Localization of spin waves in disordered quantum rotors

Abstract

We study the dynamics of excitations in a system of $O(N)$ quantum rotors in the presence of random fields and random anisotropies. Below the lower critical dimension ${d}_{\mathrm{lc}}=4$ the system exhibits a quasi-long-range order with a power-law decay of correlations. At zero temperature the spin waves are localized at the length scale ${L}_{\mathrm{loc}}$ beyond which the quantum tunneling is exponentially suppressed, $c\ensuremath{\sim}{e}^{\ensuremath{-}{(L/{L}_{\mathrm{loc}})}^{2(\ensuremath{\theta}+1)}}$. At finite temperature $T$ the spin waves propagate by thermal activation over energy barriers that scale as ${L}^{\ensuremath{\theta}}$. Above ${d}_{\mathrm{lc}}$ the system undergoes an order-disorder phase transition with activated dynamics such that the relaxation time grows with the correlation length $\ensuremath{\xi}$ as $\ensuremath{\tau}\ensuremath{\sim}{e}^{C{\ensuremath{\xi}}^{\ensuremath{\theta}}/T}$ at finite temperature and as $\ensuremath{\tau}\ensuremath{\sim}{e}^{{C}^{\ensuremath{'}}{\ensuremath{\xi}}^{2(\ensuremath{\theta}+1)}/{\ensuremath{\hbar}}^{2}}$ in the vicinity of the quantum critical point.