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

Abstract

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