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Dissipationless dynamics of randomly coupled spins at high temperatures

We develop a technique to compute the high-frequency asymptotics of spin correlators in weakly interacting disordered spin systems. We show that the dynamical spin correlator decreases exponentially at high frequencies, $< SS>_ω\sim\exp(-τ_{*}ω)$ and compute the characteristic time $τ^{*}$ of this dependence. In a typical random configuration, some fraction of spins form strongly coupled pairs, which behave as two-level systems. Their switching dynamics is driven by the high-frequency noise from the surrounding spins, resulting in low-frequency $1/f$ noise in the magnetic susceptibility and other physical quantities. We discuss application of these results to the problem of susceptibility and flux noise in superconducting circuits at mK temperatures.