The investigation of effects of short range repulsive/attractive
interaction between two particles in a random
potential with localized one-particle states was started in [69].
It gave a striking result
according to which there are states of a new type in which the particles
are located from each other on a distance of one-particle
localization length and propagate
together coherently on a much larger distance
.
Such coherent propagation takes place even in the case
of repulsive interaction.
The physical reason for appearance of such effective pairing
for repulsing particles can be understood in the following way.
In the random potential two repulsing particles which were
originally close to one another cannot diverge on a distance
much larger than
due to exponential decrease
of transition matrix elements for a distance between particles
. In some sense the localization forces the particles to stay
together. In such coupled state the particles can move one with respect
to the other. This destroys quantum interference and localization and
strongly increases the distance
on which they
propagate together, if compared to
.
This research started in [69] was then continued in papers [74-76,79-81,83,90]. The two interacting particles (TIP) effect attracted the interest of other groups who obtained a number of interesting results: Y.Imry (Weizmann); K.Frahm, A.Müller-Groeling, J.-L.Pichard and D.Weinmann (Saclay); F. von Oppen, T.Wetting and J.Müller (Heidelberg); P.Silvestrov (Novosibirsk). The case of two particles with strong long range attraction was first studied by O.Dorokhov (1990) who found that the pairs of strongly attractive particles can propagate on a larger distance.
The results for TIP can be summarized in a following way.
There is an important parameter which determines how many
noninteracting levels are mixed by interaction.
Generally,
, where
is interaction induced transition rate and
is the density of two-particle states
(here
is a strength of on site/nearby site interaction,
is intersite hopping and
is a system dimension).
The rate
has a meaning of Breit-Wigner width which determines
the shape of local density of states and a number of
unperturber states contributing to an eigenstate in the present of
interaction, which can be defined via inverse participation ratio [75].
For
the localization length for pairs is enhanced by factor
; for
the enhancement is
exponentially strong
and
for
the pairs are delocalized for
while
all one-particle states are localized [80]. In
there is a logarithmically
slow growth of pair size that slightly decreases the diffusion
rate of pair propagation. The above picture was
confirmed by extensive numerical simulations for different models
which are discussed in [69,74,76,81] and by results of other groups.
However, due to the fact that particles are moving in the same random
potential further studies are still desirable to understand in a better way
the effects of correlations and approximate selection rules
(see e.g. [80,83] and Refs. therein).
However, the most interesting questions are related to the TIP effect
at a constant density of particles. Following the suggestion of Y.Imry
it was studied in [90] in the Cooper approximation for quasiparticles
above the frozen Fermi sea. Indeed, due to the proximity to the Fermi level
a density of two-particle states is reduced:
,
where
is the TIP energy counted from the Fermi level
and
is one-particle level spacing in a block of size
.
At a first glance this should also reduce the Breit-Wigner width
. However, in a localized regime with
a return probability is enhanced comparing to a ballistic motion of particles
that finally does not lead to a strong reduction in
at small
. As a result for
the enhancement parameter becomes
for
[90]. This indicates that
delocalization of pairs can take place quite close to the Fermi level.
In this situation the effects of interaction between larger number of particles
should be taken into account to model a real situation in which the
Fermi sea is not frozen.
This direction of research becomes especially interesting in a light of recent experiments of Kravchenko et al. who definitely demonstrated the existence of metal-insulator transition for strongly interacting electrons in two-dimensional random potential. It is possible that this transition can be related to the TIP effect.
The results obtained in [75,83,85,86,90] form the basis of the thesis of Ph.Jacquod (Univ. de Neuchâtel, 1997).