Effective Hamiltonian for protected edge states in graphene

Abstract

Edge states in topological insulators (TIs) disperse symmetrically about one of the time-reversal invariant momenta Λ in the Brillouin zone (BZ) with protected degeneracies at Λ. Commonly TIs are distinguished from trivial insulators by the values of one or multiple topological invariants that require an analysis of the bulk band structure across the BZ. We propose an effective two-band Hamiltonian for the electronic states in graphene based on a Taylor expansion of the tight-binding Hamiltonian about the time-reversal invariant M point at the edge of the BZ. This Hamiltonian provides a faithful description of the protected edge states for both zigzag and armchair ribbons though the concept of a BZ is not part of such an effective model. We show that the edge states are determined by a band inversion in both reciprocal and real space, which allows one to select Λ for the edge states without affecting the bulk spectrum.A topological insulator (TI) is an insulator in the bulk with topologically protected edge states that cross the gap so that the edges are conducting. This concept was first introduced by Kane and Mele using a simple tightbinding (TB) model for the band structure of graphene [1,2]. Since then a wide range of materials with these properties have been identified in two and three dimensions (2D and 3D) [3,4]. TIs can be distinguished from trivial insulators without topological edge states by the values of one or multiple topological invariants that require an analysis of the bulk band structure across the Brillouin zone (BZ). In that sense TIs are considered conceptually different from other problems in solid state physics that permit a description local in k space.The first experimental verification of topologically protected edge states was achieved for HgTe/CdTe quantum wells (QWs) [5] following a theoretical proposal by Bernevig, Hughes and Zhang [6] based on a simple effective Hamiltonian, today known as BHZ model. Since then the BHZ model has been used in a wide range of studies. Liu et al. showed [7] that it also describes the edge states in InAs/GaSb QWs. Zhou et al. demonstrated [8] that the BHZ model can be solved exactly, yielding analytical expressions for the edge states in HgTe/CdTe QWs, see also Ref. 9. We do not question the deep insights that have emerged from the classification of solids based on topological invariants. But Zhou's work [8] raises the question to what extent TIs permit a description local in k space [10]. Is the concept of a BZ a necessary prerequisite for protected edge states in a TI? Graphene with its simple TB description [11] has served as an archetype for TIs [1,2,12], despite the fact that its intrinsic SOC has been found to be small [13]. We show here that a Taylor expansion of the graphene TB model about the time-reversal invariant M point of the BZ (with M ≡ −M) yields an effective Hamiltonian that provides a faithful description local in k space of the protected edge states in both zigzag and armchair graphene ribbons. While the proposed model is quite di...

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Intrinsic spin-orbit coupling gap and the evidence of a topological state in graphene

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Topological phase-diagram of time-periodically rippled zigzag graphene nanoribbons

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