Strained bilayer graphene, emergent energy scales, and moiré gravity

Twisted bilayer graphene is a rich condensed matter system, which allows one to tune energy scales and electronic correlations. The low-energy physics of the resulting moiré structure can be mathematically described in terms of a diffeomorphism in a continuum formulation. We stress that twisting is just one example of moiré diffeomorphisms. Another particularly simple and experimentally relevant transformation is a homogeneous isomorphic strain of one of the layers, which gives rise to a nearly identical moiré pattern (rotated by ${90}^{\circ }$ relative to the twisted structure) and potentially flat bands. We further observe that low-energy physics of the strained bilayer graphene takes the form of a theory of fermions tunneling between two curved space-times. Conformal transformation of the metrics results in emergent “moiré energy scales,” which can be tuned to be much lower than those in the native theory. This observation generalizes to an arbitrary space-time dimension with or without an underlying lattice or periodicity and suggests a family of toy models of “moiré gravity” with low emergent energy scales. Motivated by these analogies, we present an explicit toy construction of moiré gravity, where the effective cosmological constant can be made arbitrarily small. We speculate about possible relevance of this scenario to the fundamental vacuum catastrophe in cosmology.

Figure 1

(a) Generators of rotation and expansion drawn simultaneously: Blue arrows demonstrate the vector field ${\xi }_s$, while red arrows demonstrate that of ${\xi }_t$. Figures (b) and (c) are twisted and stretched bilayer graphene, respectively. The parameters of both transformations are set to $\theta \approx 0.0192$. Moiré patterns of these bilayers are exactly the same but ${90}^{\circ }$ rotated. Hexagonals (unit cells) of undeformed layers have their largest diameter along the vertical axis $y$. Figures (d) and (e) schematically show Brillouin zones of twisted and stretched bilayer, respectively (blue hexagonals); both give rise to their corresponding moiré reciprocal lattices (red hexagonals).

Figure 2

From left to right: Tunneling matrix for twisted bilayer, tunneling matrix for stretched bilayer, and vanishing of the renormalized Fermi velocity at $K$ valley for magic scale $\theta \approx 0.0192$. Green designates regions where $AB$ tunnelings are dominant and blue shows that of $AA$ tunnelings.

Figure 3

(a) Single curved layer of graphene. (b) A bilayer constructed from two equally curved layers with unequal graphene charts. One chart is slightly shrunken relative to the other, which has resulted in the familiar moiré patterns.