Stability of the spiral spin liquid in MnSc$_2$S$_4$

We investigate the stability of the spiral spin liquid phase in MnSc$_2$S$_4$ against thermal and quantum fluctuations as well as against perturbing effects of longer-range interactions. Employing $ab~initio$ density functional theory (DFT) calculations we propose a realistic Hamiltonian for MnSc$_2$S$_4$, featuring second ($J_2$) and third $(J_3)$ neighbor Heisenberg interactions on the diamond lattice that are considerably larger than previously assumed. We argue that the combination of strong $J_2$ and $J_3$ couplings reproduces the correct magnetic Bragg peak position measured experimentally. Calculating the spin-structure factor within the pseudofermion functional-renormalization group technique we find that close to the magnetic phase transition the sizable $J_3$ couplings induce a strong spiral selection effect, in agreement with experiments. With increasing temperature the spiral selection becomes weaker such that in a window around three to five times the ordering temperature an approximate spiral spin liquid is realized in MnSc$_2$S$_4$.