30,147 research outputs found
Singular Fermi Surfaces I. General Power Counting and Higher Dimensional Cases
We prove regularity properties of the self-energy, to all orders in
perturbation theory, for systems with singular Fermi surfaces which contain Van
Hove points where the gradient of the dispersion relation vanishes. In this
paper, we show for spatial dimensions that despite the Van Hove
singularity, the overlapping loop bounds we proved together with E. Trubowitz
for regular non--nested Fermi surfaces [J. Stat. Phys. 84 (1996) 1209] still
hold, provided that the Fermi surface satisfies a no-nesting condition. This
implies that for a fixed interacting Fermi surface, the self-energy is a
continuously differentiable function of frequency and momentum, so that the
quasiparticle weight and the Fermi velocity remain close to their values in the
noninteracting system to all orders in perturbation theory. In a companion
paper, we treat the more singular two-dimensional case.Comment: 48 pages LaTeX with figure
Perturbation Theory around Non-Nested Fermi Surfaces I. Keeping the Fermi Surface Fixed
The perturbation expansion for a general class of many-fermion systems with a
non-nested, non-spherical Fermi surface is renormalized to all orders. In the
limit as the infrared cutoff is removed, the counterterms converge to a finite
limit which is differentiable in the band structure. The map from the
renormalized to the bare band structure is shown to be locally injective. A new
classification of graphs as overlapping or non-overlapping is given, and
improved power counting bounds are derived from it. They imply that the only
subgraphs that can generate factorials in the order of the
renormalized perturbation series are indeed the ladder graphs and thus give a
precise sense to the statement that `ladders are the most divergent diagrams'.
Our results apply directly to the Hubbard model at any filling except for
half-filling. The half-filled Hubbard model is treated in another place.Comment: plain TeX with postscript figures in a uuencoded gz-compressed tar
file. Put it on a separate directory before unpacking, since it contains
about 40 files. If you have problems, requests or comments, send e-mail to
[email protected]
A Rigorous Proof of Fermi Liquid Behavior for Jellium Two-Dimensional Interacting Fermions
Using the method of continuous constructive renormalization group around the
Fermi surface, it is proved that a jellium two-dimensional interacting system
of Fermions at low temperature remains analytic in the coupling constant
for where is some numerical constant
and is the temperature. Furthermore in that range of parameters, the first
and second derivatives of the self-energy remain bounded, a behavior which is
that of Fermi liquids and in particular excludes Luttinger liquid behavior. Our
results prove also that in dimension two any transition temperature must be
non-perturbative in the coupling constant, a result expected on physical
grounds. The proof exploits the specific momentum conservation rules in two
dimensions.Comment: 4 pages, no figure
Constructive Field Theory and Applications: Perspectives and Open Problems
In this paper we review many interesting open problems in mathematical
physics which may be attacked with the help of tools from constructive field
theory. They could give work for future mathematical physicists trained with
the constructive methods well within the 21st century
The Anderson Model as a Matrix Model
In this paper we describe a strategy to study the Anderson model of an
electron in a random potential at weak coupling by a renormalization group
analysis. There is an interesting technical analogy between this problem and
the theory of random matrices. In d=2 the random matrices which appear are
approximately of the free type well known to physicists and mathematicians, and
their asymptotic eigenvalue distribution is therefore simply Wigner's law.
However in d=3 the natural random matrices that appear have non-trivial
constraints of a geometrical origin. It would be interesting to develop a
general theory of these constrained random matrices, which presumably play an
interesting role for many non-integrable problems related to diffusion. We
present a first step in this direction, namely a rigorous bound on the tail of
the eigenvalue distribution of such objects based on large deviation and
graphical estimates. This bound allows to prove regularity and decay properties
of the averaged Green's functions and the density of states for a three
dimensional model with a thin conducting band and an energy close to the border
of the band, for sufficiently small coupling constant.Comment: 23 pages, LateX, ps file available at
http://cpth.polytechnique.fr/cpth/rivass/articles.htm
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