Determinant Bounds and the Matsubara UV Problem of Many-Fermion Systems

TL;DR

This paper addresses the UV problem in many-fermion systems by establishing new determinant bounds that enable convergence proofs without frequency cutoffs, even for strong short-range interactions.

Contribution

It proves that standard covariances lack Gram representations and introduces a stronger determinant bound applicable to many-fermion systems, facilitating nonperturbative analysis.

Findings

Established nonperturbative bounds on all scales.

Proved convergence of the first integration step for strong interactions.

Provided bounds without imposing Matsubara frequency cutoffs.

Abstract

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 Omega (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 Omega 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.