Fracton-elasticity duality in twisted moiré superlattices

We formulate a fracton-elasticity duality for twisted moiré superlattices, taking into account that they are incommensurate crystals with dissipative phason dynamics. From a dual tensor-gauge formulation, as compared to standard crystals, we identify twice the number of conserved charges that describe topological lattice defects, namely, disclinations and another type of defects that we dub discompressions. The key implication of these conservation laws is that both glide and climb motions of lattice dislocations are suppressed, indicating that dislocation networks may become exceptionally stable. We also generalize our results to other planar incommensurate crystals and quasicrystals.

Figure 1

Density profile for the triangular moiré superlattice, as given by Eq. (2), in the cases without rotation [panel (a), ${\varrho }_{\theta =0}\left(\mathit{r}\right)$] and with a rotation of $\theta /\left(2\pi \right)=0.03$ [panel (b), ${\varrho }_{\theta }\left(\mathit{r}\right)$]. The phason excitation is illustrated by the difference in density profiles, ${\delta }_{}{\varrho }_{\theta }\left(\mathit{r}\right)={\varrho }_{\theta }\left(\mathit{r}\right)-{\varrho }_{\theta =0}\left(\mathit{r}\right)$, which is plotted in panel (c), and whose absolute value is plotted in panel (d). The underlying triangular lattice is a guide to the eye.

Figure 2

Various defects for a triangular lattice and the relations between them. (a) Positive disclination ($b_2=2/\sqrt{3}$). (b) Negative disclination ($b_2=-2/\sqrt{3}$). (c) Dislocation as a dipole of disclinations. (d) Positive discompression ($b_1=1/2$). (e) Negative discompression ($b_1=-1/2$). (f) Dislocation as a dipole of discompressions. The table summarizes the fusion rules that explain how to represent charges in terms of Dirac strings of dipoles of other charges.