Symmetry breaking, strain solitons and mechanical edge modes in two dimensional crystals

The bistability of AB and BA stackings of bilayer graphene provides an unusual mechanical example of spontaneous symmetry breaking. We develop a continuum elasticity model in which the basic degree of freedom is the relative displacement of the two layers. Within our model, domain walls between regions of AB and BA stacking are solitons; we are able to give a quantitative description of these domain walls in agreement with experiments. Our model also predicts a zoo of point-like defects that are analogous to vortices and some of which have been observed in experiments. We also study the dynamics of the relative displacement field and find that the associated vibrational modes are gapped in the bulk but there are gapless modes that propagate along the domain walls. These modes are reminiscent of the edge electronic modes of topological insulators. For a simpler model that may be applicable to monolayer antimonene we are able to make this connection precise by showing that the mechanical model is the square of a Dirac Hamitonian and then drawing upon the well established topological classification of Dirac Hamiltonians. We surmise that low-scale symmetry breaking, domain walls and gapless edge modes are common features of two dimensional materials, not unique to bilayer graphene.