Collective spinon spin wave in a magnetized U(1) spin liquid

We study the transverse dynamical spin susceptibility of the two dimensional U(1) spinon Fermi surface spin liquid in a small applied Zeeman field. We show that both short-range interactions, present in a generic Fermi liquid, as well as gauge fluctuations, characteristic of the U(1) spin liquid, qualitatively change the result based on the frequently assumed non-interacting spinon approximation. Short-range interaction leads to a new collective mode: a "spinon spin wave" which splits off from the two-spinon continuum at small momentum and disperses downward. Gauge fluctuations renormalize the susceptibility, providing non-zero power law weight in the region outside the spinon continuum and giving the spin wave a finite lifetime, which scales as momentum squared. We also study the effect of Dzyaloshinskii-Moriya anisotropy on the zero momentum susceptibility, which is measured in electron spin resonance (ESR), and obtain a resonance linewidth linear in temperature and varying as $B^{2/3}$ with magnetic field $B$ at low temperature. Our results form the basis for a theory of inelastic neutrons scattering, ESR, and resonant inelastic x-ray scattering (RIXS) studies of this quantum spin liquid state.