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Interacting Fermi liquid in two dimensions at finite temperature, Part II: Renormalization
"... Using the method of continuous renormalization group around the Fermi surface, we prove that a twodimensional interacting system of Fermions at low temperature T remains a Fermi liquid (analytic in the coupling constant λ) for λ ≤ c/  log T  where c is some numerical constant. This bound is a ste ..."
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Cited by 17 (9 self)
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Using the method of continuous renormalization group around the Fermi surface, we prove that a twodimensional interacting system of Fermions at low temperature T remains a Fermi liquid (analytic in the coupling constant λ) for λ ≤ c/  log T  where c is some numerical constant. This bound is a step in the program of rigorous (nonperturbative) study of the BCS phase transition for many Fermions systems; it proves in particular that in dimension two the transition temperature (if any) must be nonperturbative in the coupling constant. The proof is organized into two parts: the present paper deals with the convergent contributions, and a companion paper (Part II) deals with the renormalization of dangerous two point subgraphs and achieves the proof. I
The two dimensional Hubbard Model at halffilling: I. Convergent Contributions, Journ. Stat. Phys. Vol 106
 Ann. Henri Poincaré
, 2002
"... We prove analyticity theorems in the coupling constant for the Hubbard model at halffilling. The model in a single renormalization group slice of index i is proved to be analytic in λ for λ  ≤ c/i for some constant c, and the skeleton part of the model at temperature T (the sum of all graphs wit ..."
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Cited by 10 (7 self)
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We prove analyticity theorems in the coupling constant for the Hubbard model at halffilling. The model in a single renormalization group slice of index i is proved to be analytic in λ for λ  ≤ c/i for some constant c, and the skeleton part of the model at temperature T (the sum of all graphs without two point insertions) is proved to be analytic in λ for λ  ≤ c/log T  2. These theorems are necessary steps towards proving that the Hubbard model at halffilling is not a Fermi liquid (in the mathematically precise sense of Salmhofer). I
Interacting Fermi liquid in three dimensions at finite temperature: Part I: Convergent Contributions
, 2008
"... In this paper we complete the first step, namely the uniform bound on completely convergent contributions, towards proving that a three dimensional interacting system of Fermions is a Fermi liquid in the sense of Salmhofer. The analysis relies on a direct space decomposition of the propagator, on a ..."
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Cited by 10 (5 self)
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In this paper we complete the first step, namely the uniform bound on completely convergent contributions, towards proving that a three dimensional interacting system of Fermions is a Fermi liquid in the sense of Salmhofer. The analysis relies on a direct space decomposition of the propagator, on a bosonic multiscale cluster expansion and on the Hadamard inequality, rather than on a Fermionic expansion and an angular analysis in momentum space, as was used in the recent proof by two of us of Salmhofer’s criterion in two dimensions.
A Two Dimensional Fermi Liquid. Part 1: Overview
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2004
"... In a series of ten papers (see the flow chart at the end of §I), of which this is the first, we prove that the temperature zero renormalized perturbation expansions of a class of interacting many–fermion models in two space dimensions have nonzero radius of convergence. The models have “asymmetric ..."
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Cited by 5 (3 self)
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In a series of ten papers (see the flow chart at the end of §I), of which this is the first, we prove that the temperature zero renormalized perturbation expansions of a class of interacting many–fermion models in two space dimensions have nonzero radius of convergence. The models have “asymmetric ” Fermi surfaces and short range interactions. One consequence of the convergence of the perturbation expansions is the existence of a discontinuity in the particle number density at the Fermi surface. Here, we present a self contained formulation of our main results and give an overview of the methods used to prove them.
Single scale analysis of many Fermion systems, Part 1: insulators
, 2003
"... For a twodimensional, weakly coupled system of fermions at temperature zero, one principal ingredient used to control the composition of the associated renormalization group maps is the careful counting of the number of quartets of sectors that are consistent with conservation of momentum. A simila ..."
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Cited by 4 (3 self)
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For a twodimensional, weakly coupled system of fermions at temperature zero, one principal ingredient used to control the composition of the associated renormalization group maps is the careful counting of the number of quartets of sectors that are consistent with conservation of momentum. A similar counting argument is made to show that
Constructive Field Theory and Applications: Perspectives and Open Problems
, 2000
"... In this paper we review many interesting open problems in mathematical physics which may be attacked with the help of tools from constructive field theory. They could give work for future mathematical physicists trained with the constructive methods well within the 21st century. I ..."
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Cited by 1 (1 self)
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In this paper we review many interesting open problems in mathematical physics which may be attacked with the help of tools from constructive field theory. They could give work for future mathematical physicists trained with the constructive methods well within the 21st century. I
August 8, 2003 5:22 WSPC/Trim Size: 10in x 7in for Proceedings feldknoerrer CONSTRUCTION OF A 2D FERMI LIQUID
"... The temperature zero renormalized perturbation expansions of a class of interacting manyfermion models in two space dimensions have nonzero radius of convergence. The models have “asymmetric ” Fermi surfaces and short range interactions. One consequence of the convergence is the existence of a disc ..."
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The temperature zero renormalized perturbation expansions of a class of interacting manyfermion models in two space dimensions have nonzero radius of convergence. The models have “asymmetric ” Fermi surfaces and short range interactions. One consequence of the convergence is the existence of a discontinuity in the particle number density at the Fermi surface. Here we describe the main results and highlight some of the strategy of the construction. The concept of a Fermi liquid was introduced by L. D. Landau in [16, 17, 18] and has become the generally accepted explanation for the unexpected success of the independent electron approximation. An elementary sketch of Landau’s well known physical arguments can be found in [4, pp 345–351]. More thorough and technical discussions are presented in [1, pp
Constructive Renormalization Theory
, 1999
"... These notes are the second part of a common course on Renormalization Theory given with Professor P. da Veiga 1. I emphasize here the rigorous nonperturbative or constructive aspects of the theory. The usual formalism for the renormalization group in field theory or statistical mechanics is reviewe ..."
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These notes are the second part of a common course on Renormalization Theory given with Professor P. da Veiga 1. I emphasize here the rigorous nonperturbative or constructive aspects of the theory. The usual formalism for the renormalization group in field theory or statistical mechanics is reviewed, together with its limits. The constructive formalism is introduced step by step. Taylor forest formulas allow to perform easily the cluster and Mayer expansions which are needed for a single step of the renormalization group in the case of Bosonic theories. The iteration of this single step leads to further difficulties whose solution is briefly sketched. The second part of the course is devoted to Fermionic models. These models are easier to treat on the constructive level, so they are very well suited to beginners in constructive theory. It is shown how the Taylor forest formulas allow to reorganize perturbation theory nicely in order to construct the GrossNeveu2 model without any need for cluster or Mayer expansions. Finally applications of this technique to condensed matter and renormalization group around Fermi surface are briefly reviewed. 1 The Renormalization Group: an overview 1.1 Functional Integration and its problems In this section we restrict ourselves to the bosonic φ4 field theory in d Euclidean space time dimensions. The model, introduced in P. Da Veiga’s lectures, is defined by the (formal) measure dµC(φ)e −S(φ) ∫, S(φ) = λ φ 4 (x)d d x (1.1) where dµC(φ) represents the Gaussian measure for the free field. Gaussian measures are characterized by their covariance, or propagator, which for a massive theory is, in Fourier space: C(p) = (p 2 + m 2) −1, (1.2) and S is the (bare) action. In dimension d = 2,3 the model is superrenormalizable, and its rigorous construction was the first major achievement of constructive theory [GJ]. In