Convergence of Perturbation Expansions in Fermionic Models. Part 2: Overlapping Loops

Feldman, Joel; Knörrer, Horst; Trubowitz, Eugene

doi:10.1007/s00220-004-1040-8

Cited by 9 publications

(23 citation statements)

References 10 publications

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“…Remark that Fermionic field theory has undergone quietly this tree revolution almost 2 decades ago. After a long period of maturation [9][10][11], it lead to full constructive results such as the rigorous definition of differential renormalization group equations for Fermions [12] and to a flurry of theorems on condensed matter [13][14][15][16][17][18][19][20][21][22][23][24][25]. However, the full power of the idea of basing QFT on trees was still not recognized at that time because Bosonic theories could not be brought into that form.…”

Section: Vol 10 (2009) Tree Quantum Field Theory 869mentioning

confidence: 99%

Tree Quantum Field Theory

Gurău

Magnen

Rivasseau

2009

Ann. Henri Poincaré

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We propose a new formalism for quantum field theory 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 non-perturbatively {\it differential} renormalization group equations. It accommodates any general stable polynomial Lagrangian. It can equally well treat noncommutative models or matrix models such as the Grosse-Wulkenhaar model. Perhaps most importantly it removes the space-time background from its central place in QFT, paving the way for a nonperturbative definition of field theory in noninteger dimension.Comment: 20 pages, 6 figure

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Section: Vol 10 (2009) Tree Quantum Field Theory 869mentioning

confidence: 99%

Tree Quantum Field Theory

Gurău

Magnen

Rivasseau

2009

Ann. Henri Poincaré

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“…This is a major technical difficulty, since the geometry of the Fermi surface and special effects like "overlapping loops" and ladder estimates are naturally formulated in momentum space. We use the cancellation scheme developed in [FMRT,FKTcf,FKTr1,FKTr2]. It is sufficiently close to the graphical picture to allow us to implement overlapping loops in collections of diagrams within which cancellations take place.…”

Section: Integrating Out a Scalementioning

confidence: 99%

“…The combinatorics for treating diagrams of arbitrary size was developed in [FMRT,FKTcf,FKTr1,FKTr2]. We sketch it here.…”

Section: Cancellation Between Diagramsmentioning

confidence: 99%

“…See subsection 4 of §II and the introduction to [FKTr2]. However to ensure the presence of and to detect sufficiently many overlapping loops, we need that the two-legged part of W(0, ψ) vanishes.…”

Section: Formal Renormalization Group Mapsmentioning

confidence: 99%

“…However to ensure the presence of and to detect sufficiently many overlapping loops, we need that the two-legged part of W(0, ψ) vanishes. See the end of subsection 9 of §II and Remark VI.7 of [FKTr2]. Therefore, we wish that the formal interaction triple (W, G, u) input to Ω j be an element of W m,n with…”

Section: Formal Renormalization Group Mapsmentioning

confidence: 99%

See 2 more Smart Citations

A Two Dimensional Fermi Liquid. Part 1: Overview

Feldman

Knörrer

Trubowitz

2004

Communications in Mathematical Physics

Self Cite

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Abstract.In a series of ten papers, of which this is the first, we prove that 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 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.

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Parametric Cutoffs for Interacting Fermi Liquids

Disertori

Magnen

Rivasseau

2012

Ann. Henri Poincaré

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This paper is a sequel to [1]. We introduce a new multiscale decomposition of the Fermi propagator based on its parametric representation. We prove that the corresponding sliced propagator obeys the same direct space bounds than the decomposition used in [1]. Therefore the non perturbative bounds on completely convergent contributions of [1] still hold. In addition the new slicing better preserves momenta, hence should become an important new technical tool for the rigorous analysis of condensed matter systems. In particular it should allow to complete the proof that a three dimensional interacting system of Fermions with spherical Fermi surface is a Fermi liquid in the sense of Salmhofer's criterion.

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Convergence of Perturbation Expansions in Fermionic Models. Part 2: Overlapping Loops

Cited by 9 publications

References 10 publications

Tree Quantum Field Theory

Tree Quantum Field Theory

A Two Dimensional Fermi Liquid. Part 1: Overview

Parametric Cutoffs for Interacting Fermi Liquids

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