Localization, Anomalous Diffusion, and Slow Relaxations: A Random Distance Matrix Approach

Abstract

We study the spectral properties of a class of random matrices where the matrix elements depend exponentially on the distance between uniformly and randomly distributed points. This model arises naturally in various physical contexts, such as the diffusion of particles, slow relaxations in glasses, and scalar phonon localization. Using a combination of a renormalization group procedure and a direct moment calculation, we find the eigenvalue distribution density (i.e., the spectrum), for low densities, and the localization properties of the eigenmodes, for arbitrary dimension. Finally, we discuss the physical implications of the results.PACS numbers: 02.10. Yn, 71.23.Cq, Application of the theory of random matrices whose elements are independent Gaussian variables has proven to be rich mathematically and relevant for many physical systems [1]. In this Letter we study a different class of random matrices where the i, j'th element is a function of the Euclidian distance r ij between pairs of points whose positions are chosen randomly and uniformly in a d-dimensional space. It is natural that in cases where the matrix element is related to an overlap between localized quantum-mechanical wavefunctions, the dependence on the distance will be exponential, i.e., A ij = e −rij /ξ , with ξ being the localization length [2].The exponential matrix is an appropriate model for various physical systems, in this Letter we will concentrate on its application to glasses relaxing to equilibrium, a particle diffusing in random environment and localization of phonons. Most of the results are derived at the low density limit, when ǫ = ξ/r nn ≪ 1, with r nn being the average nearest neighbor distance. To understand the properties of these systems one need to find out the distribution density P (λ) of the eigenvalues λ as well as the structure of the eigenmodes. An intuitive picture of the problem arises in the application to phonon localization with springs constants K ij that depend exponentially on the Euclidean distances between the masses; we therefore use the phonon terminology: eigenmode.The low density limit allows us to find P (λ) analytically employing a direct calculation of its moments, see Eq. (2) and the Supplementary Material (SM). We find that P (λ) ∼ 1/λ in all dimensions over a broad range of λ's. While in one dimension the normalization of P (λ) is assured by an integrable power-law divergence at eigenvalues close to zero, for higher dimensions there is a peak related to a finite cutoff, cf Fig. 1. We use a logarithmic scale to plot P [log(−λ/2)] in order emphasize the deviations from the 1/λ distribution.To comprehend the structure of the eigenmodes we use a renormalization group (RG) approach for random systems that was developed in the context of spin chains [3][4][5][6]. At each RG step, we choose the stiffest spring. Since the spring is large by construction, after finding the eigenvalue associated with the stiffest spring we can glue together the two masses at its ends creating a larger mass. At the next RG ste...

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