Crumpling transition of the triangular lattice without open edges: Effect of a modified folding rule

Abstract

Folding of the triangular lattice in a discrete three-dimensional space is investigated by means of the transfer-matrix method. This model was introduced by Bowick and co-workers as a discretized version of the polymerized membrane in thermal equilibrium. The folding rule (constraint) is incompatible with the periodic-boundary condition, and the simulation has been made under the open-boundary condition. In this paper, we propose a modified constraint, which is compatible with the periodic-boundary condition; technically, the restoration of translational invariance leads to a substantial reduction in the transfer-matrix size. Treating the cluster sizes L<or=7, we analyze the singularities of the crumpling transitions for a wide range of the bending rigidity K. We observe a series of the crumpling transitions at K=0.206(2), -0.32(1), and -0.76(10). At each transition point, we estimate the latent heat as Q=0.356(30), 0.08(3), and 0.05(5), respectively.

Folding transitions in three-dimensional space with defects

“…In order to further analyze the character of the crumpling transitions of the triangular network in the fcc lattice, a modified folding rule allowing defects in the network has been introduced in [35]. Transfer-matrix calculations of [35] confirmed the order of the transitions given by DMRG.…”

Mathematical modeling and simulations for large-strain J-shaped diagrams of soft biological tissues

“…We comment on the reason why we use the FG model instead of standard or canonical models, which are defined without the variable σ [30][31][32][33][34]. The problem is whether or not the results of the canonical 2D and 3D models are identical to those of the FG models.…”

Crumpling transition of the discrete planar folding in the negative-bending-rigidity regime

“…In this paper, we utilized the transfer-matrix method [27] for the system sizes L ≤ 14. We implemented a modified folding rule [29], Eq. (5), which enables us to impose the periodicboundary condition.…”

“…Here, the variables Q and Q 3 denote the latent heat for the planar-and three-dimensional-2 folding models, respectively. A number of results, (K 3 , Q 3 ) = (−0.852, 0) [32], (−0.76(1), 0.03(2)) [33], and (−0.76(10), 0.05(5)) [29], have been obtained via the CVM, density-matrix renormalization group, and exact-diagonalization analyses, respectively. The nature of its transition at K 3 ≈ −0.8 is not fully clarified, because the three-dimensional folding is computationally demanding.…”

“…In this paper, following Ref. [29], we make a modification as to the constraint (prefactor 3 of Eq. (3)) to surmount the difficulty.…”