3 FIG. 1: Illustration of the notion of an aperiodic metastable state as a dilated crystal that was allowed to relax. a standard fashion, the surface tension implies that the dependence of the free energy on the extent of dilation exhibits a non-concave portion. 20 (By “concave,” we will consistently mean concave- up .) We can see qualitatively how the non-concavity arises by estimating the free energy of the solid using a simple non-linear Einstein-oscillator model. Suppose the configurational partition function of each oscilla- tor is ∫ ∞ g d $^{3}$r e − kr $^{2}$/ 2 k$_{B}$$^{T}$, where we incorporate the non- linearity qualitatively by assuming a hard lower cut-off in the configurational integral, at a displacement g . k is the force constant in the harmonic limit. Denote the force constants in the crystal and the expanded state as k$_{X}$ and k$_{G}$ respectively. Clearly, k$_{G}$ < k$_{X}$ . We sketch the free energies of the states as functions of g in Fig.1, taking into account that the curve corresponding to the expanded state is shifted upward from the crystal state by the free energy cost of expansion. According to Fig.1, under sufficient negative pressure (essentially equal to the negative slope of the common tangent of the two curves), the crystal will indeed begin converting into a collection of non-linear oscillators. The vast majority of the configura- tions of these oscillators are of course aperiodic, implying the metastable phase is aperiodic too. This qualitative free energy graph tentatively confirms the view above of a quenched-melt/frozen-glass as a sufficiently expanded crystal that was allowed to relax. The graph also sug- gests an appropriate order parameter for the transition, namely the amplitude of short-wavelength, strongly an- harmonic motions. In writing a formal ansatz for a solid that allows short length anharmonic motions, we first point out that local mechanical instabilities are perfectly compat- ible even with macroscopic rigidity, examples includ- ing orientational glasses 21 and ferroelectrics of order- disorder type. 22–24 The latter systems consist of rigid, periodic scaffolds where a subset of local motions could be marginally stable (resulting in a vanishing frequency of transverse optical phonons $^{22}$) or even multistable. In contrast, we wish to consider here a situation where all short length motions are allowed to be multistable. The resulting material may or may not be macroscopically stable, as will be determined self-consistently. We will limit ourselves to a dynamical regime, in which relative distortions of individual bonds during lattice reconfigura- tions are only within 10% or so, 1,6,7 as appropriate both for activated transport and stable crystals. Such small distortions imply the harmonic approximation for local stress is semi-quantitative. Take the Landau local free en- ergy density as a function of the local deformation profile u$_{ij}$ : f = 1 $_{2}$u$_{ij}$ Λ$_{ijkl}$ u$_{kl}$ , where Λ$_{ijkl}$ is the standard elas- tic tensor and the Einstein’s summation convention for Latin indeces is implied. 25 ( u$_{ij}$ ≡ 1 $_{2}$( ∂u$_{i}$/∂x$_{j}$ + ∂u$_{j}$/∂x$_{i}$ ), i, j = 1 , 2 , 3, where u is the local displacement relative to a steady-state reference state.) Decompose the deformation tensor into long- and short- wave length components: u$_{ij}$ = u < ij + d$_{ij}$ , where u < ij only includes Fourier components with wave-vectors within a certain cut-off surface in k -space, while d$_{ij}$ con- tains only Fourier components outside of the cut-off sur- face. Let us evaluate the free energy of the system sub- ject to a constraint on the self-energy of the local motions d$_{ij}$ Λ$_{ijkl}$ d$_{kl}$ = g $^{2}$, thus forcing the short-length motions to be essentially anharmonic Einstein oscillators. If the cut- off surface in the Fourier space is generically at k ≃ k$_{micro}$ , the anharmonic oscillators fill the space, approximately one per volume ( π/k$_{micro}$ ) $^{3}$, but not necessarily in a peri- odic fashion. The detailed lattice in which the oscillators are arranged depends on the precise shape of the cut-off surface. Since the presence of strongly anharmonic de- grees of freedom generally requires expanding the lattice from the reference state, owing to steric repulsion, we must include in the full free energy of our solid a penalty for increasing g , which we denote as F$_{ex}$ ( g ). The pre- cise form of this function is not essential for the majority of our conclusions; clearly it is a monotonically increas- ing, concave function of g , to ensure the compressibility is positive. As a result, the full free energy is: where the free energy of the ansatz F$_{0}$ is computed by summing over all configurations of local stress at temper- ature T ≡ 1 /k$_{B}$β : subject to the constraint The local free energy density f ( r ) of a particular stress configuration at site r is We have dropped the superscript “ < ” from the long- wavelength, “acoustic” component of the total deforma- tion u $^{<}$, to simplify notation. For the sake of argument, F ( g ) = F$_{0}$ ( g ) + F$_{ex}$ ( g ) , (3) e − βF$_{0}$ ( g ) = ∑ { u ( r ) , d ( r ) } e − β ∫ dV f ( r $^{)}$, (4) d$_{ij}$ ( r )Λ$_{ijkl}$ d$_{kl}$ ( r ) = g $^{2}$( r ) . (5) f ( r ) = 1 2 ( u$_{ij}$ + d$_{ij}$ ) Λ$_{ijkl}$ ( u$_{kl}$ + d$_{kl}$ ) . (6) G X 2 2 k g /2 dilated aperiodic phase stable reference state F g k g /2