Stripe, gossamer, and glassy phases in systems with strong nonpairwise interactions

We study structure formation in systems of classical particles in two dimensions with long-range attractive short-range repulsive two-body interactions and repulsive three-body interactions. Stripe, gossamer, and glass phases are found as a result of nonpairwise interaction.

Figure 1

Depiction of the model studied. The contour plot is of the three-body interaction energy experienced by a particle with position $\mathbf{r}=(x,y)$ by two other particles fixed at positions ${\mathbf{r}}_1=(-0.125,0)$, ${\mathbf{r}}_2=(0.125,0)$ (indicated by black dots), i.e., a plot of $f(\mathbf{r},{\mathbf{r}}_1,{\mathbf{r}}_2)$ with $f$ defined in Eq. (4). The model parameters are ${\varepsilon }_{\text{3B}}=1.0$, $\alpha =2.0$, $w=0.8$. The inset shows the two-body interaction potential (2) against particle separation with ${\varepsilon }_{\text{2B}}=1.0$, $a=3.0$, $b=0.2$, $c=3.0$, and $d=0.60$.

Figure 2

The existence of a stripe phase for $N=3$. Displayed is the difference $\Delta E$ of an equilateral triangle of side $\Delta$ and a straight equidistant line with spacing $\Delta$, as a function of $\Delta$ and ${\varepsilon }_{\text{3B}}$. In the region to the left, where $\Delta E>0$, the three-body repulsion is sufficiently strong to cause the system to energetically favor a line over a triangle. The inset confirms this by showing the ground-state value of ${\Psi }_{\text{S}}$ versus ${\varepsilon }_{\text{3B}}$ obtained from MC simulations with three particles, where the ground state is obtained by slow cooling.

Figure 3

By increasing the three-body interaction strength ${\varepsilon }_{\text{3B}}$, various phases are induced in systems of many particles. Displayed in panels (a)–(d) are structure formation for increasing ${\varepsilon }_{\text{3B}}$ at constant density $\rho =0.3$ with $N=400$. Panel (e) shows how ${\Psi }_{\text{S}}$ depends on the strength of the three-body interaction for various densities. The inset of (a) shows normalized distributions of nearest-neighbor distances for the cases ${\varepsilon }_{\text{3B}}=0.0$ and ${\varepsilon }_{\text{3B}}=1.0$. For comparison, the green triangle in the lower-left corner of panels (a)–(d) has side equal to the minimum separation of the two-body potential.

Figure 4

By increasing density for the case of competing two-body and three-body interactions (left column) and a dominating three-body interaction (right column), various structural phases are obtained. Here $N=400$.

Figure 5

A glassy phase occurs for a strong three-body repulsion at high density. Here $\rho =0.9$ with $N=72$. Panels (a) and (b) show configurations for ${\varepsilon }_{\text{3B}}=1.0$ obtained by parallel tempering and fast cooling, respectively. Panels (c) and (d) are for ${\varepsilon }_{\text{3B}}=5.0$. Panel (e) shows the evolution of the total internal energy per particle during four fast-cooling simulations for each system, with ground-state energies obtained by parallel tempering shown with dashed lines.

Figure 6

Thermal effects of the stripe phase. Here ${\varepsilon }_{\text{3B}}=5.0$, $\rho =0.1$ with $N=400$. Displayed in panel (a) are the thermally averaged total internal energy $U$ per particle $N$ (with two- and three-body contributions as dashed and dotted lines, respectively) and ${\Psi }_{\text{S}}$ against temperature $T$. The lower panels are three snapshots of the system in the low-, intermediate-, and high-temperature phases.

Figure 7

Decreasing the two-body minimum by decreasing the parameter $b$ does not induce a clear stripe phase for the parameters studied. Panels (a)–(d) show structure formation for $\rho =0.1$, $N=400$, ${\varepsilon }_{\text{3B}}=1.0$, when decreasing $b$. Panel (e) shows how ${\Psi }_{\text{S}}$ depends on $b$.