The amorphous solid state: a locally stable thermodynamic phase of randomly constrained systems

TL;DR

This paper demonstrates that the amorphous solid state in randomly constrained systems is locally stable near the phase transition, with a positive Hessian eigenvalue spectrum except for a zero mode due to broken translational symmetry.

Contribution

It provides a detailed stability analysis of the amorphous solid state, confirming its local stability through Hessian eigenvalue examination in a class of systems undergoing a liquid to amorphous-solid transition.

Findings

Hessian eigenvalues are positive near the transition.

A zero eigenvalue corresponds to broken translational symmetry.

The amorphous solid state is confirmed to be locally stable.

Abstract

The question of the local stability of the (replica-symmetric) amorphous solid state is addressed for a class of systems undergoing a continuous liquid to amorphous-solid phase transition driven by the effect of random constraints. The Hessian matrix, associated with infinitesimal fluctuations around the stationary point corresponding to the amorphous solid state, is obtained. The eigenvalues of this Hessian matrix are all shown to be strictly positive near the transition, except for one--the zero mode associated with the spontaneously broken continuous translational symmetry of the system. Thus the local stability of the amorphous solid state is established.