Distribution and localization of the harmonic magnon modes in a one-dimensional Heisenberg spin glass.

Abstract

The harmonic magnon modes in a one-dimensional Heisenberg spin glass having nearest-neighbor exchange interactions of fixed magnitude and random sign are investigated. The Lyapounov exponent is calculated for chains of ${10}^{7}$--${10}^{8}$ spins over the interval 0\ensuremath{\le}\ensuremath{\omega}\ensuremath{\le}4J. In the low-frequency regime, \ensuremath{\omega}\ensuremath{\lesssim}0.1J, an anomalous behavior for the density of states \ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\sim}${\ensuremath{\omega}}^{\mathrm{\ensuremath{-}}1/3}$ is established, consistent with earlier results obtained by Stinchcombe and Pimentel using transfer-matrix techniques; at higher frequencies, gaps appear in the spectrum. At low frequencies, the localization length diverges as ${\ensuremath{\omega}}^{\mathrm{\ensuremath{-}}2/3}$. A formal connection is established between the spin glass and the one-dimensional discretized Schr\"odinger equation. By making use of the connection, it is shown that the theory of Derrida and Gardner, which was developed for weak potential disorder, can account quantitatively for the distribution and localization of the low-frequency magnon modes in the spin-glass model.