Topological defects, inherent structures, and hyperuniformity

A wide spectrum of disordered many-body systems that are inherent structures (i.e., local minima of potential energy landscape) was recently discovered to possess a novel hidden long-range order known as hyperuniformity, yet the mechanisms associated with the emergence of disordered hyperuniformity in such systems are not well understood. Here, we address this important fundamental problem by linking together the concepts of topological defects, inherent structures, and hyperuniformity. We consider representative examples of disordered inherent structures that are topological variants of crystals obtained by continuously introducing into the crystalline state randomly distributed topological defects such as dislocations and disclinations. Using large-scale numerical simulations and analysis based on a continuum theory, we show that the topological defects and the associated transformation pathways linking the disordered inherent structures to the crystalline state preserve hyperuniformity of the latter, as long as the displacements induced by the defects are sufficiently localized (i.e., the volume integrals of the displacements and squared displacements caused by individual defect are finite) and the displacement-displacement correlation matrix of the system is diagonalized and isotropic. Our results provide insights to the discovery, design, and generation of novel disordered hyperuniform materials.

Figure 1

Illustration of the formation of inherent structures containing bound dislocations (top panel), free dislocations (middle panel), and disclinations with associated dislocations (bottom panel) through a series of topological transformations (i.e., rearrangement of bonding network) and subsequent structural relaxation in a triangular lattice. Vertices with seven bonds are highlighted with yellow circles, and vertices with five bonds are highlighted with green circles.

Figure 2

Left column: Illustration of representative particle configurations associated with inherent structures containing bound dislocations (top), free dislocations (middle), and disclinations (bottom). Right column, top section: Representative inherent structures containing primarily bound dislocations at defect concentration $p=0.02$, 0.06, 0.10, and 0.17, respectively. Right column, bottom-left section: Representative inherent structures containing primarily free dislocations at defect concentration $p=0.02$ and 0.04, respectively. Right column, bottom-right section: Representative inherent structures containing disclinations at defect concentration $q=0.01$ and 0.015, respectively. Vertices with more than six bonds are highlighted in yellow, and vertices with less than six bonds are highlighted in green.

Figure 3

Pair statistics associated with inherent structures containing primarily bound dislocations (top panel), inherent structures containing primarily free dislocations (middle panel), and inherent structures containing disclinations (bottom panel) at different defect concentrations $p$ or $q$ with $N=10800$ particles. Left to right, Column 1: Log-log plot of structure factor $S\left(k\right)$. Here we also include the plot for the $S\left(k\right)$ of the corresponding perfect triangular lattice as a reference, which is strictly zero for $k<K$, where $K$ is the location of the first peak. Column 2: Local number variance ${\sigma }_N^2\left(R\right)$. Column 3: Log-log plot of $|g_2\left(r\right)-1|$. Column 4: Small-wave-number scaling exponent $\alpha$ of $S\left(k\right)$, extracted as the slope of linear fittings of $lnS\left(k\right)$ vs $lnk$ for $k<0.5$.

Figure 4

(a) Bond-orientational order correlation function $C_6\left(r\right)$ and (b) order metric $Q_6$ of inherent structures containing primarily bound dislocations at different defect concentrations $p$ with $N=10800$ particles. (c) Bond-orientational order correlation function $C_6\left(r\right)$ and (d) order metric $Q_6$ of inherent structures containing primarily free dislocations at different defect concentrations $p$ with $N=10800$ particles. (e) Bond-orientational order correlation function $C_6\left(r\right)$ and (f) order metric $Q_6$ of inherent structures containing disclinations at different defect concentrations $q$ with $N=10800$ particles.

Figure 5

Displacement field $\mathbf{u}$ in an inherent structure containing a single pair of bound dislocations. (a) Spatial distribution of normalized $\left|u\left(r\right)\right|/{\left|u\left(r\right)\right|}_{max}$. (b) Decay of $\left|\mathbf{u}\right(r\left)\right|$ as a function of the distance $r$ from the core of the topological defect, i.e., the center of the old broken bond (the same as the center of the new formed bond), and the red solid line is an exponential fit (note that the vertical axis is a logarithmic scale).

Figure 6

(a) Visualization of spatial distribution of the vector displacement field $\mathbf{u}\left(\mathbf{r}\right)$ and (b) plot of the displacement-displacement correlation matrix $\mathrm{\Psi }\left(r\right)$ in an inherent structure containing primarily bound dislocations at $p=0.17$. (c) Visualization of spatial distribution of the vector displacement field $\mathbf{u}\left(\mathbf{r}\right)$ and (d) plot of the displacement-displacement correlation matrix $\mathrm{\Psi }\left(r\right)$ in an inherent structure containing primarily free dislocations at $p=0.04$. (e) Visualization of spatial distribution of the vector displacement field $\mathbf{u}\left(\mathbf{r}\right)$ and (f) plot of the displacement-displacement correlation matrix $\mathrm{\Psi }\left(r\right)$ in an inherent structure containing disclinations at $q=0.015$. The visualizations are using representative portions of configurations with $N=1200$ particles, while the correlation matrices are computed using configurations with $N=10800$ particles.