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Internal phase transition induced by external forces in Finsler geometric model for membranes

We numerically study an anisotropic shape transformation of membranes under external forces for two-dimensional triangulated surfaces on the basis of Finsler geometry. The Finsler metric is defined by using a vector field, which is the tangential component of a three dimensional unit vector $σ$ corresponding to the tilt or some external macromolecules on the surface of disk topology. The sigma model Hamiltonian is assumed for the tangential component of $σ$ with the interaction coefficient $λ$. For large (small) $λ$, the surface becomes oblong (collapsed) at relatively small bending rigidity. For the intermediate $λ$, the surface becomes planar. Conversely, fixing the surface with the boundary of area $A$ or with the two point boundaries of distance $L$, we find that the variable $σ$ changes from random to aligned state with increasing of $A$ or $L$ for the intermediate region of $λ$. This implies that an internal phase transition for $σ$ is triggered not only by the thermal fluctuations but also by external mechanical forces. We also find that the frame (string) tension shows the expected scaling behavior with respect to $A/N$ ($L/N$) at the intermediate region of $A$ ($L$) where the $σ$ configuration changes between the disordered and ordered phases. Moreover, we find that the string tension $γ$ at sufficiently large $λ$ is considerably smaller than that at small $λ$. This phenomenon resembles the so-called soft-elasticity in the liquid crystal elastomer, which is deformed by small external tensile forces.