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 $P(\ln J)\sim |\ln J{|}^{-1-\alpha }$, $\alpha >1$, for large $|\ln J|$ (Lévy flight statistics). For sufficiently broad distributions, $\alpha <{\alpha }_c$, the critical behavior is controlled by a line of fixed points, where the critical exponents vary with the Lévy index, $\alpha$. In one dimension, with ${\alpha }_c=2$, 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 ${\alpha }_c\approx 4.5$. Thus in the region $2<\alpha <{\alpha }_c$, where the central limit theorem holds for $|\ln J|$ the broadness of the distribution is relevant for the 2d quantum Ising model.

10pages, 13figures, final form

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