Localization effects due to a random magnetic field on heat transport in a harmonic chain

We consider a harmonic chain of $N$ oscillators in presence disordered magnetic field. The ends the are connected to heat baths and we study effects field randomness on transport. disorder, general, causes localization normal modes due which system becomes insulating. However, for this system, length diverges as mode frequency approaches zero. Therefore, low contribute transmission, $\mathcal{T}_N(\omega)$, current goes down power law with size, $N$. This is determined by small behaviour some Lyapunov exponent, $\lambda(\omega)$, transmission thermodynamic limit, $\mathcal{T}_\infty(\omega)$. While it known that constant $\mathcal{T}_\infty(\omega)\sim \omega^{3/2},~ \omega^{1/2}$ depending boundary conditions, find exponent behaves $\lambda(\omega)\sim\omega$ $\expval{B}\neq 0$ $\lambda(\omega)\sim\omega^{2/3}$ $\expval{B}=0$. obtain different laws vs $\expval{B}$ conditions.