Folding of the Triangular Lattice in the FCC Lattice with Quenched Random Spontaneous Curvature

TL;DR

This paper investigates how a triangular lattice embedded in an FCC lattice folds under quenched random spontaneous curvature, revealing different folding behaviors depending on the randomness type and extending to a Hopfield-like model.

Contribution

It introduces a novel analysis of lattice folding with quenched randomness using the Cluster Variation Method and extends the model to include multiple stored configurations.

Findings

Different folding behaviors based on randomness type

Identification of phase transitions in the models

Extension to a Hopfield-like multi-configuration model

Abstract

We study the folding of the regular two-dimensional triangular lattice embedded in the regular three-dimensional Face Centered Cubic lattice, in the presence of quenched random spontaneous curvature. We consider two types of quenched randomness: (1) a ``physical'' randomness arising from a prior random folding of the lattice, creating a prefered spontaneous curvature on the bonds; (2) a simple randomness where the spontaneous curvature is chosen at random independently on each bond. We study the folding transitions of the two models within the hexagon approximation of the Cluster Variation Method. Depending on the type of randomness, the system shows different behaviors. We finally discuss a Hopfield-like model as an extension of the physical randomness problem to account for the case where several different configurations are stored in the prior pre-folding process.