Talk:Renormalization group for non-relativistic fermions

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Renormalization group for non-relativistic fermions

by R. Shankar

Summary.

This article is a short and very nice review of the Renormalization Group (RG) approach to the low temperature behavior of non-relativistic fermions in two or more dimensions. After a brief reminder on the ideas involved in the RG method, the author discusses its application to non-relativistic fermions in two-dimensions, clearly explaining the crucial role played by the Fermi surface and by the constraints on the structure of the relevant or marginal couplings induced by the momentum conservation rules. The tree level approximation to the beta function equations and its corrections at the one-loop level are discussed; the fixed point theory is briefly discussed, as well as extensions to three dimensions and to the investigation of quantum instabilities.

Comments.

I recommend the publication of this paper, after the consideration of the following (minor) comments.

1) In the section "Beyond the tree level" it may be useful to explain what the acronyms ZS and BCS stand for (in particular ZS, which is not as standard as BCS).

2) In the section "Extensions, uses and generalizations", first paragraph, it may be useful to explain the meaning of "marginal relevant" (a term that may not be comprehensible to those not familiar with the subject)

3) In the section "Extensions, uses and generalizations", the second and last paragraphs may be joint into a single paragraph discussing generalizations to higher dimensions.

4) In the section "Extensions, uses and generalizations", third paragraph, last four lines, it may be useful to add a reference to the few works where a rigorous version of the ideas discussed in this paper have been used to prove theorems about: (i) the convergence of the expansions for the correlation functions at temperatures larger than an exponentially small "critical temperature"; (ii) the Fermi (resp. non-Fermi) liquid behavior of a system of non-relativistic 2D fermions with convex (resp. flat and nested) Fermi surface at temperatures larger than an exponentially small one. [M. Disertori and V. Rivasseau, Phys. Rev. Lett. 85, 361 (2000) and Comm. Math. Phys. 215, 251-290 and 291-341 (2000); G. Benfatto, A. Giuliani and V. Mastropietro, Ann. Henri Poincare 7, 809--898 (2006); V. Rivasseau, Jour. Stat. Phys. 106(3): 693-722 (2002) and S. Afchain, J. Magnen and V. Rivasseau, Ann. Henri Poincare 6(3), 399-448 (2005)]

A list of further minor comments and tyops is attached below.

-) Abstract: "namely THE Fermi surface"

-) Section "The RG philosophy", Eq.(3): there is a "y" missing in the argument of F on the l.h.s.

-) Section "The RG philosophy", last few lines: "just two points, the situation" (AND should be eliminated); moreover there is a right parenthesis missing after "1991"

-) Section "Non relativistic fermions at finite density", last few lines: Gallovotti should be changed into Gallavotti (the same comment is valid for the following, see: lines following Eq.(13); Reference n.2)

-) Section "Extensions, uses and generalizations", first paragraph: "being driven BY a third coupling"

-) Section "Extensions, uses and generalizations", third paragraph: "Another IS to consider"; "Once can ask" should be changed into "One can ask"

-) References: in Ref. 2, it could be added the reference to G. Benfatto and G. Gallavotti, Jour. Stat. Phys. 59, 541-664 (1990); in the reference to Salmhofer, "Renorlamization" should be changed into "Renormalization"

Review B

This is a short but very informative introduction to the modern approach to the theory of ground states and low temperature states of fermionic systems based on the renormalization group. The Author chooses as a main example the case of fermions in an otherwise empty continuum: this is certainly a wise choice which prepares the reader to the work done in the same spirit for the case of fermions on a lattice or interacting with underlying lattice vibrations.

I cannot find any objection to the presentation and I recommend this article for publication.