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arXiv:cond-mat/9805207 · Physical review. B, Condensed matter

Fermion loops, loop cancellation, and density correlations in two-dimensional Fermi systems

Arne Neumayr, Walter Metzner

Published Dec 15, 1998• 39 citations
arXivHTMLPDF

Abstract

We derive explicit results from an exact expression for fermion loops with an arbitrary number of density vertices in two dimensions at zero temperature, which has been obtained recently by Feldman et al. [in Singularities, edited by G. M. Greuel (Birkhaeuser, Basel, 1998)]. The 3-loop is an elementary function of the three external momenta and frequencies, and the N-loop can be expressed as a linear combination of 3-loops with coefficients that are rational functions of momenta and frequencies. We show that the divergencies of single loops for low energy and small momenta cancel each other when loops with permutated external variables are summed. The symmetrized N-loop, i.e., the connected N-point density-correlation function of the Fermi gas, does not diverge for low energies and small momenta. In the dynamical limit, where momenta scale to zero at fixed finite energy variables, the symmetrized N-loop vanishes as the $(2N\ensuremath{-}2)\mathrm{th}$ power of the scale parameter.

DOI
10.1103/physrevb.58.15449

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We derive explicit results from an exact expression for fermion loops with an arbitrary number of density vertices in two dimensions at zero temperature, which has been obtained recently by Feldman et al. [in Singularities, edited by G. M. Greuel (Birkhaeuser, Basel, 1998)]. The 3-loop is an elementary function of the three external momenta and frequencies, and the N-loop can be expressed as a linear combination of 3-loops with coefficients that are rational functions of momenta and frequencies. We show that the divergencies of single loops for low energy and small momenta cancel each other when loops with permutated external variables are summed. The symmetrized N-loop, i.e., the connected N-point density-correlation function of the Fermi gas, does not diverge for low energies and small momenta. In the dynamical limit, where momenta scale to zero at fixed finite energy variables, the symmetrized N-loop vanishes as the (2N\ensuremath−2)th(2N\ensuremath{-}2)\mathrm{th}(2N\ensuremath−2)th power of the scale parameter.
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We investigate the electronic properties of the boson mode in a three-point fermion loop. In this framwork, the single-particle excitation and the many-body local (in imaginary time and momentum space) field effects are investigated in IR or UV limits with the density fluctuation induced by external potential (or bosonic frequency). While in the limit of vanishing effect of external potential, which is equivalent to UV limit of the fermionic frequency, the conserving approximation can be applied together with the Luttinger-Ward analysis, in which case the anomalous contribution to the fermion self-energy or the expectation value of many-body interaction term, which is g2⟨Δ†Δ⟩g^{2}\langle Δ^†Δ\rangleg2⟨Δ†Δ⟩ (ggg is the irreducible particle-particle vertex and ΔΔΔ is the boson field operator), vanishes, and results in a Hartree-Fock type momentum- and frequency-independent fermion self-energy. The correlator ⟨Δ†Δ⟩\langleΔ^†Δ\rangle⟨Δ†Δ⟩ is positive which can be obtained through the local moment sum rule of dynamical susceptibility.