Phase transitions of an intrinsic curvature model on dynamically triangulated spherical surfaces with point boundaries

Abstract

An intrinsic curvature model is investigated using the canonical Monte Carlo simulations on dynamically triangulated spherical surfaces of size upto N = 4842 with two fixed-vertices separated by the distance 2L. We found a first-order transition at finite curvature coefficient α, and moreover that the order of the transition remains unchanged even when L is enlarged such that the surfaces become sufficiently oblong. This is in sharp contrast to the known results of the same model on tethered surfaces, where the transition weakens to a second-order one as L is increased. The phase transition of the model in this paper separates the smooth phase from the crumpled phase. The surfaces become string-like between two point-boundaries in the crumpled phase. On the contrary, we can see a spherical lump on the oblong surfaces in the smooth phase. The string tension was calculated and was found to have a jump at the transition point. The value of σ is independent of L in the smooth phase, while it increases with increasing L in the crumpled phase. This behavior of σ is consistent with the observed scaling relation σ ∼ (2L/N ) ν , where ν ≃ 0 in the smooth phase, and ν = 0.93 ± 0.14 in the crumpled phase. We should note that a possibility of a continuous transition is not completely eliminated.

Surface tension in an intrinsic curvature model with fixed one-dimensional boundaries

“…This model leads us to calculate the macroscopic string tension σ of the surface by equating exp(−σL) ∼ Z at sufficiently large L, where Z is the partition function of the surface model [27,28]. The expected scaling relation of σ with respect to N is observed, where N is the total number of vertices of the triangulated surface [29]- [32].…”