Nonperturbative theory of effective Hamiltonians for deformations in two-dimensional materials: Moiré systems and dislocations

We demonstrate that there exists a continuum Hamiltonian H(r,p) that is formally the operator equivalent of the general tight-binding method, inheriting the associativity and Hermiticity of the latter operator. This provides a powerful and controlled method of obtaining effective Hamiltonians via Taylor expansion with respect to momentum and, optionally, deformation fields. In particular, for fundamentally nonperturbative defects, such as twist faults and partial dislocations, the method allows the deformation field to be retained to all orders, providing an efficient scheme for the generation of transparent and compact Hamiltonians for such defects. We apply the method to a survey of incommensurate physics in twist bilayers of graphene, graphdiyne, MoS2, and phosphorene. For graphene we are able to reproduce the "reflected Dirac cones" of the 30° quasicrystalline bilayer found in a recent angle-resolved photoemission spectroscopy experiment, and we show that it is an example of a more general phenomenon of coupling by the moiré momentum. We show that incommensurate physics is governed by the decay of the interlayer interaction on the scale of the single-layer reciprocal lattices, and demonstrate that incommensurate scattering effects lead to a very rapid broadening of band manifolds as the twist angle is tuned through commensurate values.