# Computation of bound states of semi-infinite matrix Hamiltonians with applications to edge states of two-dimensional materials

Journal Article

We present a novel numerical method for the computation of bound states of semi-infinite matrix Hamiltonians which model electronic states localized at edges of one and two-dimensional materials (edge states) in the tight-binding limit. The na\"{i}ve approach fails: arbitrarily large finite truncations of the Hamiltonian have spectrum which does not correspond to spectrum of the semi-infinite problem (spectral pollution). Our method, which overcomes this difficulty, is to accurately compute the Green's function of the semi-infinite Hamiltonian by imposing an appropriate boundary condition at the semi-infinite end; then, the spectral data is recovered via Riesz projection. We demonstrate our method's effectiveness by a study of edge states at a graphene zig-zag edge in the presence of defects, including atomic vacancies. Our method may also be used to study states localized at domain wall-type edges in one and two-dimensional materials where the edge Hamiltonian is infinite in both directions; we demonstrate this for the case of dimerized honeycomb structures joined along a zig-zag edge.

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### Duke Authors

### Cited Authors

- Thicke, K; Watson, A; Lu, J