Isolated Flat Bands and Spin-1 Conical Bands in Two-Dimensional Lattices

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In models for amorphous semiconductors, the Hamiltonian can be written as H = A ^ + T ^ H=\hat{A}+\hat{T} , where A ^ \hat{A} accounts for the intra-cluster hopping and satisfies A ^ 2 = A ^ \hat{A}^{2}=\hat{A} , i.e

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We speculate that the flat band arises from the property of the on-site interaction in the hexagonal Hamiltonian, B 2 = B † B^{2}=B^{\dagger} . This would be a generalization of Straley’s result for amorphous semiconductors, where the intra-cluster hopping satisfies A ^ 2 = A ^ \hat{A}^{2}=\hat{A} , although we do not have an exact proof

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