Two interacting particles effect

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 $l_{1}$ and propagate together coherently on a much larger distance $l_{c} \gg l_{1}$. 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 $l_1$ due to exponential decrease of transition matrix elements for a distance between particles $R \gg l_1$. 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 $l_c$ on which they propagate together, if compared to $l_1$.

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 $\kappa$ which determines how many noninteracting levels are mixed by interaction. Generally, $\kappa \sim \Gamma_2 \rho_2$, where $\Gamma_2 \sim U^2/V{l_1}^d$ is interaction induced transition rate and $\rho_2 \sim {l_1}^{2d}/V$ is the density of two-particle states (here $U$ is a strength of on site/nearby site interaction, $V > U$ is intersite hopping and $d$ is a system dimension). The rate $\Gamma_2$ 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 $d=1$ the localization length for pairs is enhanced by factor $l_c/l_1 \sim \kappa > 1$; for $d=2$ the enhancement is exponentially strong $\ln ({l_c/l_1}) \sim \kappa > 1$ and for $d=3$ the pairs are delocalized for $\kappa > 1$ while all one-particle states are localized [80]. In $d=3$ 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: $\rho_2 \sim \epsilon/\Delta$, where $\epsilon$ is the TIP energy counted from the Fermi level and $\Delta$ is one-particle level spacing in a block of size $l_1$. At a first glance this should also reduce the Breit-Wigner width $\Gamma_2 \propto \rho_2$. However, in a localized regime with $d=2,3$ a return probability is enhanced comparing to a ballistic motion of particles that finally does not lead to a strong reduction in $\Gamma_2$ at small $\epsilon$. As a result for $U \sim V$ the enhancement parameter becomes $\kappa > 1$ for $\epsilon > \Delta$ [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).