Phase transitions of a tethered membrane model with intrinsic curvature on spherical surfaces with point boundaries

Abstract

Abstract. We found that the order for the crumpling transition of an intrinsic curvature model changes depending on the distance between two boundary vertices fixed on the surface of spherical topology. The model is a curvature one governed by an intrinsic curvature energy, which is defined on triangulated surfaces. It was already reported that the model undergoes a first-order crumpling transition without the boundary conditions on the surface. However, the dependence of the transition on such boundary condition is yet to be studied. We have studied in this paper this problem by using the Monte Carlo simulations on surfaces up to a size N = 8412. The first-order transition changes to a second-order one if the distance increases.

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

“…It is also expected that the smooth phase is separated from a non-smooth phase by the first-order transition, where the surface is almost linear and is very similar to the branched polymer surface [47]. We also confirmed in [43] that the order of the transition changes from first-order to second-order on tethered surfaces as the distance L(N) increases. Therefore, we aimed in this paper to see how the transition changes depending on L(N) on dynamically triangulated fluid surfaces, where the surfaces are expected to be oblong at sufficiently large L(N).…”

“…Another intrinsic curvature model undergoes a first-order transition, which is independent of the surface topology [40,41,42]. It was also reported that the order of transition changes from first to second as L increases from L = L 0 to L = 2L 0 , where L 0 is the radius of the original sphere constructed to satisfy the relation S 1 /N = 1.5 [43]. However, the string tension was unable to be extracted from the simulations on the tethered surfaces, because the surfaces do not always become string-like even when L is relatively large compared to L 0 .…”

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

“…The model is defined by a Hamiltonian that is a linear combination of the Gaussian bond potential S 1 and the intrinsic curvature energy S 2 . The model is known to have first-order transitions on the surfaces without boundaries and on those with point boundaries [18,31]. However, it is non-trivial to establish whether the model undergoes a phase transition with the 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 [29,30,31,32].…”

“…The area πRL of the tube can also be used for the calculation of τ ; however, we use the projected area L 2 for simplicity. We should emphasize that it is non-trivial to establish whether the model undergoes a phase transition because the surface is fixed by one-dimensional boundaries, which include many vertices, and the size of the boundaries increases with the system size, and therefore, the phase structure of the model without boundaries [18] or with point boundaries [31] is expected to be influenced by the one-dimensional boundaries. The purpose of the study is to see whether or not the model undergoes phase transitions, and moreover, to see how the transition is reflected in the surface tension if the transition occurs in the model.…”