1 / f α noise and generalized diffusion in random Heisenberg spin systems

Abstract

We study the "flux-noise" spectrum of random-bond quantum Heisenberg spin systems using a real-space renormalization group (RSRG) procedure that accounts for both the renormalization of the system Hamiltonian and of a generic probe that measures the noise. For spin chains, we find that the dynamical structure factor S-q(f), at finite wave vector q, exhibits a power-law behavior both at high and low frequencies f, with exponents that are connected to one another and to an anomalous dynamical exponent through relations that differ at T = 0 and T = infinity. The low-frequency power-law behavior of the structure factor is inherited by any generic probe with a finite bandwidth and is of the form 1/f(alpha) with 0.5 < alpha < 1. An analytical calculation of the structure factor, assuming a limiting distribution of the RG flow parameters (spin size, length, bond strength) confirms numerical findings. More generally, we demonstrate that this form of the structure factor, at high temperatures, is a manifestation of anomalous diffusion which directly follows from a generalized spin-diffusion propagator. We also argue that 1/f-noise is intimately connected to many-body-localization at finite temperatures. In two dimensions, the RG procedure is less reliable; however, it becomes convergent for quasi-one-dimensional geometries where we find that one-dimensional 1/f(alpha) behavior is recovered at low frequencies; the latter configurations are likely representative of paramagnetic spin networks that produce 1/f(alpha) noise in SQUIDs.