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Localization of spin waves in disordered quantum rotors

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 \sim e^{-(L/L_{\mathrm{loc}})^{2(θ+1)}}$. At finite temperature $T$ the spin waves propagate by thermal activation over energy barriers that scales as $L^θ$. 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 $ξ$ as $τ\sim e^{C ξ^θ/T}$ at finite temperature and as $τ\sim e^{C' ξ^{2(θ+1)}/\hbar^2}$ in the vicinity of the quantum critical point.