Symmetry Classification of Altermagnetism and Emergence of Type-IV Magnetism in Two Dimensions

Two-dimensional (2D) magnetism, particularly 2D altermagnetism (AM), has attracted considerable interest due to its exceptional physical properties and broad application potential. However, the classification of AM undergoes a fundamental paradigm shift when transitioning from three-dimensional (3D) to 2D symmetry-enforced fully compensated collinear magnetism$-$a shift that has remained largely overlooked. Here, by extending unconventional magnetism to 2D collinear systems, we identify the symmetry conditions and electronic band characteristics of a distinct magnetic phase: type-IV magnetism. This new class lies beyond the established descriptions of ferromagnetism, conventional antiferromagnetism, and AM. Type-IV magnetism supports the successive emergence of both nonrelativistic spin-degenerate and relativistic spin-splitting phenomena, belonging strictly to neither conventional antiferromagnetism nor standard AM. We further establish a universal symmetry classification framework for 2D type-IV magnets via a mapping from the collinear spin layer group to the magnetic layer group. Monolayer MgCr$_2$O$_3$ and monolayer BaMn$_2$Ch$_3$ (Ch=Se, Te) are showcased as representative materials, exhibiting gate-tunable reversible spin textures and the quantum electric Hall effect, respectively. Our work underscores the rich functional prospects of type-IV magnets, offering a new route toward spin manipulation and anomalous transport that promises innovative designs for high-performance spintronic devices.