Emergence of zero modes in disordered solids under periodic tiling

TL;DR

This paper investigates how periodic boundary conditions in computational models can fail to accurately represent the rigidity and energy minima of infinitely repeated structures, revealing the emergence of zero modes in disordered solids.

Contribution

It provides a theoretical proof showing the limitations of periodic boundary conditions in capturing the true rigidity of repeated structures and discusses the implications for modeling disordered solids.

Findings

Periodic boundary conditions may not always reflect the rigidity of infinite structures.

Some jammed packings under periodic conditions are not at true energy minima.

Zero modes can emerge in disordered solids due to boundary condition effects.

Abstract

In computational models of particle packings with periodic boundary conditions, it is assumed that the packing is attached to exact copies of itself in all possible directions. The periodicity of the boundary then requires that all of the particles' images move together. An infinitely repeated structure, on the other hand, does not necessarily have this constraint. As a consequence, a jammed packing (or a rigid elastic network) under periodic boundary conditions may have a corresponding infinitely repeated lattice representation that is not rigid or indeed may not even be at a local energy minimum. In this manuscript, we prove this claim and discuss ways in which periodic boundary conditions succeed to capture the physics of repeated structures and where they fall short.