523 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 II. Regularity of the Moving Fermi Surface: RPA contributions
Regularity of the deformation of the Fermi surface under short-range
interactions is established for all contributions to the RPA self-energy (it is
proven in an accompanying paper that the RPA graphs are the least regular
contributions to the self-energy). Roughly speaking, the graphs contributing to
the RPA self-energy are those constructed by contracting two external legs of a
four-legged graph that consists of a string of bubbles. This regularity is a
necessary ingredient in the proof that renormalization does not change the
model. It turns out that the self--energy is more regular when derivatives are
taken tangentially to the Fermi surface than when they are taken normal to the
Fermi surface. The proofs require a very detailed analysis of the singularities
that occur at those momenta p where the Fermi surface S is tangent to S+p.
Models in which S is not symmetric under the reflection p to -p are included.Comment: 87 pages, plain TeX, ps figures. If you have problems with the
figures when TeXing, choose showfigsfalse at the beginning of the TeX file,
and request the figures from [email protected]
Single Scale Analysis of Many Fermion Systems. Part 4: Sector Counting
For a two dimensional, weakly coupled system of fermions at temperature zero,
one principal ingredient used to control the composition of the associated
renormalization group maps is the careful counting of the number of quartets of
sectors that are consistent with conservation of momentum. A similar counting
argument is made to show that particle-particle ladders are irrelevant in the
case of an asymmetric Fermi curve.Comment: 52 pages, 2 figure
Particle-Hole Ladders
A self contained analysis demonstrates that the sum of all particle-hole
ladder contributions for a two dimensional, weakly coupled fermion gas with a
strictly convex Fermi curve at temperature zero is bounded. This is used in our
construction of two dimensional Fermi liquids.Comment: 131 pages, 26 figure
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