Localization of disordered bosons and magnets in random fields

TL;DR

This paper investigates localization phenomena in disordered bosons and spins in random fields, revealing distinct behaviors at criticality and implications for phase transitions in high-dimensional and finite-dimensional systems.

Contribution

It provides a comparative analysis of localization in XY and Ising models, including non-analytic localization length behavior and activated scaling at the transition.

Findings

Localization length exhibits non-analytic behavior at omega=0 in 1D Ising chains.

Ising models show activated scaling at the phase transition.

Order in Ising systems arises from delocalization at omega=0 without a mobility gap.

Abstract

We study localization properties of disordered bosons and spins in random fields at zero temperature. We focus on two representatives of different symmetry classes, hard-core bosons (XY magnets) and Ising magnets in random transverse fields, and contrast their physical properties. We describe localization properties using a locator expansion on general lattices. For 1d Ising chains, we find non-analytic behavior of the localization length as a function of energy at \omega = 0, $\xi^{-1}(\omega) = \xi^{-1}(0) + A |\omega|^\alpha$, with $\alpha$ vanishing at criticality. This contrasts with the much smoother behavior predicted for XY magnets. We use these results to approach the ordering transition on Bethe lattices of large connectivity K, which mimic the limit of high dimensionality. In both models, in the paramagnetic phase with uniform disorder, the localization length is found to have a local maximum at \omega = 0. For the Ising model, we find activated scaling at the phase transition, in agreement with infinite randomness studies. In the Ising model long range order is found to arise due to a delocalization and condensation initiated at \omega = 0, without a closing mobility gap. We find that Ising systems establish order on much sparser (fractal) subgraphs than XY models. Possible implications of these results for finite-dimensional systems are discussed.