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43
Perturbation Theory around NonNested Fermi Surfaces  II. Regularity of the Moving Fermi Surface: RPA Contributions
 JOURNAL OF STATISTICAL PHYSICS
, 1996
"... Regularity of the deformation of the Fermi surface under shortrange interactions is established for all contributions to the RPA selfenergy (it is proven in an accompanying paper that the RPA graphs are the least regular contributions to the selfenergy). Roughly speaking, the graphs contributing ..."
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Cited by 20 (9 self)
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Regularity of the deformation of the Fermi surface under shortrange interactions is established for all contributions to the RPA selfenergy (it is proven in an accompanying paper that the RPA graphs are the least regular contributions to the selfenergy). Roughly speaking, the graphs contributing to the RPA selfenergy are those constructed by contracting two external legs of a fourlegged graph that consists of a string of bubbles. This regularity is a necessary ingredient in the proof that renormalization does not change the model. It turns out that the selfenergy is more regular when derivatives are taken tangentially to the Fermi surface than when they are taken normal to the Fermi surface. The proofs require a very detailed analysis of the singularities that occur at those momenta p where the Fermi surface S is tangent to S + p. Models in which S is not symmetric under the reflection p # p are included.
An Explicit Large Versus Small Field Multiscale
 Cluster Expansion, Rev. Math. Phys. 9
, 1997
"... We introduce a new type of cluster expansion which generalizes a previous formula of Brydges and Kennedy. The method is especially suited for performing a phasespace multiscale expansion in a just renormalizable theory, and allows the writing of explicit nonperturbative formulas for the Schwinger ..."
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Cited by 20 (13 self)
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We introduce a new type of cluster expansion which generalizes a previous formula of Brydges and Kennedy. The method is especially suited for performing a phasespace multiscale expansion in a just renormalizable theory, and allows the writing of explicit nonperturbative formulas for the Schwinger functions. The procedure is quite model independent, but for simplicity we chose the infrared φ 4 4 model as a testing ground. We used also a large field versus small field expansion. The polymer amplitudes, corresponding to graphs without almost local two and for point functions, are shown to satisfy the polymer bound. 1
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
A Representation for Fermionic Correlation Functions
 COMMUN. MATH. PHYS
, 1998
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Tree Quantum Field Theory
 Annales Henri Poincare 10 (2009) 867 [arXiv:0807.4122 [hepth
"... We propose a new formalism for quantum field theory (QFT) which is neither based on functional integrals, nor on Feynman graphs, but on marked trees. This formalism is constructive, i.e. it computes correlation functions through convergent rather than divergent expansions. It applies both to Fermio ..."
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Cited by 10 (3 self)
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We propose a new formalism for quantum field theory (QFT) which is neither based on functional integrals, nor on Feynman graphs, but on marked trees. This formalism is constructive, i.e. it computes correlation functions through convergent rather than divergent expansions. It applies both to Fermionic and Bosonic theories. It is compatible with the renormalization group, and it allows to define nonperturbatively differential renormalization group equations. It accommodates any general stable polynomial Lagrangian. It can equally well treat noncommutative models or matrix models such as the GrosseWulkenhaar model. Perhaps most importantly it removes the spacetime background from its central place in QFT, paving the way for a nonperturbative definition of field theory in noninteger dimension. 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 Single Scale Infinite Volume Expansion for Three Dimensional Many Fermion Green’s Functions
 Math. Phys. Electronic Journal
, 1995
"... Abstract In [FMRT1] we introduced a multiscale expansion for many Fermion systems in two space dimensions based on a socalled sector decomposition. In this paper a completely different expansion is introduced to treat the more difficult case of three (or more) space dimensions: it is based on an au ..."
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Abstract In [FMRT1] we introduced a multiscale expansion for many Fermion systems in two space dimensions based on a socalled sector decomposition. In this paper a completely different expansion is introduced to treat the more difficult case of three (or more) space dimensions: it is based on an auxiliary scale decomposition and the use of the Hadamard inequality. We prove that the perturbative expansion for a single scale model has a convergence radius independent of the scale. This is a typical result, already proved in the two dimensional case in [FMRT1], which we cannot obtain in three dimensions by naive extrapolation of the sector method. Although we do not treat in this paper the full (multiscale) system, we hope this new method to be a significant step towards the rigorous construction of the BCS theory of superconductivity in three space dimensions. I
Low temperature analysis of two dimensional Fermi systems with symmetric Fermi surface
, 2002
"... We prove the convergence of the perturbative expansion, based on Renormalization Group, of the two point Schwinger function of a system of weakly interacting fermions in d = 2, with symmetric Fermi surface and up to exponentially small temperatures, close to the expected onset of superconductivi ..."
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Cited by 8 (1 self)
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We prove the convergence of the perturbative expansion, based on Renormalization Group, of the two point Schwinger function of a system of weakly interacting fermions in d = 2, with symmetric Fermi surface and up to exponentially small temperatures, close to the expected onset of superconductivity.