Study of the localization-delocalization transition for phonons via transfer matrix method techniques

We use a transfer-matrix method to study the localization properties of vibrations in a `mass and spring' model with simple cubic lattice structure. Disorder is applied as a box-distribution to the force-constants $k$ of the springs. We obtain the reduced localization lengths $Λ_M$ from calculated Lyapunov exponents for different system widths to roughly locate the squared critical transition frequency $ω_{\text{c}}^2$. The data is finite-size scaled to acquire the squared critical transition frequency of $ω_{\text{c}}^2 = 12.54 \pm 0.03$ and a critical exponent of $ν= 1.55 \pm 0.002$.

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