Localization properties of harmonic chains with correlated mass and spring disorder: Analytical approach

We study the localization properties of normal modes in harmonic chains with mass and spring weak disorder. Using a perturbative approach, an expression for the localization length $L_{\text{loc}}$ is obtained, which is valid for arbitrary correlations of the disorder (mass disorder correlations, spring disorder correlations, and mass-spring disorder correlations are allowed), and for practically the whole frequency band. In addition, we show how to generate effective mobility edges by the use of disorder with long range self-correlations and cross-correlations. The transport of phonons is also analyzed, showing effective transparent windows that can be manipulated through the disorder correlations even for relative short chain sizes. These results are connected to the problem of heat conduction in the harmonic chain; indeed, we discuss the size scaling of the thermal conductivity from the perturbative expression of $L_{\text{loc}}$. Our results may have applications in modulating thermal transport, particularly in the design of thermal filters or in manufacturing high-thermal-conductivity materials.

Figure 1

Inverse localization length as a function of the normalized frequency for Case I. Symbols represent numerical simulations for a chain of length $N=2\times {10}^6$, whereas continuous, dotted, and dashed black lines correspond to the theoretical result (5.3) for $\delta =\pi /4$, $\delta =0$, and $\delta =-\pi /4$, respectively. The inset is an enlargement around the first localization window.

Figure 2

Inverse localization length as a function of the normalized frequency for Case II. Symbols represent numerical simulations for a chain of length $N=2\times {10}^6$, whereas continuous, dotted, and black lines correspond to the theoretical result (5.3) for $\delta =\pi /4$, $\delta =0$, and $\delta =-\pi /4$, respectively. The inset shows $\lambda$ in logarithmic scale in the low frequency regime for $N=2\times {10}^9$.

Figure 3

Probability phase distribution functions $\rho \left(\theta \right)$ obtained after $5\times {10}^8$ iterations of the map (3.4) with $\mu =5\times {10}^{-5}$. The first row corresponds to Case I, whereas the second row represents Case II. The horizontal black lines are the uniform distribution $\rho \left(\theta \right)=1/2\pi$.

Figure 4

Transmission coefficient averaged over 1000 chain realizations as a function of the normalized frequency for Case I. Numerical simulations are done for chains of length $N=1000$.

Figure 5

Transmission coefficient averaged over 1000 chain realizations as a function of the normalized frequency for Case II. Numerical simulations are done for chains of length $N=1000$ (main panel) and $N=2\times {10}^6$ (inset).

Figure 6

${\kappa }^{*}$ averaged over 10 disorder realizations as a function of the chain length $N$ for free and fixed boundary conditions (symbols). Black continuous lines are the best fits of the data with the function $f\left(N\right)=a{N}^b$. For free BC we got $b=0.501\pm 0.001$ ($\delta =0$) and $b=0.502\pm 0.004$ ($\delta =\pi /4$), whereas for fixed BC we obtained $b=-0.48\pm 0.03$ ($\delta =0$) and $b=-0.498\pm 0.005$ ($\delta =\pi /4$). For free BC ${\tilde{\sigma }}_m^2={\tilde{\sigma }}_k^2=0.001$, while for fixed BC ${\tilde{\sigma }}_m^2={\tilde{\sigma }}_k^2=0.003$. $k_B$ was set to 1.