Variable-Range Hopping through Marginally Localized Phonons

We investigate the effect of coupling Anderson localized particles in one dimension to a system of marginally localized phonons having a symmetry protected delocalized mode at zero frequency. This situation is naturally realized for electrons coupled to phonons in a disordered nanowire as well as for ultracold fermions coupled to phonons of a superfluid in a one-dimensional disordered trap. To determine if the coupled system can be many-body localized we analyze the phonon-mediated hopping transport for both the weak and strong coupling regimes. We show that the usual variable-range hopping mechanism involving a low-order phonon process is ineffective at low temperature due to discreteness of the bath at the required energy. Instead, the system thermalizes through a many-body process involving exchange of a diverging number $n\propto -\log T$ of phonons in the low temperature limit. This effect leads to a highly singular prefactor to Mott’s well-known formula and strongly suppresses the variable range hopping rate. Finally, we comment on possible implications of this physics in higher dimensional electron-phonon coupled systems.

Figure 1

The model of a localized electron coupled to a disordered harmonic chain with random masses and spring constants. An electron with localization length $\xi$ may hop from one localized state to another due to exponential overlap $\sim \exp (-R/\xi )$ between the states by extracting an amount of energy ${\mathrm{\Delta }}_R$ from the phonon bath, e.g., by absorbing a single phonon. However, due to the localized nature of the phonons, the one-phonon bath spectrum is discrete with a level spacing ${\delta }_{{\mathrm{\Delta }}_R}\simeq {\mathrm{\Delta }}_R^{\alpha }$. The level spacing of the phonon bath rapidly collapses as ${\mathrm{\Delta }}_R^{{\alpha }^n}$ for the multiphonon process involving $n$ phonons.

Figure 2

(a) First- and (b) second-order self-energy diagrams considered for estimating the hopping rate of the localized state $l$.