565 research outputs found
Singularity Cancellation in Fermion Loops through Ward Identities
Recently Neumayr and Metzner have shown that the connected N-point density-
correlation functions of the two-dimensional and the one-dimensional Fermi gas
at one-loop order generically vanish/are regular in the small momentum/small
energy-momentum limits. Their result is based on an explicit analysis in the
sequel of results of Feldman et al.[2]. In this note we use Ward identities to
give a proof of the same fact - in a considerably shortened and simplified way
- for any dimension of space.Comment: 11 pages, 2nd corrected and improved version, to appear in Ann. Henri
Poincar
Constructive field theory without tears
We propose to treat the Euclidean theory constructively in a simpler
way. Our method, based on a new kind of "loop vertex expansion", no longer
requires the painful intermediate tool of cluster and Mayer expansions.Comment: 22 pages, 10 figure
Renormalization of the 2-point function of the Hubbard model at half-filling
We prove that the two dimensional Hubbard model at finite temperature T and
half-filling is analytic in the coupling constant in a radius at least . We also study the self-energy through a new two-particle irreducible
expansion and prove that this model is not a Fermi liquid, but a Luttinger
liquid with logarithmic corrections. The techniques used are borrowed from
constructive field theory so the result is mathematically rigorous and
completely non-perturbative.
Together with earlier results on the existence of two dimensional Fermi
liquids, this new result proves that the nature of interacting Fermi systems in
two dimensions depends on the shape of the Fermi surface.Comment: 45 pages, 28 figure
The Hubbard model at half-filling, part III: the lower bound on the self-energy
We complete the proof that the two-dimensional Hubbard model at half-filling
is not a Fermi liquid in the mathematically precise sense of Salmhofer, by
establishing a lower bound on a second derivative in momentum of the first
non-trivial self-energy graph.Comment: 31 pages, 4 figure
Ward type identities for the 2d Anderson model at weak disorder
Using the particular momentum conservation laws in dimension d=2, we can
rewrite the Anderson model in terms of low momentum long range fields, at the
price of introducing electron loops. The corresponding loops satisfy a Ward
type identity, hence are much smaller than expected. This fact should be useful
for a study of the weak-coupling model in the middle of the spectrum of the
free Hamiltonian.Comment: LaTeX 2e document using AMS symbols, 25 pages and 32 eps figure
Scaling behaviour of three-dimensional group field theory
Group field theory is a generalization of matrix models, with triangulated
pseudomanifolds as Feynman diagrams and state sum invariants as Feynman
amplitudes. In this paper, we consider Boulatov's three-dimensional model and
its Freidel-Louapre positive regularization (hereafter the BFL model) with a
?ultraviolet' cutoff, and study rigorously their scaling behavior in the large
cutoff limit. We prove an optimal bound on large order Feynman amplitudes,
which shows that the BFL model is perturbatively more divergent than the
former. We then upgrade this result to the constructive level, using, in a
self-contained way, the modern tools of constructive field theory: we construct
the Borel sum of the BFL perturbative series via a convergent ?cactus'
expansion, and establish the ?ultraviolet' scaling of its Borel radius. Our
method shows how the ?sum over trian- gulations' in quantum gravity can be
tamed rigorously, and paves the way for the renormalization program in group
field theory
The Jacobian conjecture
The Jacobian conjecture involves the map where are
n-dimensional vectors, is a symmetric polynomial of degree for which
the Jacobian hypothesis holds: . The
conjecture states that the inverse map ( as a function of ) is also
polynomial. The proof is inspired by perturbative field theory. We express the
inverse map as a perturbative expansion which is a sum of
partially ordered connected trees.
We use the property : to
extract inductively in the index all the sub traces in the expansion of the
inverse map.
We obtain
By the Jacobian hypothesis and a
straightforward graphical argument gives that $degree \ in\ y \ of \ F(|\le
n)\le d^{2^{n} -2}
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