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the boundary term of the second Chern character, for example see [56], dcs3(a,f) = ch2(f). In addition, the Chern character can be expressed in Chern classes, in our case ch2(f) = 1[c21(f) − 2c2(f)] = −c2(f) 2 since c1(f) = 0 for time reversal invariant models. After dimensional reduction, the effective action for a 3d topological insulator Notes on topological insulators is given by P S3d = 3 d3xdtεμνρσFμνFρσ (4.12) 16π2 where the magneto-electric polarization P3 is defined as the Chern–Simons action of the Berry connection, P3(ai) = 16π2 d3kεijktr aifjk − 3ai[aj,ak] . It is related to the θ-parameter known in condensed matter physics by θ := 2πP3 ∈ {0, π} mod 2π. (4.13) In general, the Chern–Simons action is gauge invariant up to a winding number. More precisely, under a gauge transformation, ai →agi =g−1aig−g−1dig, forg:T3 →U(n) one has 11 ∆P3 = P3(agi ) − P3(ai) = 1 24π2 d3k εijktr(g−1digg−1djgg−1dkg) (4.14) the right-hand side is an integer, namely the winding number of g. It is also known as the topological WZW term. The Chern–Simons invariant υ of a time reversal invariant system is defined in [61] by the change of magneto-electric polarization under the gauge transformation induced by the time reversal symmetry modulo two. More precisely, if w is the specific gauge transformation induced by the time reversal symmetry given by the transition matrix (3.16), the Chern–Simons invariant is defined by 1 υ ≡ 2 d3k tr(w−1dw)3 (mod 2). (4.15) Now the Bloch bundle splits as before, so that we have a block form of w ∈ U (2) × · · · × U (2) ⊂ U (2N ). Furthermore, by the property of the transition matrix, the gauge group can be reduced from U(2) to SU(2), see [69], hence υ is the mod 2 version of the topological term in the SU(2) WZW model. So the chiral edge states can be modeled by some current algebra, which is a Kac–Moody algebra in conformal field theory, the interested reader may consult [73]. 24π T3 1630003-37 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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