Accessing 1D phases of matter via magnetic noise spectroscopy with single spins

The study of exotic 1D states, particularly those featuring strong spin-orbit coupling at the edges of topological materials, demand experimental probes that can access the interplay between charge and spin degrees of freedom. One potential candidate is a single spin probe, which has recently emerged as a versatile tool to probe nanoscale systems in a non-invasive fashion. Here we present a theory to describe noise magnetometry of 1D systems using single spin probes, and outline protocols to access correlations which are not available to other probes. One key observation is that, by exploiting the spin degree of freedom of the probe, it is possible to measure independently local charge and spin fluctuations in a variety of 1D systems. We describe how experimentally tunable parameters, such as temperature and probe-to-sample distance, can be used to study charge and spin excitations, e.g. whether they propagate ballistically or are pinned by disorder. We discuss some applications in the context of quantum spin Hall states and quantum Hall effects in graphene. The ability to sample both charge and spin fluctuations in a wide range of legthscales opens new pathways to bridge the large gap between atomic scale resolution of tunneling probes and global transport measurements.