Link to manuscript: M. L. Manning and A. J. Liu, “A random matrix definition of the boson peak,” accepted to Europhysics Letters, arXiv:1307.5904 (2015)

An old theory by Debye predicts that the number of sound modes in a crystal increases as the frequency of the sound mode to the power (d-1), where d is the number of dimensions. It has long been recognized that the boson peak—defined as an excess of sound modes compared to the Debye expectation—is a universal phenomenon in disordered solids.  However, even perfectly crystalline solids show an excess of modes compared to Debye at sufficiently high frequencies.   Therefore, we suggest a new definition of the boson peak, in terms of its eigenvector statistics.  This definition has two important advantages over the old one: it does not rely on a comparison to Debye scaling and it is true only for disordered solids. In addition, this manuscript identifies a new universality class in random matrices based on their eigenvector statistics, and indicates that modes of the boson peak have the same eigenvector statistics as this new universality class.  This result implies that any random matrix with disorder and a generalized global translation invariance should exhibit this universality, explaining the ubiquity of boson peaks in disordered solids. Finally, this paper identifies a class of “diagonally dominant” random matrices that capture the scaling of the frequency location of the boson peak in jammed solids.