Numerical studies of amorphous Si in harmonic approximation show that the highest 3.5% of vibrational normal modes are localized. As the vibrational frequency increases through the boundary separating localized from delocalized modes, near ω_c = 70 meV (the 'mobility edge') there is a localization-delocalization transition, similar to a second-order thermodynamic phase transition. By a numerical study on a system with 4096 atoms, we are able to see exponential decay lengths of exact vibrational eigenstates and to test whether or not these diverge at ω_3. Results are consistent with a localization length ζ which diverges above ω_c as (ω-ω_c)~(-p) where the exponent is p ≈ 1.3 ± 0.5. Below the mobility edge we find no evidence for a diverging correlation length. Such an asymmetry would contradict scaling ideas, and we suppose it is a finite-size artefact. If the scaling regime is narrower than our (approximately 1 meV) resolution, then it cannot be seen directly on our finite system.
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