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Existence of a Phase with Finite Localization Length in the Double Scaling Limit of N-Orbital Models

Among the models of disordered conduction and localization, models with $N$ orbitals per site are attractive both for their mathematical tractability and for their physical realization in coupled disordered grains. However Wegner proved that there is no Anderson transition and no localized phase in the $N \rightarrow \infty$ limit, if the hopping constant $K$ is kept fixed. Here we show that the localized phase is preserved in a different limit where $N$ is taken to infinity and the hopping $K$ is simultaneously adjusted to keep $N \, K$ constant. We support this conclusion with two arguments. The first is numerical computations of the localization length showing that in the $N \rightarrow \infty$ limit the site-diagonal-disorder model possesses a localized phase if $N\,K$ is kept constant, but does not possess that phase if $K$ is fixed. The second argument is a detailed analysis of the energy and length scales in a functional integral representation of the gauge invariant model. The analysis shows that in the $K$ fixed limit the functional integral's spins do not exhibit long distance fluctuations, i.e. such fluctuations are massive and therefore decay exponentially, which signals conduction. In contrast the $N\,K$ fixed limit preserves the massless character of certain spin fluctuations, allowing them to fluctuate over long distance scales and cause Anderson localization.