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

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.

29 pages LaTeX

References in corpus (3)

• Continuous renormalization for fermions and Fermi liquid theory
• Positivity and Convergence in Fermionic Quantum Field Theory
• Dynamical Adjustment of Propagators in Renormalization Group Flows

Cited by in corpus (22)

• Functional renormalization group approach to correlated fermion systems
• Efficient Parametrization of the Vertex Function, $\Omega$-Scheme, and the (t,t')-Hubbard Model at Van Hove Filling
• Frequency Dependent Vertex Functions of the (t,t')-Hubbard Model at Weak Coupling
• Universality of the Hall conductivity in interacting electron systems
• The 2D Hubbard model on the honeycomb lattice
• Persistence of exponential decay and spectral gaps for interacting fermions
• Gentle introduction to rigorous Renormalization Group: a worked fermionic example
• Renormalization in Condensed Matter: Fermionic Systems - from Mathematics to Materials
• Clustering of fermionic truncated expectation values via functional integration
• The algebra of Grassmann canonical anti-commutation relations (GAR) and its applications to fermionic systems
• Exponential decay of correlation functions in many-electron systems
• Functional Integral and Stochastic Representations for Ensembles of Identical Bosons on a Lattice
• Superconducting phase in the BCS model with imaginary magnetic field
• A rigorous treatment of the perturbation theory for many-electron systems
• d-Wave Pairing Driven by Bipolaric Modes Related to Giant Electron-Phonon Anomalies in High-$T_c$ Superconductors
• The ground state construction of bilayer graphene
• Large Deviations in Weakly Interacting Fermions -- Generating Functions as Gaussian Berezin Integrals and Bounds on Large Pfaffians
• Adiabatic Evolution of Low-Temperature Many-Body Systems
• Superconducting phase in the BCS model with imaginary magnetic field. III. Non-vanishing free dispersion relations
• Order, disorder and phase transitions in quantum many body systems
• A mathematical derivation of zero-temperature 2D superconductivity from microscopic Bardeen-Cooper-Schrieffer model
• Smoothing of Singular Legendre Transforms in Renormalization Group Flows