**Abstract: **

Quasiparticles of the Heisenberg spin-1/2 chain — spinons — represent the best experimentally accessible example of fractionalized excitations known to date. Dynamic spin response of the spin chain is typically dominated by the broad multi-spinon continuum that often masks subtle features, such as continuum edge singularities, induced by the interaction between spinons. This, however, is not the case in the small momentum region of the magnetized spin chain where strong interaction between spinons leads to qualitative changes to the response.

I present hydrodynamic theory of the dynamic spin susceptibility of the antiferromagnetic spin-1/2 XXZ Heisenberg chain in external magnetic field. The theory is based on equations of motion for magnetization densities and currents that follow from Kac-Moody algebra of spin current operators. We find that backscattering interaction between spinons strongly affects dynamic response and produces finite energy splitting between optical branches of excitations at small momenta.

Next, I describe experimental verification of our findings by the electron spin resonance (ESR) experiments on a model material K2CuSO4Br2. Here we exploit the unique feature of the material — the uniform DM interaction between chain’s spins — in order to access small momentum regime of the dynamic spin susceptibility. By measuring interaction-induced splitting between the two components of the ESR doublet we directly determine the magnitude of the marginally irrelevant backscattering interaction between spinons for the first time ever.

I conclude by outlining another, transport, probe of spinon dynamics via a spin Seebeck effect and describe inadequacies of the previous theoretical approximations.

Overall, our results emphasize the importance of interactions between spinons for the quantitative description of the dynamic response of the spin liquid and point out an intriguing similarity between the one-dimensional interacting liquid of neutral spinons and the Landau Fermi liquid of electrons.