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 $Lambda_M$ from calculated Lyapunov exponents for different system widths to roughly locate the squared critical transition frequency $omega_{text{c}}^2$. The data is finite-size scaled to acquire the squared critical transition frequency of $omega_{text{c}}^2 = 12.54 pm 0.03$ and a critical exponent of $ u = 1.55 pm 0.002$.