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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.
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.
I Renormalization of the 2point function of the Hubbard model at halffilling
, 2008
"... We prove that the Hubbard model at finite temperature T and halffilling is analytic in its coupling constant λ for λ  ≤ c/log T  2, where c is some numerical constant. We also bound the selfenergy and prove that the Hubbard model at halffilling is not a Fermi liquid (in the mathematically pr ..."
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Cited by 5 (3 self)
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We prove that the Hubbard model at finite temperature T and halffilling is analytic in its coupling constant λ for λ  ≤ c/log T  2, where c is some numerical constant. We also bound the selfenergy and prove that the Hubbard model at halffilling is not a Fermi liquid (in the mathematically precise sense of Salmhofer), modulo a simple lower bound on the first nontrivial selfenergy graph, which will be published in a companion paper.
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
I The Hubbard model at halffilling, part III: the lower
, 2008
"... We complete the proof that the twodimensional Hubbard model at halffilling is not a Fermi liquid in the mathematically precise sense of Salmhofer, by establishing a lower bound on a second derivative in momentum of the first nontrivial selfenergy graph. ..."
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We complete the proof that the twodimensional Hubbard model at halffilling is not a Fermi liquid in the mathematically precise sense of Salmhofer, by establishing a lower bound on a second derivative in momentum of the first nontrivial selfenergy graph.
On the ultraviolet problem for the 2D weakly interacting Fermi gas
, 2008
"... We prove that the effective potential of the twodimensional interacting continuous Fermi gas with infrared cutoff is an analytic function of the coupling strength near the origin. This is the starting point to study the infrared problem of the model without putting any ultraviolet cutoff, as usuall ..."
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Cited by 2 (0 self)
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We prove that the effective potential of the twodimensional interacting continuous Fermi gas with infrared cutoff is an analytic function of the coupling strength near the origin. This is the starting point to study the infrared problem of the model without putting any ultraviolet cutoff, as usually done in the literature. 1
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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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