Random quantum magnets with broad disorder distribution

Abstract

We study the critical behavior of Ising quantum magnets with broadly distributed random couplings (J), such that $P(\ln J) \sim |\ln J|^{-1-α}$, $α>1$, for large $|\ln J|$ (Lévy flight statistics). For sufficiently broad distributions, $α<α_c$, the critical behavior is controlled by a line of fixed points, where the critical exponents vary with the Lévy index, $α$. In one dimension, with $α_c=2$, we obtaind several exact results through a mapping to surviving Riemann walks. In two dimensions the varying critical exponents have been calculated by a numerical implementation of the Ma-Dasgupta-Hu renormalization group method leading to $α_c \approx 4.5$. Thus in the region $2<α<α_c$, where the central limit theorem holds for $|\ln J|$ the broadness of the distribution is relevant for the 2d quantum Ising model.