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where the subscript b stands for “bulk Hamiltonian”. As mentioned previously, the corresponding edge Hamiltonian is obtained by a partial Fourier transformation in the y-direction; see end of Sec. 2.1. H e = − i ∂ x σ x + k y σ y + ( M + B ∂ x2 − B k y2 ) σ z . ( 2 . 1 7 ) One introduces a domain wall at x = 0 separating two bulk systems, by letting the mass term become dependent on x. This M(x) is assumed to have stabilized limits, limx→±∞ M(x) = M±. The edge eigenstates are assumed to satisfy the eigenvalue equation of the edge Hamiltonian He(x,ky)φn(x,ky) = εn(ky)φn(x,ky). We make the usual assumption, that as |ky| gets big, then eventually εn(ky) will lie in the conduction or valence band. An edge eigenstate φn is called an edge state, if it connects the valence band and the conduction band. This means that as one varies ky, εn(ky) hits both the valence and the conduction band. Hence for any edge state φn(ky), εn(ky) intersects with the Fermi level EF = 0 (after a possible shift). Weletsf(φn)betheintersectionnumberofεn(ky)withEF =0,i.e.thespectral flow of the edge state φn. If the intersection is transversal and there are only finitely many points of intersection, i.e. points where εn(ky) = 0, then the spectral flow of the edge state φn is sf(φn)=n+−n− ∈{−1,1} (2.18) where n+ is the number of times that the sign of εn changes from negative to positive and n− is the number of times that the sign of εn changes from positive to negative when increasing ky. Note that sf makes sense for all the edge eigenstates, but is 0 for the ones that are not edge states. Set ν = n sf(φn) = N+ − N− where the sum is either over all eigenstates or over the edge states, N+ is the number of edge states going from the valence to the conduction band as ky increases and N− being the number of states going from the conduction to the valence band. In order to identify the spectral flow with an analytical index, one introduces an extended Dirac Hamiltonian [36] H ̃e = −ivτ1∂ky + τ2He (2.19) where τi are the Pauli matrices and v is a constant parameter representing a velocity. This is a block 4 × 4 matrix Hamiltonian with the diagonal blocks being zero. The anti-diagonal blocks represent the two processes corresponding to N+ and N−. In this form it looks like the Dirac operator in the Atiyah–Patodi–Singer index theorem [6]. Based on the chiral symmetry of H ̃e({H ̃e,τ3} = 0) one can compute 1630003-11 Notes on topological insulators Rev. Math. Phys. 2016.28. Downloaded from www.worldscientific.com by PURDUE UNIVERSITY on 08/11/17. For personal use only.PDF Image | Notes on topological insulators
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