Crystalline order on Riemannian manifolds with variable Gaussian curvature and boundary

We investigate the zero-temperature structure of a crystalline monolayer constrained to lie on a two-dimensional Riemannian manifold with variable Gaussian curvature and boundary. A full analytical treatment is presented for the case of a paraboloid of revolution. Using the geometrical theory of topological defects in a continuum elastic background, we find that the presence of a variable Gaussian curvature, combined with the additional constraint of a boundary, gives rise to a rich variety of phenomena beyond that known for spherical crystals. We also provide a numerical analysis of a system of classical particles interacting via a Coulomb potential on the surface of a paraboloid.

Figure 1

(Color online) The function ${\Gamma }_s\left(r\right)$ for different values of $\kappa ∊[1,2]$.

Figure 2

(Color online) The quantity $U_{\kappa ,R}$ as a function of $R$ for different values of $\kappa ∊[1,6]$.

Figure 3

(Color online) Energy density ${\Gamma }^2\left(\mathit{x}\right)$ for two different defect distributions. The energy density corresponding to an isolated $+1$ disclination at the origin is shown on the top. The defect distribution used for the figure on the bottom has been obtained from the numerical minimization of the Coulomb energy of a system of $V=50$ charges (see Sec. 3).

Figure 4

(Color online) Two examples of $Y_{b,n}$ triangulations of the paraboloid ($Y_{4,7}$ on the top and $Y_{5,10}$ on the bottom). Plaquettes with disclinations are highlighted in red for $+1$ disclinations and green for $+2$ disclinations.

Figure 5

(Color online) Phase diagram in the large core energy regime. For small $\kappa$, the lattice preserves the sixfold rotational symmetry of the flat case. As the curvature at the origin increases, the system undergoes a transition to the $Y_5$ phase.

Figure 6

(Color online) Defect phase diagram for a paraboloidal crystal of radius $R=1$. The two phase boundaries that separate the isolated disclination (ID) regime from the coexistence regime and the coexistence regime from the scar phase correspond to the solutions of Eq. (31) for $K_0=K_{\min }$ and $K_0=K_{\max }$, respectively.

Figure 7

(Color online) Voronoi lattices and Delaunay triangulations for five selected systems from numerical simulations with $R=1$. The first row corresponds to $V=200$ and $\kappa =1.6$, while the second row is for $V=200$ and $\kappa =0.8$ (see Sec. 3B for a discussion). In the bottom two rows $V=100$, 80, and 50 with $\kappa =1.6$.

Figure 8

(Color online) CPU time and speedup (i.e., the time employed by $n$ processors to accomplish a given number of iterations divided by the time employed by a single machine to achieve the same task).