Symmetry-Driven Multiferroic Altermagnetism in Two-Dimensional Materials

Altermagnetism, characterized by momentum-dependent spin polarization in collinear antiparallel spins with vanishing net magnetization, represents a distinct magnetic phase beyond conventional ferromagnetic and antiferromagnetic classifications. This phenomenon arises from unique spin group symmetries that decouple spin and spatial degrees of freedom, enabling nonrelativistic spin-split electronic bands. Integrating this phenomenon with multiferroicity in two-dimensional (2D) materials offers unprecedented opportunities for quantum state manipulation. However, a unified theoretical framework for such multifunctional materials remains underdeveloped. Here, we establish a symmetry-driven framework identifying four point group species (¹4̅²2²mF²m²m¹2, ²4̅¹2²mF²m²m¹2, ²4̅F²2, and ²2²2¹2F²2) that can simultaneously host altermagnetism, ferroelasticity, and out-of-plane ferroelectricity, termed altriferroicity. First-principles calculations further validate this framework in Fe₂WS₂Se₂ and half-fluorinated Cr-based metal-organic frameworks, revealing robust spin-lattice-charge coupling. Our work establishes symmetry-guided design as a powerful approach for unlocking emergent quantum phenomena in 2D materials for spintronic and valleytronic applications.