Phasons of moiré superlattices
These days I am paying a lot of attention to the field of moiré-patterned materials. These are compounds formed by slightly misaligned atomic layers stacked on top of each other. At low temperatures many of these systems exhibit macroscopic quantum behavior: correlated insulating phases, superconductivity, and quantized Hall responses together with magnetism of orbital origin.
A prominent example is magic-angle graphene, which consists of two graphene layers with a relative twist between them. Within the pitch of the moiré pattern the system explores all the possible stacking configurations, giving rise to a quasi-periodic modulation of the electron tunneling between layers.
For twists around the magic angle (~ 1 deg) the moiré pitch is about 14 nm, two orders of magnitude larger than the atomic bonds. For this reason one can assume a coarse-grained description in which, effectively, the system explores all the stacking configurations already within a moiré period.
Phasons in a nutshell
Incommensurability suggests the existence of soft modes associated with the sliding motion of one layer with respect to the other. However, the coupling between the two layers make things more difficult (and interesting!).
The coarse-grained or continuum description of the problem reduces to a two-dimensional version of the Frenkel-Kontorova model. This theory describes the lateral relaxation that minimizes the energy cost in adhesion energy: The bilayer forms regions of partial commensuration at the cost of creating stacking domain walls in between.
Gradually, for smaller and smaller twist angles, the moiré pattern can be envisioned as a network of strings under tension, as I explained in this article.
The acoustic vibrations of this network are called phasons. In the continuum theory, they can be understood as the Goldstone modes associated with translations of the domain walls.
Why do they matter?
There is a crucial difference between phasons and acoustic phonons, though: The former always involve the relative motion of a group of ions with respect to other and, therefore, they are subjected to frictional forces. The stick-slip nature of solid friction is related to the fact that arbitrary forms of disorder always pin the moiré pattern, so positional order in the moiré superlattice is always lost beyond a finite length scale. We demonstrated this in a recent article in Physical Review Letters.
Phasons are then very effective in propagating tensions generated during the fabrication of the devices, which is the origin of twist angle disorder. I recently showed how these strains accumulated between the layers favor different forms of symmetry breaking in twisted bilayer graphene.
Moreover, from a mechanical point of view, the moiré pattern is more akin to a fluid (or a glass) than to a crystalline solid. The reason is that phasons are always overdamped due to these frictional forces between the layers. The dynamics of these modes are not protected by a microscopic conservation law and the relative momentum of the layers experience "ohmic" losses. At low frequencies the response of the moiré is diffusive.
The associated transfer of spectral weight to lower energies in order to build a broad diffusive peak in the response makes phason scattering a very efficient channel for entropy production. We believe that these soft modes are maybe responsible for some of the anomalies in thermodynamic and transport properties observed in moiré materials at low temperatures; for example, the survival of a very large, linear-in-T resistivity down to very low temperatures, as we explain in this recent preprint.