Localization-Delocalization Transition and Current Fractalization

TL;DR

This paper develops an analytical theory describing the transition from localized to delocalized states in a disordered Bose system, revealing how temperature and disorder influence charge transport and turbulence onset.

Contribution

It introduces a novel analytical framework for understanding the localization-delocalization transition in disordered superconducting systems, linking turbulence and spectral excitation dynamics.

Findings

At T < E_c, Coulomb blockade suppresses transport except at resonant voltages.

At T > E_c, transport becomes thermally activated due to environment spectrum becoming quasi-continuous.

The transition occurs at T = E_c, corresponding to turbulence onset in environmental excitations.

Abstract

We develop an analytical theory of the localization-delocalization transition for a disordered Bose system, focusing on a Cooper-pair insulator. We consider a chain of small superconducting granules coupled via Josephson links and show that the low-temperature tunnelling transport of Cooper pairs is mediated by a self-generated environment of dipole excitations comprised of the same particles as the tunnelling charge carriers in accord with the early notion by Fleishman, Licciardello, and Anderson. We derive an analytical expression for the current-voltage characteristic and find that at temperatures, T, below the the charging energy of a single junction, E_c, the dc transport is completely locked by Coulomb blockade effect at all voltages except for a discrete set of resonant ones. At T>E_c the combined action of disorder and temperature unlocks the charge transport, since the environment excitation spectrum becomes quasi-continuous according to a Landau-Hopf-like scenario of turbulence, and the conductivity acquires an Arrhenius-like thermal activation form. The transition from the localized to delocalized behaviour occurs at T=E_c which corresponds to the onset of turbulence in the spectral flow of environmental excitations with Reynolds number Re=(k_B T/E_c)=1. The proposed theory breaks ground for a quantitative description of dynamic and quantum phase transitions in a wealth of physical systems ranging from cold atoms in optical lattices, through disordered films and wires to granular and nanopatterned materials.