LETTER TO THE EDITOR: A single defect approximation for localized states on random lattices

Geometrical disorder is present in many physical situations giving rise to eigenvalue problems. The simplest case of diffusion on a random lattice with fluctuating site connectivities is studied analytically and by exact numerical diagonalizations. Localization of eigenmodes is shown to be induced by geometrical defects, that is sites with abnormally low or large connectivities. We expose a 'single defect approximation' (SDA) scheme founded on this mechanism that provides an accurate quantitative description of both extended and localized regions of the spectrum. We then present a systematic diagrammatic expansion allowing to use SDA for finite-dimensional problems, e.g. to determine the localized harmonic modes of amorphous media. Since Anderson's fundamental work (1), physical systems in the presence of disorder are well known for exhibiting localization effects (2). While most attention has been paid so far to Hamiltonians with random potentials (e.g. stemming from impurities), there are situations in which disorder also originates from geometry. Of particular interest among these are the harmonic vibrations of amorphous materials such as liquids, colloids, glasses, etc around random particle configurations. Recent experiments on sound propagation in granular media (3, 4) have stressed the possible presence of localization effects, highly correlated with the microscopic structure of the sample. The existing theoretical framework for calculating the density of harmonic modes in amorphous systems was developed in liquid theory. In this context, microscopic configurations are not frozen but instantaneous normal modes (INM) give access to short time dynamics (5). Wu and Loring (6) and Wan and Stratt (7) have calculated good estimates of the density of INM for Lennard-Jones liquids, averaged over instantaneous particle configurations. However, localization-delocalization properties of the eigenvectors have not been considered. Diffusion on random lattices is another problem where geometrical randomness plays a crucial role (8). Long-time dynamics is deeply related to the small eigenvalues of the Laplacian on the lattice and therefore to its spectral dimension. Campbell suggested that diffusion on a random lattice could also mimic the dynamics taking place in a complicated phase space, e.g. for glassy systems (9). From this point of view, sites on the lattice represent microscopic configurations and edges represent allowed moves from one configuration to another. At low temperatures, most edges correspond to very improbable jumps and may be erased. The tail of the density of states of the Laplacian on random graphs was studied by means of heuristic arguments by Bray and Rodgers (10). Localized eigenvectors, closely related to metastable states are of particular relevance for asymptotic dynamics.