Rigidity percolation by next-nearest-neighbor braces on generic and regular isostatic lattices

TL;DR

This paper investigates how adding next-nearest-neighbor bonds affects the rigidity transition in 2D isostatic lattices, revealing different behaviors in regular versus generic lattices and providing both numerical and analytical insights.

Contribution

It distinguishes the rigidity percolation transitions in regular and generic lattices, showing different critical points and transition types, with new analytical models for each case.

Findings

Regular lattices show a transition at ~L ln L braces with mixed transition features.

Generic lattices transition near L braces, exhibiting a sharp first-order-like transition.

The study develops analytical theories matching numerical simulation results.

Abstract

We study rigidity percolation transitions in two-dimensional central-force isostatic lattices, including the square and the kagome lattices, as next-nearest-neighbor bonds ("braces") are randomly added to the system. In particular, we focus on the differences between regular lattices, which are perfectly periodic, and generic lattices with the same topology of bonds but whose sites are at random positions in space. We find that the regular square and kagome lattices exhibit a rigidity percolation transition when the number of braces is $\sim L\ln L$, where $L$ is the linear size of the lattice. This transition exhibits features of both first order and second order transitions: the whole lattice becomes rigid at the transition, whereas there exists a diverging length scale. In contrast, we find that the rigidity percolation transition in the generic lattices occur when the number of braces is very close to the number obtained from the Maxwell's law for floppy modes, which is $\sim L$. The transition in generic lattices is a very sharp first-order-like transition, at which the addition of one brace connects all small rigid regions in the bulk of the lattice, leaving only floppy modes on the edge. We characterize these transitions using numerical simulations and develop analytic theories capturing each transition. Our results relate to other interesting problems including jamming and bootstrap percolation.