Spectral statistics and dynamics of Lévy matrices

We study the spectral statistics and dynamics of a random matrix model where matrix elements are taken from power-law tailed distributions. Such distributions, labeled by a parameter μ, converge on the L\'evy basin, giving the matrix model the label ``L\'evy matrix'' [P. Cizeau and J. P. Bouchaud, Phys. Rev. E $50,$ 1810 (1994)]. Such matrices are interesting because their properties go beyond the Gaussian universality class and they model many physically relevant systems such as spin glasses with dipolar or Ruderman-Kittel-Kasuya-Yosida interactions, electronic systems with power-law decaying interactions, and the spectral behavior at the metal insulator transition. Regarding the density of states we extend previous work to reveal the sparse matrix limit as →μ0. Furthermore, we find for 2×2 L\'evy matrices that geometrical level repulsion is not affected by the distribution's broadness. Nevertheless, essential singularities particular to L\'evy distributions for small arguments break geometrical repulsion and make it μ dependent. Level dynamics as a function of a symmetry breaking parameter gives new insight into the phases found by Cizeau and Bouchaud (CB). We map the phase diagram drawn qualitatively by CB by using the Δ₃ statistic. Finally we compute the conductance of each phase by using the Thouless formula, and find that the mixed phase separating conducting and insulating phases has a unique character.