The magnetic fields generated by spins and currents provide a unique window into the physics of correlatedelectron materials and devices. Proposed only a decade ago, magnetometry based on the electron spin of nitrogen-vacancy (NV) defects in diamond is emerging as a platform that is excellently suited for probing condensed matter systems: it can be operated from cryogenic temperatures to above room temperature, has a dynamic range spanning from DC to GHz, and allows sensor-sample distances as small as a few nanometres. As such, NV magnetometry provides access to static and dynamic magnetic and electronic phenomena with nanoscale spatial resolution. Pioneering work focused on proof-of-principle demonstrations of its nanoscale imaging resolution and magnetic field sensitivity. Now, experiments are starting to probe the correlatedelectron physics of magnets and superconductors and to explore the current distributions in low-dimensional materials. In this Review, we discuss the application of NV magnetometry to the exploration of condensed matter physics, focusing on its use to study static and dynamic magnetic textures, and static and dynamic current distributions. Box 1| Measuring static fieldsHere we describe elementary considerations for the use of nitrogen-vacancy (NV) centres for imaging magnetic fields generated by static magnetic textures and current distributions. Reconstructing a vector magnetic field by measuring a single field componentBecause the NV electron spin resonance splitting is first-order sensitive to the projection of the magnetic field B on the NV spin quantization axis, B||, this is the quantity typically measured in an NV magnetometry measurement 16 . It is therefore convenient to realize that the full vector field B can be reconstructed by measuring any of its components in a plane positioned at a distance d from the sample, where d is the NV-sample distance (provided this component is not parallel to the measurement plane). This results from the linear dependence of the components of B̂ in Fourier space 24,25 , which follows from the fact that B can be expressed as the gradient of a scalar magnetostatic potential. Moreover, by measuring B||(x, y; z = d) we can reconstruct B at all distances d + h through the evanescent-field analogue of Huygens' principle, a procedure known as upward propagation 24 . As an example, the out-of-plane stray field component Bz(x, y; z = d + h) can be reconstructed from B||(x, y; z = d) using ̂( ; + ℎ) = − ℎ̂| | ( ; ) NV •

Nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI) are wellestablished techniques that provide valuable information in a diverse set of disciplines but are currently limited to macroscopic sample volumes. Here we demonstrate nanoscale NMR spectroscopy and imaging under ambient conditions of samples containing multiple nuclear species, using nitrogen-vacancy (NV) colour centres in diamond as sensors. With single, shallow NV centres in a diamond chip and samples placed on the diamond surface, we perform NMR spectroscopy and one-dimensional MRI on few-nanometre-sized samples containing 1 H and 19 F nuclei. Alternatively, we employ a high-density NV layer near the surface of a diamond chip to demonstrate wide-field optical NMR spectroscopy of nanoscale samples containing 1 H, 19 F, and 31 P nuclei, as well as multi-species two-dimensional optical MRI with sub-micron resolution. For all diamond samples exposed to air, we identify a ubiquitous 1 H NMR signal, consistent with a ∼ 1 nm layer of adsorbed hydrocarbons or water on the diamond surface and below any sample placed on the diamond. This work lays the foundation for nanoscale NMR and MRI applications such as studies of single proteins and functional biological imaging with subcellular resolution, as well as characterization of thin films with sub-nanometre resolution.

The spin chemical potential characterizes the tendency of spins to diffuse. Probing this quantity could provide insight into materials such as magnetic insulators and spin liquids and aid optimization of spintronic devices. Here we introduce single-spin magnetometry as a generic platform for nonperturbative, nanoscale characterization of spin chemical potentials. We experimentally realize this platform using diamond nitrogen-vacancy centers and use it to investigate magnons in a magnetic insulator, finding that the magnon chemical potential can be controlled by driving the system's ferromagnetic resonance. We introduce a symmetry-based two-fluid theory describing the underlying magnon processes, measure the local thermomagnonic torque, and illustrate the detection sensitivity using electrically controlled spin injection. Our results pave the way for nanoscale control and imaging of spin transport in mesoscopic systems.