Numéro |
J. Phys. I France
Volume 3, Numéro 2, February 1993
|
|
---|---|---|
Page(s) | 625 - 678 | |
DOI | https://doi.org/10.1051/jp1:1993153 |
J. Phys. I France 3 (1993) 625-678
Some aspects of singular interactions in condensed Fermi systems
P.C.E. StampPhysics Department, University of British Columbia, 6224 Agricultural Road, Vancouver, B.C., Canada V6T 1Z1
(Received 30 August 1992, accepted in final form 5 November 1992)
Abstract
This article gives a fairly detailed survey of some of the problems raised when the interaction energy
between 2 fermionic quasiparticles (in 2 dimensions) is singular when
. Before dealing with singular interactions, it is shown how a non-singular
leads to a 2-dimensional Fermi liquid theory, which is internally consistent, at least as far as its infrared properties
are concerned. The quasiparticle properties are calculated in detail. The question of whether singular interactions arise
for the dilute Fermi gas, with short-range repulsive interactions, is investigated perturbatively. One finds a weak singularity
in
, when the dimensionality
D = 2, but it does not destabilize the Fermi liquid. A more sophisticated analysis is then given, to all orders in the interaction,
using the Lippman-Schwinger equation as well as a phase shift analysis for a finite box. The conclusion is that any breakdown of Fermi liquid theory must come from non-perturbative effects. An examination
is then made of some of the consequences arising if a singular interaction is introduced - the form proposed by Anderson is
used as an example. A hierarchy of singular terms arise in all quantities - this is shown for the self-energy, and also the
3-point and 4-point scattering functions. These may be summed in a perfectly consistent manner. Most attention is given to
the particle-hole channel, since it appears to lead to results different from those of Anderson. Nevertheless it appears that
it is possible to derive a sensible theory starting from a singular effective Hamiltonian - although Fermi Liquid theory breaks
down, all fermionic quantities may be calculated consistently. Finally, the effect of a magnetic field (which cuts off the
infrared divergences) is investigated, and the de Haas-van Alphen amplitude calculated, for such a singular Fermionic system.
67.40D - 67.50 - 71.25H
© Les Editions de Physique 1993