Figure 1

(a) The projection of $H^2$ associated with the Poincaré disk model: atoms are represented by disks whose diameter contracts as they move away from the center of the Poincaré disk. The geodesics are arcs of circles perpendicular to the disk boundary. We also display the octagonal tiling (in which 8 octagons meet at each vertex) used in the PBC. (b) $F_s(k,t)$ versus $t$ for $\kappa \sigma =0.2$ and for different temperatures $T/T^{*}$ ranging from 3.9 to 0.1 (left to right): for $T\sim T^{*}$ and below, stretched exponential behavior, $\exp [-(t/\tau {)}^{\beta }]$, is observed with $\beta$ decreasing from 1 at the highest $T$ to 0.54 at the lowest equilibrated $T$; for $T<0.35T^{*}$ the system is out of equilibrium and ages (last three curves).Reuse & Permissions

Figure 2

Arrhenius plot of the translational relaxation time $\tau$ versus $T^{*}/T$ for several curvatures, where $T^{*}$ is the (approximate) location of the ordering transition in the Euclidean plane. The particle density is $\rho {\sigma }^2=0.85$. Note the deviation from the simple Arrhenius dependence shown by the dashed line. As the frustration $\kappa \sigma$ decreases, the deviation is stronger; i.e., the fragility gets larger. The thick blue line above $T^{*}$ corresponds to the Euclidean case. The continuous lines are fits to the VFT formula, $\tau ={\tau }_0\exp [D{T}_0/(T-T_0)]$: as frustration decreases, $D$ is found to decrease from 10.8 to 4.6 while $T_0/T^{*}$ increases from 0.12 to 0.33, which again indicates an increase in fragility.Reuse & Permissions

Figure 3

(a),(b) Atomic trajectories followed for a time interval during which the average distance traveled by the atoms is roughly $\sigma$ at (a) $T/T^{*}=2.4$ and (b) $T/T^{*}=0.52$. Whereas at high $T$ all atoms seem to move by a comparable amount and the dynamics is thus spatially homogeneous, a strikingly different picture is obtained at low $T$: most atoms hardly move or just rattle in the cage formed by their neighbors and mobility is concentrated in rare localized regions, illustrating the spatial heterogeneity of the dynamics. (c),(d) Same atomic configurations as in (a) and (b) (respectively), with the color indicating the coordination number for each atom. Black: 6 neighbors (local hexagonal order), red: 7 neighbors (disclination of topological charge $-\pi /3$), blue: 5 neighbors (disclination of charge $+\pi /3$); at $T/T^{*}=2.4$ (c), there are also defects with larger charges (orange: 8 neighbors, cyan: 4 neighbors). Note the correspondence at low $T$ between the rare localized defective regions in (d) and the regions of high mobility in (b). The frustration is $\kappa \sigma =0.05$ and the particle density $\rho {\sigma }^2=0.85$.Reuse & Permissions