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A fundamental fact in solids is that the frequencies of elastic waves vanishas the wave number approaches zero\cite{6}. Here we theoretically show thatthis fact is overturned when studying the lattice vibration of skyrmioncrystals (SkX), i.e., periodic alignment of topologically nontrivial spinsolitons called magnetic skyrmions. As emergent crystals, SkX possesscollective excitations called "emergent phonons", which describe dynamics ofSkX caused by lattice vibration (resembling acoustical branches of ordinaryphonons) and in-lattice vibration (resembling optical branches of ordinaryphonons). We find that lattice vibration and in-lattice vibration of theemergent phonons in SkX are coupled even at long wavelength limit, such thatmultiple types of "emergent elastic waves" (modes causing lattice vibration ofSkX) with finite frequencies exist. This phenomenon, which originates from theBerry phase form of kinetic energy, is generally true for emergent crystallinestates of spins. Our results show that the dynamics of magnetic emergentcrystals are intrinsically different from that of ordinary crystals.

Emergent elastic waves in skyrmion crystals with finite frequencies at long wavelength limit