Fig. 4: The Ginzburg–Landau model with linearly coupled structural and magnetic degrees of freedom.

a Static OLD polarization rotation \({\varphi }_{{{{{{\rm{LD}}}}}}}\) and monoclinic angle change δβ in FePS₃ as a function of temperature. The inset shows the linear relation of these two parameters. b Evolution of Δ\({\varphi }_{{{{{{\rm{LD}}}}}}}\) (i.e., \({\eta }^{2}\left(t\right)-{\eta }^{2}\left(t \, < \, 0\right)\)), Δβ (i.e., δβ − 0.1°, 0.1° is the β change across T_N.), and Δα close to T_N. Solid lines: coupled relaxation of Δ\({\varphi }_{{{{{{\rm{LD}}}}}}}\) and Δβ based on a model with a linearly coupled term (\({\eta }^{2}\delta \beta\)). Dashed lines: decoupled relaxation of Δ\({\varphi }_{{{{{{\rm{LD}}}}}}}\) and Δα based on a model with a quadratically coupled term (\({\eta }^{2}\delta {\alpha }^{2}\), see “Methods”). c The relaxation time of the interlayer shearing (\({\tau }_{\beta }\)) agrees with the numerical results based on the Ginzburg–Landau model (solid curve). d Schematic of the relaxation process of OLD polarization rotation and interlayer shear. The potential energy surface is generated from Eq. (1), shown as a function of intralayer magnetic order parameter and monoclinic angle. The system is excited from the ground state (open red circle) to the excited state (solid red dot) and relaxes to the ground state by a coupled recovery of magnetic order and monoclinic angle. The green solid dot represents the partially recovered intermediate state.