Goldstone-type fluctuations and their implications for the amorphous solid state

In sufficiently high spatial dimensions, the formation of the amorphous (i.e., random) solid state of matter, e.g., upon sufficent crosslinking of a macromolecular fluid, involves particle localization and, concomitantly, the spontaneous breakdown of the (global, continuous) symmetry of translations. Correspondingly, the state supports Goldstone-type low energy, long wavelength fluctuations, the structure and implications of which are identified and explored from the perspective of an appropriate replica field theory. In terms of this replica perspective, the lost symmetry is that of relative translations of the replicas; common translations remain as intact symmetries, reflecting the statistical homogeneity of the amorphous solid state. What emerges is a picture of the Goldstone-type fluctuations of the amorphous solid state as shear deformations of an elastic medium, along with a derivation of the shear modulus and the elastic free energy of the state. The consequences of these fluctuations—which dominate deep inside the amorphous solid state—for the order parameter of the amorphous solid state are ascertained and interpreted in terms of their impact on the statistical distribution of localization lengths, a central diagnostic of the state. The correlations of these order parameter fluctuations are also determined, and are shown to contain information concerning further diagnostics of the amorphous solid state, such as spatial correlations in the statistics of the localization characteristics. Special attention is paid to the properties of the amorphous solid state in two spatial dimensions, for which it is shown that Goldstone-type fluctuations destroy particle localization, the order parameter is driven to zero, and power-law order-parameter correlations hold.

Figure 1

A classical state in replicated real space: a hill in $x$ space with its ridge aligned in the $x_{\lambda }$ direction and passing through the origin. The thickness of the lines is intended to suggest the amplitude of the order parameter, the thicker the line the larger the amplitude. Symmetry-related classical states follow from rigid displacements of the hill perpendicular to the ridge, i.e., in the $x_{\tau }$ direction.

Figure 2

Goldstone-distorted state in replicated real space: the hill is displaced perpendicular to the ridge to an extent $u_{\tau }$ that varies with position $x_{\lambda }$ along the ridge. Note that the scale of this figure is much larger than that used for Fig. 1: the thick line lies along the ridge of the classical state, but now it is the width of the line that indicates the width of the hill ${\xi }_{\mathrm{typ}}$. The Goldstone-type fluctuations occur on wavelengths longer than ${\xi }_{\mathrm{typ}}$.

Figure 3

Molecular bound state view of the classical and Goldstone-distorted states in replicated real space. The full circles within a border represent the replicas of a given monomer location. Repetitions of them along a row indicate that the center of mass of the bound states is distributed homogeneously. Upper bars, one classical state; the probability density is peaked at the shown configurations, in a manner independent of the location of the center of mass of the bound state of the replicated particles. Middle bars, another classical state, obtained by the former one via a relative translation of the replicas that does not vary with the location of the center of mass. Lower bars, a Goldstone-distorted state; the probability density is peaked in a manner that varies with the location of the center of mass.