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We provide a method for the generation of effective continuum Hamiltoniansthat goes beyond the well known k.p method in being equally effective in bothhigh, and low (or no) symmetry situations. Our approach is based on asurprising exact map of the two-centre tight-binding method onto a compactcontinuum Hamiltonian, with a precise condition given for the hermiticity ofthe latter object. We apply this method to a broad range of low dimensionalsystems of both high and low symmetry: graphene, graphdiyne, {\gamma}-graphyne,6,6,12-graphyne, twist bilayer graphene, and partial dislocation networks inBernal stacked bilayer graphene. For the single layer systems the method yieldsHamiltonians for the ideal lattices, as well as a systematic theory forcorrections due to deformation. In the case of bilayer graphene we provide acompact expression for an effective field capable of describing any stackingdeformation of the bilayer; twist bilayer graphene, as well as the partialdislocation network in AB stacked graphene, emerge as special cases of thisfield. For the latter system we find (i) charge pooling on the mosaic of AB andAC segments near the Dirac point and (ii) localized current carrying states onthe partials with the current density characterized by both intralayer andinterlayer components.

Going beyond k.p theory: a general method for obtaining effective Hamiltonians in both high and low symmetry situations