Scaling theory of heat transport in quasi-one-dimensional disordered harmonic chains

We introduce a variant of the banded random matrix ensemble and show, using detailed numerical analysis and theoretical arguments, that the phonon heat current in disordered quasi-one-dimensional lattices obeys a one-parameter scaling law. The resulting $\beta$ function indicates that an anomalous Fourier law is applicable in the diffusive regime, while in the localization regime the heat current decays exponentially with the sample size. Our approach opens a new way to investigate the effects of Anderson localization in heat conduction based on the powerful ideas of scaling theory.

Figure 1

(top) Temperature and (bottom) heat current profiles for bandwidths of $b=6$ (black) and $b=30$ [red (gray)] in a system of size $N=1000$. The flatness of the heat current profile indicates a nonequilibrium steady state.

Figure 2

Participation number vs. frequency for $b=5$. Various finite system sizes are delineated by different colors. Colored stripes indicate the groups of the most extended modes, for which $P_2\ge 0.6P_2^{\max }$. A normalized count of selected states yield a scaling $I\sim N^{-\gamma }$ for the integrated density of states, as shown in the inset, for three representative bandwidths: $b=5$ (circles), $b=10$ (squares), and $b=12$ (diamonds). The best fit indicates $\gamma \approx 0.1\pm 0.05$ (black solid line).

Figure 3

The disordered and spectral (over $\delta \omega$) averaged PN ${\xi }_N$ against $1/N$ for various $b$ values. Saturation of ${\xi }_N$ as $N\to \infty$ is observed for moderate bandwidths. For larger values of $b$ a fitting of the data to a rational function of fifth-order polynomials (dashed and dot-dashed lines) allows us to extract the limit ${\xi }_{\infty }$. The inset shows the parametric dependence of ${\xi }_{\infty }$ on $b$. The dashed line is the power relation $b^{3.5}$.

Figure 4

Rescaled heat current $\tilde{J}$ vs $\Lambda ={\xi }_{\infty }/N$, where ${\xi }_{\infty }\sim b^{3.5}$, as was found from Fig. 3. The dashed line is a fit to the analytical curve of Eq. (10). The inset shows the resulting $\beta$ function of Eq. (1). The dashed lines correspond to the asymptotic limits of $\beta =1.28+0.94\ln \tilde{J}$ for $N\to \infty$ and $\beta =-\nu$ for $N\to 0$.