Heat conduction in harmonic chains with Lévy-type disorder

We consider heat transport in a one-dimensional harmonic chain attached at its ends to Langevin heat baths. The harmonic chain has mass impurities where the separation d between any two successive impurities is randomly distributed according to a power-law distribution P(d)∼1/d^{α+1}, being α>0. In the regime where the first moment of the distribution is well defined (1<α<2) the thermal conductivity κ scales with the system size N as κ∼N^{(α-3)/α} for fixed boundary conditions, whereas for free boundary conditions κ∼N^{(α-1)/α} if N≫1. When α=2, the inverse localization length λ scales with the frequency ω as λ∼ω^{2}lnω in the low-frequency regime, due to the logarithmic correction, the size scaling law of the thermal conductivity acquires a nonclosed form. When α>2, the thermal conductivity scales as in the uncorrelated disorder case. The situation α<1 is only analyzed numerically, where λ(ω)∼ω^{2-α}, which leads to the following asymptotic thermal conductivity: κ∼N^{-(α+1)/(2-α)} for fixed boundary conditions and κ∼N^{(1-α)/(2-α)} for free boundary conditions.