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As one of the first proposed topologically protected states, the quantum spinHall effect in graphene relies critically on the existence of a spin-dependentgap at the K/K' points of the Brillouin zone. Using a tight-binding formulationbased on the method of invariants, we identify the origin of such an intrinsicgap as the three-center interaction between the pi-orbitals caused byspin-orbit interactions. This methodology incorporates all symmetry compliantinteractions previously neglected and has wider applications for comparisonsbetween first-principle calculations and the tight-binding method. It alsoidentifies a correction to the Haldane model and its generalization, whichincorporates the spin degrees of freedom and reproduces all the salientfeatures required for the quantum spin Hall effect in graphene.

Topological insulators, spin, and the tight-binding method