Collective modes of magnetized spin liquids.

We show that Zeeman magnetic field enhances the interaction between spinons in spin-conserving U(1) spin liquids. This interaction shifts the two-spinon continuum up in energy and leads to the appearance of the collective spin-1 mode in the transverse dynamic susceptibility at small momenta. This general effect is checked by detailed analytical and numerical calculations for the best-understood spin liquid — the spin-1/2 magnetized Heisenberg chain. We show that antiferromagnetic next-nearest neighbor exchange interaction can be used to tune the spin chain between the interacting spinon liquid and non-interacting spinon gas regimes at a small magnetic field. We describe how the Dzyaloshinskii-Moriya interaction can be used to detect the interaction-induced splitting of transverse spin modes at small momentum in the ESR experiments.