Lagrangian formulation for emergent elastic waves in magnetic emergent crystals

Magnetic emergent crystals are periodic alignment of "particle-like" spin textures that emerge in magnets. Instead of focusing on an individual spin or a macroscopic magnetization field, we analyze the dynamical behaviors of these novel states by taking a solid-state point of view. Based on variational principles, the basic equations for lattice dynamics of any emergent crystalline states appearing in magnetic materials is established. For small amplitude emergent elastic waves propagating in emergent crystals, the basic equations reduce to an eigenvalue problem, from which the dispersion relation and vibrational patterns for all emergent phonons are determined at the long wavelength limit.