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Magnetic emergent crystals are periodic alignment of "particle-like" spintextures that emerge in magnets. Instead of focusing on an individual spin or amacroscopic magnetization field, we analyze the dynamical behaviors of thesenovel states by taking a solid-state point of view. Based on variationalprinciples, the basic equations for lattice dynamics of any emergentcrystalline states appearing in magnetic materials is established. For smallamplitude emergent elastic waves propagating in emergent crystals, the basicequations reduce to an eigenvalue problem, from which the dispersion relationand vibrational patterns for all emergent phonons are determined at the longwavelength limit.

Lagrangian formulation for emergent elastic waves in magnetic emergent crystals