Phonons in disordered harmonic lattices
arxiv: Condensed Matter::Disordered Systems and Neural Networks
This work explores the nature of the normal modes of vibration for harmonic\ud lattices with the inclusion of disorder in one-dimension (1D) and three-dimensions\ud (3D). The model systems can be visualised as a `ball' and `spring' model in simple\ud cubic configuration, and the disorder is applied to the magnitudes of the masses, or\ud the force constants of the interatomic `springs' in the system.\ud With the analogous nature between the electronic tight binding Hamiltonian\ud for potential disordered electronic systems and the isotropic Born model for\ud phonons in mass disordered lattices we analyse in detail a transformation between\ud the normal modes of vibration throughout a mass disordered harmonic lattice and\ud the electron wave function of the tight-binding Hamiltonian. The transformation\ud is applied to density of states (DOS) calculations and is also particularly useful for\ud determining the phase diagrams for the phonon localisation-delocalisation transition\ud (LDT). The LDT phase boundary for the spring constant disordered system is\ud obtained with good resolution and the mass disordered phase boundary is verified\ud with high precision transfer-matrix method (TMM) results. High accuracy critical\ud parameters are obtained for three transitions for each type of disorder by finite size\ud scaling (FSS), and consequently the critical exponent that characterises the transition\ud is found as = 1:550+0:020\ud -0:017 which indicates that the transition is of the same\ud orthogonal universality class as the electronic Anderson transition.\ud With multifractal analysis of the generalised inverse participation ratio (gIPR)\ud for the critical transition frequency states at spring constant disorder width k = 10\ud and mass disorder width m = 1:2 we confirm that the singularity spectrum is the\ud same within error as the electronic singularity spectrum at criticality and can be considered to be universal.\ud We further investigate the nature of the modes throughout the spectrum of\ud the disordered systems with vibrational eigenstate statistics. We find deviations of\ud the vibrational displacement \ud uctuations away from the Porter-Thomas distribution\ud (PTD) and show that the deviations are within the vicinity of the so called `bosonpeak'\ud (BP) indicating the possible significance of the BP.
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