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We report simulation results of skyrmions on fluctuating 2D lattices, wherethe vertices ${\bf r}_i (\in {\bf R}^3)$ are treated as a dynamical variableand, hence, there is no crystalline structure. On the fluctuating surfaces, anexternal magnetic field perpendicular to the surface, Dzyaloshinskii-Moriya andferromagnetic interactions are assumed in addition to the Helfrich-PolyakovHamiltonian for membranes. The surface (or frame) tension $\tau$ is calculatedunder both isotropic and uniaxial strain conditions, and this calculationclarifies a non-trivial dependence of $\tau$ on the skyrmion, stripe, andferromagnetic phases. We find that the variation of $\tau$ with respect to theapplied magnetic field in the skyrmion phase is accompanied by a variation ofthe total number of skyrmions. Moreover, we find that this total numbervariation is qualitatively consistent with a recent experimental result for thecreation/annihilation of skyrmions of 3D crystalline material under uniaxialstress conditions. It is also found that the stripe phase is significantlyinfluenced by uniaxial strains, while the skyrmion phase remains unchanged.These results allow us to conclude that the skyrmion phase is stable even onfluctuating surfaces.

Skyrmions on 2D Elastic Surfaces with Fixed Boundary Frame