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transform. A detailed treatment of twisted equivariant K-theory can be found for example in [30]. 5.1. Tenfold way Generalizing the standard Wigner–Dyson threefold way, namely, symmetries in quantum mechanics described by unitary, orthogonal and symplectic groups, Alt- land and Zirnbauer [1] further proposed a tenfold way in random matrix theory, which implies that free fermionic systems can be classified by ten symmetry classes. Indeed, there are three types of discrete (pseudo)symmetries in topological insu- lators and BdG Hamiltonians: the time reversal symmetry T , the particle-hole symmetry P and the chiral symmetry C, and the combinations of these three sym- metries give ten classes in total. Based on a time reversal invariant Dirac Hamiltonian H, which means that T HT −1 = H, and additionally T 2 = ±1 depending on the spin being integer or half-integer, the time reversal symmetry (TRS) induces three classes +1 ifTH(k)T−1 =H(−k), T2 =+1 TRS=−1 ifTH(k)T−1 =H(−k), T2 =−1 0 if T H(k)T −1 ̸= H(−k). Similarly, the particle hole symmetry (PHS) also gives three classes, (5.1) (5.2) +1 PHS = −1 0 if PH(k)P−1 = −H(k), P2 = +1 if PH(k)P−1 = −H(k), P2 = −1 if PH(k)P−1 ̸= −H(k). The chiral symmetry can be defined by the product C = T · P, sometimes also referred to as the sublattice symmetry. Since T and P are anti-unitary, C is a unitary operator. If both T and P are non-zero, then the chiral symmetry is present, i.e. C = 1. On the other hand, if both T and P are zero, then C is allowed to be either 0 (type A or unitary class, according to the classification of [1], see e.g., [62]) or 1 (type AIII or chiral unitary class). In sum, there are 3 × 3 + 1 = 10 symmetry classes. In particular, the half-spin Hamiltonian with time reversal symmetry falls into type AII or symplectic class, which is the case we are mostly interested in, for more details about the 10-fold way and other symmetry classes see [62]. In [51], the above particle-hole symmetry P is replaced by a U(1)-symmetry Q representing the charge or particle number. Roughly, the parity P is related to the charge by P = (−1)Q. So sometimes topological insulators are also referred to as U(1) and time reversal Z2 symmetry protected phases. 5.2. Nonlinear σ-model One classification scheme is given by the homotopy groups of nonlinear sigma mod- els or target spaces of the Brillouin torus [62]. Assume that there are N occupied 1630003-45 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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