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We systematically studied the spin-lattice couplings in the CrF$_3$monolayer. Our study reveals that the spin exchange constants between thenearest neighbors are notably affected by these couplings. Specifically, thecouplings arise predominantly from three distinct phonon modes, namely thecovariant, rotation, and stretch of the Cr-F-Cr-F rhombus. By integrating outthe phonon degrees of freedom, we derived an effective spin Hamiltonianfeaturing four-spin product terms, which yields a remarkably intricate magneticphase diagram. Significantly, numerous plateau states characterized byfractional magnetizations, including 1/2, 1/3, 2/3, 1/4, 1/5, 5/8, 1/9, and2/9, emerge in the vicinity of the phase transition boundary separatingferromagnetic and antiferromagnetic states. These findings show the profoundinfluence of spin-lattice couplings on magnetic properties near the magneticphase boundaries, and the predicted plateau states are expected to beobservable in future experiments.

Spin-Lattice Coupling Induced Rich Magnetic States in CrF$_3$ monolayer