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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 effectsof 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 onthe diamond lattice that are considerably larger than previously assumed. Weargue that the combination of strong $J_2$ and $J_3$ couplings reproduces thecorrect magnetic Bragg peak position measured experimentally. Calculating thespin-structure factor within the pseudofermion functional-renormalization grouptechnique we find that close to the magnetic phase transition the sizable $J_3$couplings induce a strong spiral selection effect, in agreement withexperiments. With increasing temperature the spiral selection becomes weakersuch that in a window around three to five times the ordering temperature anapproximate spiral spin liquid is realized in MnSc$_2$S$_4$.

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