Folding of the triangular lattice in a discrete three-dimensional space: Crumpling transitions in the negative-bending-rigidity regime

Abstract

Folding of the triangular lattice in a discrete three-dimensional space is studied numerically.Such "discrete folding" was introduced by Bowick and co-workers as a simplified version of the polymerized membrane in thermal equilibrium. According to their cluster-variation method (CVM) analysis, there appear various types of phases as the bending rigidity K changes in the range −∞ < K < ∞. In this paper, we investigate the K < 0 regime, for which the CVM analysis with the single-hexagon-cluster approximation predicts two types of (crumpling) transitions of both continuous and discontinuous characters. We diagonalized the transfer matrix for the strip widths up to L = 26 with the aid of the density-matrix renormalization group. Thereby, we found that discontinuous transitions occur successively at K = −0.76(1) and −0.32(1). Actually, these transitions are accompanied with distinct hysteresis effects. On the contrary, the latent-heat releases are suppressed considerably as Q = 0.03(2) and 0.04(2) for respective transitions. These results indicate that the singularity of crumpling transition can turn into a weak-first-order type by appreciating the fluctuations beyond a meanfield level.

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

“…(1) and (2). Making use of K 3 = −0.76(1) [33] and the present result K = 0.270(2), we arrive at K 3 /K = 2.815(43)(∼ 3). The relation (1) appears to hold satisfactorily; a slight deviation indicates that the truncation of the configuration space is not exactly validated.…”

“…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.…”

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

“…(A.1), is too restrictive to adopt the periodic-boundary condition. So far, the numerical simulation has been performed under the open-boundary condition [26,28,32,33].…”

“…A peculiarity of this Ising magnet is that the spin variables are subjected to a local constraint (folding rule), which is incompatible with the periodic-boundary condition. Because of this difficulty, the open-boundary condition has been implemented so far [26,28,32,33]. With the full diagonalization method, the finite clusters with the sizes L ≤ 6 were considered [26,28].…”

Folding transitions in three-dimensional space with defects

“…A sequence of transitions from the flat to the piled-up phase through the partially folded octahedral and tetrahedral phases was found by varying the bending rigidity from ∞ to −∞. The density-matrix renormalization group (DMRG) calculations of [33,34] have confirmed the occurrence of these transitions. Both CVM and DMRG have predicted first-order flat-octahedral and octahedral-tetrahedral transitions while the character of the third transition is controversial.…”