Topological order in densely packed anisotropic colloids

The existence of topological order is frequently associated with strongly coupled quantum matter. Here, we demonstrate the existence of topological phases in classical systems of densely packed, hard, anisotropic polyhedrally shaped colloidal particles. We show that previously reported transitions in dense packings lead to the existence of topologically ordered thermodynamic phases, which we show are stable away from the dense packing limit. Our work expands the library of known topological phases, whose experimental realization could provide new means for constructing plasmonic materials that are robust in the presence of fluctuations.

Figure 1

Surfaces (solid blue and green lines) of maximal packing density (${\varphi }_d$) as a function of particle shape ($\alpha$) have been shown [19], in general, to exhibit nonanalytic behavior at the point $({\alpha }_{\vee },{\varphi }_{\vee })$ that is associated with a change in the topology of contacts between adjacent particles. Within the blue and green triangular regions, dense packings exhibit topological distinction by particle contacts. However, within the gray region bounded above by the dashed blue and green lines, it is unknown, in general, whether topological contact types persist (top panel). We show that there is a first-order phase transition, indicated by the divergence of the so-called alchemical potential [20] near the valley packing discontinuity (bottom panel).

Figure 2

Example dense packing structures (where packing ${\lambda }_{A=52}$ is blue, above, and packing ${\lambda }_{B=58}$ is green, below) of anisotropic shapes, including “exploded” views that show the location and orientation of neighboring particles, and densely packed units.

Figure 3

Example dense packing structures (where packing ${\lambda }_{A=74}$ is blue, above, and packing ${\lambda }_{B=69}$ is green, below) of anisotropic shapes, including “exploded” views that show the location and orientation of neighboring particles, and densely packed units.

Figure 4

Panel (a) shows the curves of maximal packing density at ${\varphi }_{\vee }$, outlining the two protected packing regions, where packing ${\lambda }_{A=52}$ is blue and packing ${\lambda }_{B=58}$ is green. Lower curves indicate computed free energies at three packing densities (0.85, 0.80, 0.75). Darker shaded colors indicate protected regions, while lighter shaded colors indicate free energy preferred regions of the ($\alpha$, $\varphi$) phase diagram. The gray region is a region where the preferred phase is unknown. Panels (b), (d), (f) indicate topological order evaluated using the intersection equations for ${\lambda }_{A=52}$ and panels (c), (e), (g) indicate the same using the intersection equations for ${\lambda }_{B=58}$. The dotted black line roughly demarcates boundaries between thermodynamically preferred packings as a function of packing density, and is meant to guide the eye.

Figure 5

Panel (a) shows the curves of maximal packing density at ${\varphi }_{\vee }$, outlining the two protected packing regions, where packing ${\lambda }_{A=74}$ is blue and packing ${\lambda }_{B=69}$ is green. Lower curves indicate computed free energies at three packing densities (0.85, 0.80, 0.75). Darker shaded colors indicate protected regions, while lighter shaded colors indicate free energy preferred regions of the ($\alpha$, $\varphi$) phase diagram. The gray region is a region where the preferred phase is unknown. Panels (b), (d), (f) indicate topological order evaluated using the intersection equations for ${\lambda }_{A=74}$ and panels (c), (e), (g) indicate the same using the intersection equations for ${\lambda }_{B=69}$. The dotted black line roughly demarcates boundaries between thermodynamically preferred packings as a function of packing density, and is meant to guide the eye.

Figure 6

Panel (a) shows the vector components ($a_x$ (blue circles), $a_y$ (green triangles), $a_z$ (red squares)) for the simulation box (connected by dashed lines) and the extracted unit cell (connected by solid lines) of ${\lambda }_{B=69}$. The box vectors are scaled uniformly to match the unit cell vectors. In panel (a), the $a_y$ and $a_z$ fall directly onto one another, so both cannot be seen. Panels (b) and (c) show the same but instead for ($b_x$, $b_y$, $b_z$) and ($c_x$, $c_y$, $c_z$) respectively. Panel (d) shows the comparison of the order parameters ${\theta }_{AA}$ (blue pluses) and ${\theta }_{BA}$ (green diamonds) for both the system with box moves (connected by solid lines) and without box moves (connected by dashed lines). Panel (e) shows the same but for ${\theta }_{AB}$ (blue pluses) and ${\theta }_{BB}$ (green diamonds).