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arXiv:math-ph/0209045 · Communications in Mathematical Physics

Convergence of Perturbation Expansions in Fermionic Models. Part 1: Nonperturbative Bounds

Joel Feldman, Horst Kn�rrer, Eugene Trubowitz B

Published Apr 7, 2004• 17 citations
arXivHTMLPDF
DOI
10.1007/s00220-004-1039-1

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We consider a dilute Fermion system in continuum two spatial dimensions with short-range interaction. We prove nonperturbatively that at low temperature the renormalized perturbation expansion has non-zero radius of convergence. The convergence radius shrinks when the energy scale goes to the infrared cutoff. The shrinking rate of the convergence radius is established to be dependent of the sign of the coupling constant by a detailed analysis of the so-called ladder contributions. We prove further that the self-energy of the model is uniformly of C1, in the analytic domain of the theory. The proofs are based on renormalization of the Fermi surface and multiscale analysis employing mathematical renormalization group technique. Tree expansion is introduced to reorganize perturbation expansion nicely. Finally we apply these techniques to construct a half-filled Hubbard model on honeycomb bilayer lattice with local interaction.
It is known that perturbation theory converges in fermionic field theory at weak coupling if the interaction and the covariance are summable and if certain determinants arising in the expansion can be bounded efficiently, e.g. if the covariance admits a Gram representation with a finite Gram constant. The covariances of the standard many–fermion systems do not fall into this class due to the slow decay of the covariance at large Matsubara frequency, giving rise to a UV problem in the integration over degrees of freedom with Matsubara frequencies larger than some Ω (usually the first step in a multiscale analysis). We show that these covariances do not have Gram representations on any separable Hilbert space. We then prove a general bound for determinants associated to chronological products which is stronger than the usual Gram bound and which applies to the many–fermion case. This allows us to prove convergence of the first integration step in a rather easy way, for a short–range interaction which can be arbitrarily strong, provided Ω is chosen large enough. Moreover, we give – for the first time – nonperturbative bounds on all scales for the case of scale decompositions of the propagator which do not impose cutoffs on the Matsubara frequency.