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R. M. Kaufmann, D. Li & B. Wehefritz-Kaufmann invariant is identified with the spectral flow modulo two, υ≡sf(H,w−1Hw) mod2. (4.23) For manifolds with boundary, a well-defined Dirac operator involves the APS boundary condition, see [6, 50]. The analytic η-invariant of a Dirac operator D is defined by 1∞ −sD2 ds η(D) := √π 0 tr(De )√s. (4.24) If DP (t) is a family of Dirac operators defining a good boundary value problem, P is a spectral projection, then the spectral flow of DP(t) can be computed as variations of the eta invariant, see [37, 50], sf(DP(t))=η ̃(DP(t))|10−11 dη(DP(t))dt 2 0 dt where η ̃ is the reduced eta invariant. As a remark, if a superconnection A is used in (4.24) instead of D, then it would define the eta form, which has important applications in the family index theorem and Witten’s holonomy theorem [19]. 4.2.3. Green’s function In this subsection, we define the Chern–Simons invariant of time reversal invariant topological insulators by Green’s functions following [70]. This approach has the advantage that such topological invariant is still valid even in interacting or disor- dered systems. The idea of applying Green’s functions to the study of domain walls in D-branes or edge states in topological insulators can be traced back to early works by Volovik, for example see [67]. By evaluating one loop Feynman diagrams, the second Chern class (4.11) in the 4d action (4.10) is equivalently defined by fermionic propagators [38], i.e. Green’s functions, c2 = π2 d4kdωεμνρστtr[G∂μG−1G∂νG−1G∂ρG−1G∂σG−1G∂τG−1] (4.25) 15 (2π)5 where the imaginary-time single-particle Green’s function of the Hamiltonian H is defined as G(iω, k) := (iω − H(k))−1. For 3d time reversal invariant topological insulators, the electro-magnetic polar- ization P3 is again obtained by dimensional reduction, π 1 d3kdω μνρσ −1 −1 −1 −1 −1 P3 := 6 du (2π)4 ε tr[G∂μG G∂νG G∂ρG G∂σG G∂uG ]. 0 (4.26) By a similar argument as the WZW extension problem, which will be explained in the next subsection, 2P3 is well defined up to certain integer belonging to the 1630003-40 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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