Absence of hyperuniformity in amorphous hard-sphere packings of nonvanishing complexity

We relate the structure factor $S(\mathbf{k} \to \mathbf{0})$ in a system of jammed hard spheres of number density $\rho$ to its complexity per particle $\Sigma(\rho)$ by the formula $S(\mathbf{k} \to \mathbf{0})=-1/ [\rho^2\Sigma''(\rho)+2\rho\Sigma'(\rho)]$. We have verified this formula for the case of jammed disks in a narrow channel, for which it is possible to find $\Sigma(\rho)$ and $S(\mathbf{k})$ analytically. Hyperuniformity, which is the vanishing of $S(\mathbf{k} \to \mathbf{0})$, will therefore not occur if the complexity is nonzero. An example is given of a jammed state of hard disks in a narrow channel which is hyperuniform when generated by dynamical rules that produce a non-extensive complexity.Comment: 5 pages, 3 figure

Understanding the ideal glass transition: Lessons from an equilibrium study of hard disks in a channel

We use an exact transfer-matrix approach to compute the equilibrium properties of a system of hard disks of diameter $\sigma$ confined to a two-dimensional channel of width $1.95\,\sigma$ at constant longitudinal applied force. At this channel width, which is sufficient for next-nearest-neighbor disks to interact, the system is known to have a great many jammed states. Our calculations show that the longitudinal force (pressure) extrapolates to infinity at a well-defined packing fraction $\phi_K$ that is less than the maximum possible $\phi_{\rm max}$, the latter corresponding to a buckled crystal. In this quasi-one-dimensional problem there is no question of there being any \emph{real} divergence of the pressure at $\phi_K$. We give arguments that this avoided phase transition is a structural feature -- the remnant in our narrow channel system of the hexatic to crystal transition -- but that it has the phenomenology of the (avoided) ideal glass transition. We identify a length scale $\tilde{\xi}_3$ as our equivalent of the penetration length for amorphous order: In the channel system, it reaches a maximum value of around $15\,\sigma$ at $\phi_K$, which is larger than the penetration lengths that have been reported for three dimensional systems. It is argued that the $\alpha$-relaxation time would appear on extrapolation to diverge in a Vogel-Fulcher manner as the packing fraction approaches $\phi_K$.Comment: 17 pages, 16 figure

Disappearance of the de Almeida-Thouless line in six dimensions

We show that the Almeida-Thouless line in Ising spin glasses vanishes when their dimension d -> 6 as h_{AT}^2/T_c^2 = C(d-6)^4(1- T/T_c)^{d/2 - 1}, where C is a constant of order unity. An equivalent result which could be checked by simulations is given for the one-dimensional Ising spin glass with long-range interactions. It is shown that replica symmetry breaking also stops as d -> 6.Comment: Additional text and one figure adde