Folding of the triangular lattice in a discrete three-dimensional space: Density-matrix-renormalization-group study

TL;DR

This study uses density-matrix renormalization group methods to analyze the discrete folding of a triangular lattice in three dimensions, revealing a strong, discontinuous crumpling transition with larger latent heat than previous estimates.

Contribution

It introduces a novel application of DMRG to treat larger system widths, providing more accurate insights into the crumpling transition of discretely folded membranes.

Findings

Discontinuous crumpling transition with large latent heat.

Transition is more pronounced with higher embedding dimensions.

Folding entropy aligns with previous analytical bounds.

Abstract

Folding of the triangular lattice in a discrete three-dimensional space is investigated numerically. Such ``discrete folding'' has come under through theoretical investigation, since Bowick and co-worker introduced it as a simplified model for the crumpling of the phantom polymerized membranes. So far, it has been analyzed with the hexagon approximation of the cluster variation method (CVM). However, the possible systematic error of the approximation was not fully estimated; in fact, it has been known that the transfer-matrix calculation is limited in the tractable strip widths L \le 6. Aiming to surmount this limitation, we utilized the density-matrix renormalization group. Thereby, we succeeded in treating strip widths up to L=29 which admit reliable extrapolations to the thermodynamic limit. Our data indicate an onset of a discontinuous crumpling transition with the latent heat substantially larger than the CVM estimate. It is even larger than the latent heat of the planar (two dimensional) folding, as first noticed by the preceding CVM study. That is, contrary to our naive expectation, the discontinuous character of the transition is even promoted by the enlargement of the embedding-space dimensions. We also calculated the folding entropy, which appears to lie within the best analytical bound obtained previously via combinatorics arguments.