The small-world model has been recently suggested [1] as a description of some unusual topological properties in communication (social) networks (Internet, airline routes, power grids, etc.). This model consists of  nodes regularly connected on a regular lattice with a small concentration of  random (irregular) connections (rewiring), and thus is an intermediate case between the regular lattice (large world) and a random graph. Very recently,  disordered quantum small-world-like networks have been introduced to study the localization properties of electrons in varying dimensions [2]. The aim of the project is to apply a similar formalism  to study classical atomic vibrations in   small-world-like networks. Classical atomic vibrations of atoms situated on a regular lattice (of different dimensionalities, e.g. 1D linear chain, 2D square lattice etc.) and connected by harmonic springs regularly to  nearest (and next nearest) neighbours and randomly with other possible remote sites will be investigated.

Two properties are of particular interest:

(i) Localization-delocalization transition for atomic vibrations in  systems of varying dimensions (irregular springs change the dimensionality of the problem) studied by multi-fractal analysis. This is the mainly computational part of the project. The codes for the multi-fractal analysis (box-counting and wavelet analysis) are available in the group. We have expertise in  such studies for disordered lattices [3].

(ii) Analytical study (supported by simulations) of the vibrational spectrum for such   models  by mean-field approaches in the frequency range where localization effects are not present. The problem can be formulated using the Hamiltonian formalism and studied using  Green's function techniques. We have expertise in such studies [4].

These two aspects of the project can be considered as two independent projects, or as a single project if an applicant feels strong enough to cope with both of them.