Theory of non-hermitian localization in one dimension: Localization length and eigenergies
We recall some basic aspects of the pinning of flux lines in a superconducting cylindrical shell subjected to a depinning magnetic field, as well as its description by the quantum mechanics of a disordered ring with an imaginary vector potential proportional to the depinning field (N Hatano and D R Nelson, Phys. Rev.B56, 8651 (1997)). We then discuss our recent analysis of the pinning-depinning transition in terms of an explicit solution for the inverse localization length of the eigenstates of the non-hermitian quantum system for weak disorder. Our results as to the nature of the non hermitian quantum states, differ qualitatively from earlier studies which did not examine the detailed properties of the localization length. Nevertheless we obtain a well-defined simple picture for the pinning-depinning transition of flux lines. We discuss furthermore a new exact calculation of localized state eigenenergies for weak disorder, which we compare with previous analytic and numerical results.
Volume 94, 2020
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