Magnetic symmetry of low-dimensional multiferroics and ferroelastics

Ferroics are materials with two or more orientation states with the ability to switch between them via an applied field. Based on spatial inversion and time reversal symmetry, there are four types of primary ferroics: ferroelectrics described by polarization P (a polar vector with broken spatial inversion symmetry), ferromagnets described by magnetization M (an axial vector with broken time reversal symmetry), ferrotoroidics described by torodization T (an axio-polar vector with both spatial inversion and time reversal symmetries broken) and ferroelastics described by strain (a symmetric second rank polar tensor with neither spatial inversion nor time reversal symmetry broken but with broken rotational symmetry). Crystals exhibiting simultaneous ferroelectricity and magnetism are called magnetoelectrics. We analyze the magnetic symmetry of one- and two-dimensional (1D and 2D) multiferroics and enumerate 1D and 2D magnetoelectric point groups. We study a representative phase transition to such a point group and suggest possible composition of magnetoelectric chains and planar multiferroics. We also enumerate all ferroelastic species for the 2D magnetic groups.