Maksimilian Usoltcev

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A characteristic feature of the  hyperbolic plane, $\mathbb{H}_2$, is that its effective dimensionality changes with scale: from approximately flat, two-dimensional  at short, to high-dimensional at large
scales. It shares the feature of scale-dependent dimensionality with the exponentially expanding geometries of Fock spaces, but offers a simpler setting for an analytical treatment of localization. We here derive and analyze an effective theory of Anderson localization on the hyperbolic plane, demonstrating that it requires an extension of the celebrated one-parameter scaling paradigm of localization by a second parameter tracking the scale-dependent effective dimensionality. Our main
result is a flow diagram in this two-parameter space, divided by a critical line into a metallic and an insulating phase.

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University of Cologne
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Contact: Karim Chahine