Euclidean random matrices, the glass transition and the Boson peak

TL;DR

This paper explores how Euclidean random matrices relate to the glass transition and Boson peak, revealing a phase transition that explains features observed in glasses at low temperatures.

Contribution

It connects the phonon-saddle phase transition in Euclidean matrices to the dynamical transition in glasses, providing a new theoretical perspective.

Findings

Identifies a phase transition in Euclidean matrices between phonon and saddle phases.

Links the glass dynamical transition to the phonon-saddle transition in the Hessian.

Suggests the Boson peak is a remnant of this phase transition.

Abstract

In this paper I will describe some results that have been recently obtained in the study of random Euclidean matrices, i.e. matrices that are functions of random points in Euclidean space. In the case of translation invariant matrices one generically finds a phase transition between a phonon phase and a saddle phase. If we apply these considerations to the study of the Hessian of the Hamiltonian of the particles of a fluid, we find that this phonon-saddle transition corresponds to the dynamical phase transition in glasses, that has been studied in the framework of the mode coupling approximation. The Boson peak observed in glasses at low temperature is a remanent of this transition.