Glass phenomenology in the hard matrix model

Abstract

We introduce a new toy model for the study of glasses: the hardmatrix model (HMM). This may be viewed as a single particle moving on SO(N ), where there is a potential proportional to the 1-norm of the matrix. The ground states of the model are "crystals" where all matrix elements have the same magnitude. These are the Hadamard matrices when N is divisible by four. Just as finding the latter has been a difficult challenge for mathematicians, our model fails to find them upon cooling and instead shows all the behaviors that characterize physical glasses. With simulations we have located the first-order crystallization temperature, the Kauzmann temperature where the glass has the same entropy as the crystal, as well as the standard, measurement-time dependent glass transition temperature. A new feature in this glass model is a "disorder parameter" ρ 0 that is zero for any of the crystal phases and in the liquid/glass, where it is non-zero, corresponds to the density of matrix elements at the maximum in their contribution to the energy. We conclude with speculation on how a quantum extension of the HMM, with the backdrop of current work on many-body localization, might advance the understanding of glassy dynamics.Much of statistical mechanics is the study of "toy models," minimalistic distillations of physical systems that capture particular phenomena. The simplest model of liquids, mono-disperse hard spheres, is also much used as a model of glassy behavior. In three dimensions, and when compressed rapidly, this system produces jammed structures with a reproducible packing