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43
Convergence of Perturbation Expansions in Fermionic Models. Part 2: Overlapping Loops
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2004
"... We improve on the abstract estimate obtained in Part 1 by assuming that there are constraints imposed by ‘overlapping momentum loops’. These constraints are active in a two dimensional, weakly coupled fermion gas with a strictly convex Fermi curve. The improved estimate is used in another paper to ..."
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Cited by 6 (2 self)
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We improve on the abstract estimate obtained in Part 1 by assuming that there are constraints imposed by ‘overlapping momentum loops’. These constraints are active in a two dimensional, weakly coupled fermion gas with a strictly convex Fermi curve. The improved estimate is used in another paper to control everything but the sum of all ladder contributions to the thermodynamic Green’s functions.
Constructive field theory in zero dimension
, 2009
"... Constructive field theory can be considered as a reorganization of perturbation theory in a convergent way. In this pedagogical note we propose to wander through five different methods to compute the number of connected graphs of the zerodimensional φ 4 field theory, in increasing order of sophisti ..."
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Cited by 6 (3 self)
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Constructive field theory can be considered as a reorganization of perturbation theory in a convergent way. In this pedagogical note we propose to wander through five different methods to compute the number of connected graphs of the zerodimensional φ 4 field theory, in increasing order of sophistication and power.
The Anderson Model as a Matrix Model
, 2008
"... In this paper we describe a strategy to study the Anderson model of an electron in a random potential at weak coupling by a renormalization group analysis. There is an interesting technical analogy between this problem and the theory of random matrices. In d = 2 the random matrices which appear are ..."
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Cited by 5 (3 self)
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In this paper we describe a strategy to study the Anderson model of an electron in a random potential at weak coupling by a renormalization group analysis. There is an interesting technical analogy between this problem and the theory of random matrices. In d = 2 the random matrices which appear are approximately of the free type well known to physicists and mathematicians, and their asymptotic eigenvalue distribution is therefore simply Wigner’s law. However in d = 3 the natural random matrices that appear have nontrivial constraints of a geometrical origin. It would be interesting to develop a general theory of these constrained random matrices, which presumably play an interesting role for many nonintegrable problems related to diffusion. We present a first step in this direction, namely a rigorous bound on the tail of the eigenvalue distribution of such objects based on large deviation and graphical estimates. This bound allows to prove regularity and decay properties
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.
Improved Power Counting and Fermi Surface Renormalization
, 1996
"... : The naive perturbation expansion for manyfermion systems is infrared divergent. One can remove these divergences by introducing counterterms. To do this without changing the model, one has to solve an inversion equation. We call this procedure Fermi surface renormalization (FSR). Whether or not ..."
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Cited by 5 (3 self)
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: The naive perturbation expansion for manyfermion systems is infrared divergent. One can remove these divergences by introducing counterterms. To do this without changing the model, one has to solve an inversion equation. We call this procedure Fermi surface renormalization (FSR). Whether or not FSR is possible depends on the regularity properties of the fermion selfenergy. When the Fermi surface is nonspherical, this regularity problem is rather nontrivial. Using improved power counting at all orders in perturbation theory, we have shown sufficient differentiability to solve the FSR equation for a class of models with a nonnested, nonspherical Fermi surface. I will first motivate the problem and give a definition of FSR, and then describe the combination of geometric and graphical facts that lead to the improved power counting bounds. These bounds also apply to the fourpoint function. They imply that only ladder diagrams can give singular contributions to the fourpoint func...
How are Feynman graphs resummed by the Loop Vertex Expansion, arXiv 1006
"... The purpose of this short letter is to clarify which set of pieces of Feynman graphs are resummed in a Loop Vertex Expansion, and to formulate a conjecture on the φ4 theory in noninteger dimension. LPT20XXxx ..."
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Cited by 4 (4 self)
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The purpose of this short letter is to clarify which set of pieces of Feynman graphs are resummed in a Loop Vertex Expansion, and to formulate a conjecture on the φ4 theory in noninteger dimension. LPT20XXxx
A Class Of Fermi Liquids
 IN TWOSPACE DIMENSIONS &QUOT;, IN &QUOT;CONSTRUCTIVE PHYSICS&QUOT;, LECTURE NOTES IN PHYSICS 446 (SPRINGER
, 1995
"... ..."
Infrared Analysis of the Tridimensional Gross Neveu Model: Pointwise Bounds for the Effective Potential
 Ann. Inst. Henri Poincar'e
, 1997
"... Within the context of renormalization group analysis, we describe how to get a minute control of the effective potential theory for some fermionic systems using the tree expansion technique. We consider the tridimensional GrossNeveu model (with smooth ultraviolet cutoff) and present pointwise boun ..."
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Cited by 3 (3 self)
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Within the context of renormalization group analysis, we describe how to get a minute control of the effective potential theory for some fermionic systems using the tree expansion technique. We consider the tridimensional GrossNeveu model (with smooth ultraviolet cutoff) and present pointwise bounds for all the kpoint kernels of the effective potential after n steps of the renormalization group transformation. We still prove, for these kernels, the analycity in the initial interaction parameters (with an upper bound for the convergence radius independent on the volume), and the polynomial decay (in a well precise sense) as the distance between points becomes large. We use these results to study the kpoint truncated correlation functions of the model in a separated paper [PPO]. Running title: Effective potential for the tridimensional GrossNeveu model 1 Introduction It is well known the importance of renormalization group (RG) method to the progress of mathematical physics in se...
Supersymmetric Analysis of a Simplified Two Dimensional Anderson Model at Small Disorder
, 2002
"... Abstract This work proposes a very simple random matrix model, the Flip Matrix Model, liable to approximate the behavior of a two dimensional electron in a weak random potential. Its construction is based on a phase space analysis, a suitable discretization and a simplification of the true model. Th ..."
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Abstract This work proposes a very simple random matrix model, the Flip Matrix Model, liable to approximate the behavior of a two dimensional electron in a weak random potential. Its construction is based on a phase space analysis, a suitable discretization and a simplification of the true model. The density of states of this model is investigated using the supersymmetric method and shown to be given, in the limit of large size of the matrix by the usual Wigner’s semicircle law. I