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two since this definition of ν is not gauge invariant, indeed, the integral of a over the two circles always changes by an even number under a gauge transformation. An edge state explanation of the invariant is as follows: the uI,II can be viewed as chiral edge states. At any fixed point these cross each other and, after fixing the Fermi energy to be EF = 0, change signs. By the time reversal symmetry, it would be enough to consider one chiral edge state, since the other one can be easily recovered by reflection. The Pfaffian would produce a negative sign if the spectral flow of the chosen chiral edge state is changed by +1 or −1 under the gauge transformation wn. Thus ν keeps track of the parity of the change of signs of the Pfaffian under the time reversal symmetry represented by wn, which can be interpreted as the evenness or oddness of the spectral flow or a mod 2 analytical index. This is made precise below. In physical terms, a negative sign of the Pfaffian also corresponds to the exis- tence of an unpaired Majorana zero mode. Under the bosonization procedure, two Majorana zero modes pair together to create an effective composite boson, so only the parity of Majorana zero modes matters. For ν ≡ 1, the effective theory consists of a collection of composite bosons (or none) and an unpaired Majorana zero mode such as a single Dirac cone. 3.2.5. Strong Z2 invariant We have defined the Kane–Mele invariant or the analytical Z2 index above precisely in 2d. One can generalize it to 3d [33] and higher dimensions using the formula (3.17), which is called the strong Z2 invariant. But, one can also consider various subtori and their Z2 invariants, which are called weak Z2 invariants. This in turn allows one to define the strong invariant rigorously via recursion. We will go through this in detail for a 3d topological insulator, whose momentum space (as the bulk space) is assumed to be T3 with T (k) = −k. There are eight time reversal invariant points in total: (n1, n2, n3) : ni ∈ {0, π}. The three standard emdeddings of T2 → T3 which have constant i-th coordi- nate equal 0,π each contain 4 fixed points and give rise to six Z2 weak invariants νni=0,νni=π,i = 1,2,3, computed by the restriction to the particular T2 as a 2d invariant. These 2d Z2 invariants are not independent and the relation between them is given by (−1)νn1=0 (−1)νn1=π = (−1)νn2=0 (−1)νn2=π = (−1)νn3=0 (−1)νn3 =π =: (−1)ν0 which collects complete information from eight fixed points for different decompo- sitions of T3 into the strong invariant ν0. In practice, physicists use four Z2 invariants (ν0;ν1ν2ν3) to characterize a 3d topological insulator, where ν0 is the 3d Z2 invariant and νi(i = 1,2,3) are 2d Z2 invariants. For 3d, physicists often use the momenta ki to lie in [0, 2] mod 2 and the mul- tiplicative form of Z2 instead of the additive form, i.e. ν ≡ ±1 instead of ν ≡ 0, 1. 1630003-23 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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