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Quantum diffusion of the random Schrödinger evolution in the scaling limit II. The recollision diagrams
"... We consider random Schrödinger equations on R d for d ≥ 3 with a homogeneous AndersonPoisson type random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0. The space and time variables scale as x ∼ λ −2−κ/2,t ∼ λ −2−κ with 0 < κ < κ0(d). We prove that, in t ..."
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Cited by 37 (6 self)
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We consider random Schrödinger equations on R d for d ≥ 3 with a homogeneous AndersonPoisson type random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0. The space and time variables scale as x ∼ λ −2−κ/2,t ∼ λ −2−κ with 0 < κ < κ0(d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψt converges weakly to the solution of a heat equation in the space variable x for arbitrary L 2 initial data. The proof is based on analyzing the phase cancellations of multiple scatterings on the random potential by expanding the propagator into a sum of Feynman graphs. In this paper we consider the nonrecollision graphs and prove that the amplitude of the nonladder diagrams is smaller than their “naive size ” by an extra λ c factor per non(anti)ladder vertex for some c> 0. This is the first rigorous result showing that the
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
Regularity of Interacting Nonspherical Fermi Surfaces: The Exact SelfEnergy
 SelfEnergy, Communications on Pure and Applied Mathematics
"... Regularity of the deformation of the Fermi surface under shortrange interactions is established to all orders in perturbation theory. The proofs are based on a new classification of all graphs that are not doubly overlapping. They turn out to be generalized RPA graphs. This provides a simple extens ..."
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Cited by 9 (7 self)
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Regularity of the deformation of the Fermi surface under shortrange interactions is established to all orders in perturbation theory. The proofs are based on a new classification of all graphs that are not doubly overlapping. They turn out to be generalized RPA graphs. This provides a simple extension to all orders of the regularity theorem of the Fermi surface movement proven in [FST2]. Models in which S is not symmetric under the reflection p ! \Gammap are included. 1 feldman@math.ubc.ca, http://www.math.ubc.ca/¸feldman/ 2 manfred@math.ethz.ch, http://www.math.ethz.ch/¸manfred/manfred.html 3 trub@math.ethz.ch 1. Introduction This paper is a continuation of [FST1,FST2], which we refer to as I and II in the following. It completes the proof of the regularity of the counterterm function K that describes the deformation of the Fermi surface under the interaction. In II, we proved regularity of the randomphaseaproximation (RPA) selfenergy by a detailed, and delicate, analysi...
Fermionic functional integrals and the renormalization group
 CRM Monograph Series, 16, Providence RI, AMS
, 2002
"... Abstract The Renormalization Group is the name given to a technique for analyzing the qualitative behaviour of a class of physical systems by iterating a map on the vector space of interactions for the class. In a typical nonrigorous application of this technique one assumes, based on one’s physica ..."
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Cited by 7 (0 self)
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Abstract The Renormalization Group is the name given to a technique for analyzing the qualitative behaviour of a class of physical systems by iterating a map on the vector space of interactions for the class. In a typical nonrigorous application of this technique one assumes, based on one’s physical intuition, that only a certain finite dimensional subspace (usually of dimension three or less) is important. These notes concern a technique for justifying this approximation in a broad class of Fermionic models used in condensed matter and high energy physics. These notes expand upon the Aisenstadt Lectures given by J. F. at the Centre de
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.
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...
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...
Renormalization of the Fermi Surface
 XII TH INTERNATIONAL CONGRESS OF MATHEMATICAL PHYSICS
, 1999
"... We review the role that renormalization plays in generating welldefined perturbation expansions for fermionic manybody models, particularly in the absence of rotation invariance. ..."
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Cited by 3 (3 self)
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We review the role that renormalization plays in generating welldefined perturbation expansions for fermionic manybody models, particularly in the absence of rotation invariance.