Anderson localization in disordered systems with competing channels

Hongyi Dr. Xie (1)

1 Condensed Matter and Statistical Physics

The work in this thesis is motivated by the problem of localization of interacting particles. The qualitative investigation of Thouless-type arguments in Chap.~\ref{ch:interacting-particles} lead us to consider the question of competition between alternative propagation channels, a question which we studied in great detail in the form of a single particle problem with two parallel, coupled channels. The theory also naturally applies for the Anderson localization of hybrid particles such as polaritons. These systems have a common feature: Two or more propagating channels with parametrically different transport properties are coupled and compete with each other. The principal question is: What happens to the localization properties when a less localized lattice is coupled to a more localized one? Will the less localized lattice dominate the localization of the system or the more localized? The qualitative answer to this question depends on the dimensionality of the system. Correspondingly, we exactly solved the Anderson models on a two-leg ladder ($D=1$) and on a two-layer Bethe lattice (formally $D=\infty$). In one dimension, weak disorder has a strong localization effect. In the \emph{weak disorder limit} we have found that under \emph{resonance} conditions the localization lengths of two coupled chains are of the order of the localization length of the more localized, uncoupled leg. We may interpret this phenomenon as a manifestation of the fact that in one dimension the mean free path is the relevant length scale that sets the localization length. It is not surprising that the backscattering rate, and thus the ''worst'' leg of the chains determines the localization properties of a coupled system. If away from resonance the two legs are hardly affected by each other. However, the close relation (proportionality) between mean free path and localization length is special for one-dimensional systems. On coupled Bethe lattices, weak disorder is irrelevant to localization. The localization effect is significant only if the disorder is intermediate or strong. Therefore, resonance conditions, which require weak disorder as compared to the hopping, can not be achieved. In general, we found that the less disordered lattice is not affected much by the more disordered lattice in the presence of coupling, except in the case where the less disordered (delocalized) lattice is very close to the transition and the more disordered lattice is strongly localized, in which case the more disordered lattice can push the less disordered lattice to a localized phase. We believe that these trends persist in high dimensions ($D>2$) where the metal-insulator transition takes places at strong disorder. In two dimensions, the localization length becomes parametrically larger than the mean free path at weak disorder. However, since the proliferation of weak-localization and backscattering leads to complete localization (in the absence of special symmetries), we expect that a well propagating channel becomes more strongly localized upon resonant coupling to a more disordered channel, similarly as in one dimension. It might be interesting to investigate this numerically. Investigating the localization properties of few- or many-particle systems is more complicated. First, we should map an interacting Hamiltonian to the Anderson model in the few- many-particle Fock space [cf. Eq.~\ref{ipfham}]. Thereby, the interaction provides effective hopping among the Fock states. This hopping in Fock space can be organized into channels with rather different propagation characteristics [e.g. Fig.~\ref{four-par} for four particles], namely, faster channels and slower channels. According to our analysis, the slow channel dominates only if it is \emph{resonantly} coupled to the fast channel. If the two channels are away from resonance, the fast channel essentially dominates the localization properties. For the few-particle problems discussed in Chap.~\ref{ch:interacting-particles} we expect that the fast channel, that is, the hierarchical structure we predicted, dominates the delocalization of the interacting particles, since the resonance between the fast and slow channels should be an exception rather than a rule. At this stage, this remains a conjecture which needs to be tested further.

Mots clés: Anderson localization; Anderson transition; Few particles; Polariton

Domaines: Physique Quantique [quant-ph]

Hongyi Dr. Xie. Anderson localization in disordered systems with competing channels. Quantum Physics [quant-ph]. Scuola Internazionale Superiore di Studi Avanzati, 2012. English. ⟨NNT : ⟩. ⟨tel-00752118⟩