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R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann 4. Z2 Invariant as an Index Theorem In this section, we show how the Z2 invariant can be viewed both as an analytical index based on the Maslov index and as a topological index by the Chern–Simons invariant. We have explained that the Kane–Mele invariant counts the parity of the spectral flow of edge states in Sec. 3.2.4 and that this in turn, in the language of index theory, is the analytical Z2 index. The topological Z2 index is the focus of the second subsection. In physics, it corresponds to the Chern–Simons invariant, more precisely, the mod 2 version of the topological term in the Wess–Zumino– Witten (WZW) model. The last subsection is a summary of the relations between the different interpretations of the topological Z2 invariant. 4.1. Analytical Z2 index The mod 2 analytical index was first introduced by Atiyah and Singer [8, 9] for real skew-adjoint elliptic operators P , ind2(P ) ≡ dim ker(P ) (mod 2) (4.1) which can be equivalently described by a mod 2 spectral flow. We argue that the Kane–Mele invariant can be interpreted as a mod 2 analytical index of the effective Hamiltonian of Majorana zero modes, which is explained clearly in our work [47]. For the original Kane–Mele invariant, a possible connection was first observed by Tony Pantev, see reference [14, 44]. In the quantum spin Hall system, the Kane–Mele invariant ν ≡ 0 is equivalent to the fact that the Fermi level intersects a chiral edge state even times, while ν ≡ 1 means the intersection number is odd. This is exactly the definition of the mod 2 analytical index (4.1). Here we put it into the symplectic setting and give an equivalent definition of the Kane–Mele invariant using the Maslov index as in [10], which gives a geometric realization of the analytical index. 4.1.1. Maslov index Assume that the Fermi level sits inside the band gap between the conduction and valence bands, in this subsection we view the Fermi level as a fixed reference Lagrangian subspace R ⊂ R2. From the helical edge state, we pick one chiral edge state and model it by a continuous piecewise-smooth real-valued function, which is also modeled by one Lagrangian subspace or several Lagrangians in R2. If we count the intersection points of the chosen chiral edge state with the Fermi level, then the 2d Z2 invariant can be equivalently defined by the Lagrangian intersection number or Maslov index as an edge Z2 index [10]. It is well known that the Maslov index can be characterized by a set of axioms, which has different geometric and analytical realizations such as eta invariant or spectral flow. In this subsection, we recall the relations between the Maslov index, eta invariant and spectral flow within R2 following [21]. More generally, on manifolds 1630003-32 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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