We determine the ‘flux noise’ spectrum of random-bond Heisenberg spin-chains using a real-space renormalization group (RSRG) procedure that accounts for both the renormalization of the system Hamiltonian and a generic probe that measures this noise. In particular, we focus on harmonic probes that couple to spins with a modulation wave-vector \(q\); these probes measure the dynamical structure factor \(S_q (\omega)\) associated with spin-waves in the system. At both high and low temperatures we find that the system exhibits a finite dynamical exponent \(z \neq 1\),\(2\) that governs the high and low frequencies (\(\omega \gg q^z\), \(\omega \ll q^z\) respectively) power-laws \(\alpha\), \(\alpha'\) in the frequency-dependent behavior of the structure factor. These power laws are shown to have non-trivial relations amongst each other and with \(z\), that differ at \(T = 0\) and \( T = \infty\). We prove these relations using an analytical calculation of the structure factor assuming a limiting distribution of the RG flow parameters (spin sizes, length, bond strengths) and verify them via a numerical implementation of the RG. We show that these relations hold generally in the infinite-temperature case by postulating a generalized form of the spin-difussion propagator and showing that it captures all features of the structure-factor with minimal assumptions. We also provide arguments as to why \(1/f\)-noise is intimately connected to many-body-localization at finite-temperatures.
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