Anomalous dynamics in two- and three- dimensional Heisenberg-Mattis spin glasses

TL;DR

This study explores the spectral and localization characteristics of unmagnetized Heisenberg-Mattis spin glasses in two and three dimensions, revealing localization in 2D and specific density-of-states behaviors in 3D.

Contribution

It provides numerical analysis of spectral properties and localization in spin glasses, confirming theoretical predictions and highlighting differences between 2D and 3D cases.

Findings

All states are localized in 2D with diverging localization length at zero energy.

Density of states in 3D matches theoretical predictions for antiferromagnetic spin waves.

Logarithmic corrections to density of states in 2D align with theoretical models.

Abstract

We investigate the spectral and localization properties of unmagnetized Heisenberg-Mattis spin glasses, in space dimensionalities $d=2$ and 3, at T=0. We use numerical transfer-matrix methods combined with finite-size scaling to calculate Lyapunov exponents, and eigenvalue-counting theorems, coupled with Gaussian elimination algorithms, to evaluate densities of states. In $d=2$ we find that all states are localized, with the localization length diverging as $\omega^{-1}$, as energy $\omega \to 0$. Logarithmic corrections to density of states behave in accordance with theoretical predictions. In $d=3$ the density-of-states dependence on energy is the same as for spin waves in pure antiferromagnets, again in agreement with theoretical predictions, though the corresponding amplitudes differ.