Figure 1

Sketch of the equilibrium configuration of two crystallites in contact with liquid. The configuration is mechanically stable only if the condition (1) is met.Reuse & Permissions

Figure 2

Initial setup for the simulation of a grain boundary in contact with a liquid. Two truncated pyramids are placed on top of each other. The basal plane of both pyramids is a square with size $L_x\times L_y=L\times L$, with $L=24$ [12]. The upper and lower pyramid have different random orientations and the height of both pyramids is $L_z=L/2$. The upper facets of the truncated pyramids are squares of initial size $L/2\times L/2$, and form a grain boundary between the two crystallites, indicated by the blue square. Liquid fills the volume outside the crystallites. Periodic boundaries are used in the $x$ and $y$ directions, while atoms in the $z=0$ and $z=L$ plane, drawn in red, are pinned. See the auxiliary material for more details [13].Reuse & Permissions

Figure 3

Condensate map of the two-pyramid sample. The map represents (by the density of points) the condensate wave function in the slice $x\in [0.4L,0.6L]$ averaged over the $x$ direction. The initial setup is shown by dashed lines.Reuse & Permissions

Figure 4

Sketch of the initial sample of two cuboid crystallites with different random orientations (see Ref. 13 for more details) placed on top of each other. The basal plane of both cuboids is a square with size $L_x\times L_y=L\times L$, with $L=12$. The height of both cuboids is $Lz/2=7$, yielding a grain boundary in the $z=0$ plane (light blue) and a grain boundary in the $z=Lz/2$ plane (dark blue). Motion across the $xz$ boundary was suppressed by pinning all atoms at a distance smaller than 0.75 from the $y=0(y=L_y)$ boundary indicated in red. This ensures that any superfluid response in the $x$-direction is due to the superfluidity of two horizontal grain boundaries [14]. An additional grain boundary due to periodic boundary conditions arises for each crystallite at the $x=0(x=L_x)$ plane (yellow and orange for the lower and the upper cuboid, respectively), but these do not affect the superfluid response in the $x$ direction.Reuse & Permissions

Figure 5

Phase-coherence properties (winding-cycles maps) of grain boundaries in the ${}^4\mathrm{He}$ sample shown in Fig. 4. Upper panel: projection on the $xy$ plane of the data for the upper half of the sample containing one of the two grain boundaries. Lower panel: ($z$-shifted) projection of all the data points on the $xz$ plane. Note that the $x=0$ ($x=L_x$) grain boundary (induced by boundary conditions) of one of the two crystallites turned out to be insulating (see Fig. 6).Reuse & Permissions

Figure 6

The base plane of an insulating grain boundary between two crystallites that differ only by a rotation about the axis perpendicular to the base plane. Outside the box we show the periodic continuation of the sample. Even with this moderate system size, it is clearly established that the grain boundary is not superfluid. Decreasing the angle between the two crystallites strengthens the insulating character, due to the better match between the atoms of different crystallites. Since this grain boundary is insulating and can be viewed as a wall of sparsely spaced edge dislocations, we conclude that edge dislocations along the $c$ axis are insulating as well.Reuse & Permissions