Title: Waveguide QED at the onset of spin-spin correlations

Abstract: We experimentally explore the competition between light-mediated and direct matter-matter interactions in waveguide quantum electrodynamics. For this, we couple a superconducting line to a model magnetic material, made of organic free radical DPPH molecules with a spin $S=1/2$ and a $g_{S}$ factor very close to that of a free electron. The microwave transmission has been measured in a wide range of temperatures ($0.013$ K $\leq T \leq 2$ K), magnetic fields ($0\leq B \leq 0.5$ T) and frequencies ($0 \leq \omega/2 \pi \leq 14$ GHz). We find that molecules belonging to the crystal sublattice B form one-dimensional spin chains. Temperature then controls intrinsic spin correlations along the chain in a continuous and monotonic way. In the paramagnetic region ($T > 0.7$ K), the microwave transmission shows evidences for the collective coupling of quasi-identical spins to the propagating photons, with coupling strengths that reach values close to the dissipation rates. As $T$ decreases, the growth of intrinsic spin correlations, combined with the anisotropy in the spin-spin exchange constants, break down the collective spin-photon coupling. In this regime, the temperature dependence of the spin resonance visibility reflects the change in the nature of the dominant spin excitations, from single spin flips to bosonic magnons, which is brought about by the magnetic correlation growth.