Monodisperse cluster crystals: Classical and quantum dynamics

We study the phases and dynamics of a gas of monodisperse particles interacting via soft-core potentials in two spatial dimensions, which is of interest for soft-matter colloidal systems and quantum atomic gases. Using exact theoretical methods, we demonstrate that the equilibrium low-temperature classical phase simultaneously breaks continuous translational symmetry and dynamic space-time homogeneity, whose absence is usually associated with out-of-equilibrium glassy phenomena. This results in an exotic self-assembled cluster crystal with coexisting liquidlike long-time dynamical properties, which corresponds to a classical analog of supersolid behavior. We demonstrate that the effects of quantum fluctuations and bosonic statistics on cluster-glassy crystals are separate and competing: Zero-point motion tends to destabilize crystalline order, which can be restored by bosonic statistics.

Figure 1

Phase diagram as a function of scaled temperature $T/(V_0/R_c^6)$ and density $R_c^2\rho$ for classical monodisperse particles with soft-core interactions [see Eq. (1)] at equilibrium. Here $N/N_s$ is the average number of particles per cluster in the ground state (upper $x$ axis). The dashed vertical line marks $R_c^2\rho \simeq 1.1$ (see Fig. 2).

Figure 2

Cluster short-range orientational order ${\mathrm{\Psi }}_6$ as a function of $T$ at equilibrium for $R_c^2\rho \simeq 1.1$. The insets show the static structure factor $S\left(\mathbf{k}\right)$ at the same rescaled density for (a) $T/(V_0/R_c^6)=0.02$ and (b) $T/(V_0/R_c^6)=0.09$. In the former case the system is a cluster crystal, in the latter, as indicated by the extremely small value of ${\mathrm{\Psi }}_6$ and the essentially featureless $S\left(\mathbf{k}\right)$, a liquid.

Figure 3

(a) Self-intermediate scatter function $F_s(k^{*},t)$ vs time $t$ for $T/(V_0/R_c^6)\times {10}^2=0$, 3.3, 4.4, 4.95, 5.5, 6.05, 7.15,7.70, 8.25, and 13.2 (top to bottom). (b) Mean square displacement (in units of $R_c^2$) $\text{MSD}\left(t\right)$ vs $t$; symbols are the same as in (a). (c) Relaxation time ${\tau }_{\alpha }$ defined via $F_s(k^{*},{\tau }_{\alpha })=e^{-1}$. The dashed line is an exponential fitting function (see the text). Data refer to the scaled density $R_c^2\rho =1.1$.

Figure 4

Snapshots of the sequential evolution of a portion of the system, starting at $t=t_0$. Particles belonging to sites involved in a hopping process have been highlighted in red and blue.

Figure 5

Non-Gaussian parameter ${\alpha }_2\left(t\right)$ vs $t$ for $T/(V_0/R_c^6)=0.03$ (black squares), $0.05$ (red circles), $0.07$ (green stars), and $0.08$ (blue triangles). Data refer to the scaled density $R_c^2\rho =1.1$.

Figure 6

(a) Function $F_s(k^{*},t)$ computed by path-integral Langevin dynamics. Data are for $R_c^2\rho =0.8119$ and $m{V}_0/\left({\hslash }^2{R}_c^4\right)=80$ (circles) and 60 (triangles). The $T$ values are 1.6 and 2.6 (full and empty symbols, respectively), in units of ${\hslash }^2/\left(m{R}_c^2\right)$. (b) Function $g\left(r\right)$ computed via path-integral Monte Carlo simulations with and without bosonic quantum exchanges (solid and dashed lines, respectively). Here $T/[{\hslash }^2/\left(m{R}_c^2\right)]=10,m{V}_0/\left({\hslash }^2{R}_c^4\right)=35$, and $R_c^2\rho =2.029$.

Figure 7

Snapshot of a portion of the system in a cluster crystal configuration. (a) Individual particles. (b) Schematic outcome after the hierarchical clustering technique: Blue circles are centered in cluster centroids, dashed blue segments connect nearest-neighbor clusters, and ${\theta }_{jl}$ is the angle between the $x$ axis and the line joining cluster $j$ with its neighbor $l$.