Abstract
We examine the nature and properties of the “exponentiated random walk” one-dimensional wavefunction Ψ0=exp[−x(x)], previously introduced in the context of the supersymmetric mappings of a classical Langevin random field problem. Three main results are presented. The first is that the state Ψ0 is extended, although it is the exact groundstate of a disordered one-dimensional quantum problem. The second is that in that problem supersymmetry is neither truly unbroken, or truly broken, we call this a situation of marginal unbroken supersymmetry and identify a class of other problems with the same property. The third result is obtained by studying the local behaviour of the wave function Ψ0 by means of generalized Lyapunov exponents. Locally, Ψ0 exhibits exponential localization, with a localization length identical to that of weak localization in the 1-dimensional Anderson problem.
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Tosatti, E., Zannetti, M. & Pietronero, L. Exponentiated random walks, supersymmetry and localization. Z. Physik B - Condensed Matter 73, 161–166 (1988). https://doi.org/10.1007/BF01305733
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DOI: https://doi.org/10.1007/BF01305733