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 $\sigma$ 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 $\sigma$ with the interaction coefficient $\lambda$. For large (small) $\lambda$, the surface becomes oblong (collapsed) at relatively small bending rigidity. For the intermediate $\lambda$, 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 $\sigma$ changes from random to aligned state with increasing of $A$ or $L$ for the intermediate region of $\lambda$. This implies that an internal phase transition for $\sigma$ 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 $\sigma$ configuration changes between the disordered and ordered phases. Moreover, we find that the string tension $\gamma$ at sufficiently large $\lambda$ is considerably smaller than that at small $\lambda$. This phenomenon resembles the so-called soft-elasticity in the liquid crystal elastomer, which is deformed by small external tensile forces.