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 -infinity <K<infinity. 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 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 the 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 mean-field level.