Emergent elasticity of skyrmion crystal in chiral magnets: the constitutive equations

Here we derive the linear constitutive equations governing the emergent elastic behavior of skX under applied mechanical forces as well as other effective fields. The linear coefficients appearing in the equations are expressed analytically in terms of the thermodynamic parameters describing the magnetoelastic interaction of the underlying chiral magnet. Through the deduction of equations, we clarify that two types of coupled deformation may occur for skX, that is the internal deformation, described by variation of the Fourier magnitudes, and the lattice deformation, described by the emergent elastic strains. We also find that besides mechanical loads, bias magnetic field, spatially periodic magnetic field, spatially periodic mechanical loads, and emergent stresses can all lead to deformation of the skX. When homogeneous stresses are applied only, the emergent elastic strains are linearly related to the elastic strains through the emergent strain ratio matrix {\lambda}. We calculate {\lambda} for the skX in bulk MnSi and discuss its variation with the temperature and magnetic field. In particular, we find that all components of {\lambda} are sensitive to variation of magnetic field b, such that they change sign as b increases. Moreover, as b approaches the critical magnetic field of the skX-ferromagnetic phase transition, the dominant components of {\lambda} diverges. The constitutive equations are of fundamental significance in emergent elasticity, a new area studying the variation of internal forces and deformation of emergent crystals and their relation when exposed to effective external fields.