Field Theory for Amorphous Solids

Glasses at low temperature fluctuate around their inherent states; glassy anomalies reflect the structure of these states. Recently, there have been numerous observations of long-range stress correlations in glassy materials, from supercooled liquids to colloids and granular materials, but without a common explanation. Herein it is shown, using a field theory of inherent states, that long-range stress correlations follow from mechanical equilibrium alone, with explicit predictions for stress correlations in two and three dimensions. “Equations of state” relating fluctuations to imposed stresses are derived, as well as field equations that fix the spatial structure of stresses in arbitrary geometries. Finally, a new holographic quantity in 3D amorphous systems is identified.

Figure 1

Illustration of response to a localized force dipole in a model glass, courtesy of Lerner and Bouchbinder [21]. Plotted on the right is $|\overrightarrow{r}||\overrightarrow{u}|$, where $\overrightarrow{u}$ is the displacement response decaying as $\sim 1/|\overrightarrow{r}|$, where $|\overrightarrow{r}|$ is the distance to the source. Beyond a few particle diameters, the response is well represented by a disordered continuum field [22]. In field theory, the glass (left) is replaced by a continuum, whose structure is characterized by one or several smooth structural fields.