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

TL;DR

This study numerically investigates the crumpling transitions of a discrete triangular lattice in three-dimensional space, revealing successive weak-first-order transitions in the negative-bending-rigidity regime with fluctuation effects beyond mean-field predictions.

Contribution

It provides the first numerical confirmation of multiple crumpling transitions, including weak-first-order types, in the negative-bending-rigidity regime of discrete lattice folding.

Findings

Discontinuous transitions at K=-0.76 and -0.32 with hysteresis.

Suppressed latent heat indicating weak-first-order transitions.

Fluctuations beyond mean-field theory influence transition nature.

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 -infty < K < infty. 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.