.

  9. 9.

    The question of whether and how to remove that ultraviolet cutoff has been discussed extensively in the literature, but we consider it as unphysical for a non-relativistic model of condensed matter, which is certainly an effective theory at best.

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Acknowledgements

I thank J. Magnen, M. Disertori, M. Smerlak and L. Gouba for contributing various aspects of this work.

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  1. Laboratoire de physique théorique, Université Paris-Sud, 91405, Orsay, France

    Vincent Rivasseau

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Rivasseau, V. (2012). Introduction to the Renormalization Group with Applications to Non-relativistic Quantum Electron Gases. In: Quantum Many Body Systems. Lecture Notes in Mathematics(), vol 2051. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29511-9_1

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