Localization of spin waves in disordered quantum rotors

We study the dynamics of excitations in a system of $O\left(N\right)$ 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, $c\sim e^{-{(L/L_{\mathrm{loc}})}^{2(\theta +1)}}$. At finite temperature $T$ the spin waves propagate by thermal activation over energy barriers that scale as $L^{\theta }$. 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 $\xi$ as $\tau \sim e^{C{\xi }^{\theta }/T}$ at finite temperature and as $\tau \sim e^{C^{\prime }{\xi }^{2(\theta +1)}/{\hslash }^2}$ in the vicinity of the quantum critical point.

Figure 1

The $\pi$-periodic FP solution ${\tilde{R}}^{\prime \prime }\left(\varphi \right)$ describing the QLRO phase in the three-dimensional (3D) $O\left(3\right)$ RA model. The dashed line is the zero-$\Gamma$ FP, and the solid green and blue lines with a rounded cusp are the finite $\Gamma$ FPs for $\Gamma =0.05$ and $\Gamma =0.1$.

Figure 2

The renormalized spin-wave velocity in the 3D $O\left(3\right)$ RA model as a function of $\ell -{\ell }_c$ for different $T$ and the initial condition for the bare coupling constant $g_0=c_0{\tilde{\hslash }}_0=10$. The dashed line corresponds to ${\tilde{T}}_0=0$; the solid black, green, and blue lines to ${\tilde{T}}_0=0.2,0.5,1$.