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On the $$C^*$$-algebraic approach to topological phases for insulators. (English) Zbl 1382.82045
The $$C^*$$-algebraic approach to topological insulators is based on the non-commutative topology of the $$C^*$$-algebra of observables of the insulator in the one-particle approximation. The author relates the symmetries of insulators to graded real structures on the observable $$C^*$$-algebra and classifies the topological phases using A. van Daele’s formulation of $$K$$-theory [Q. J. Math., Oxf. II. Ser. 39, No. 154, 185–199 (1988; Zbl 0657.46050)].

##### MSC:
 82D20 Statistical mechanical studies of solids 81V70 Many-body theory; quantum Hall effect 46L80 $$K$$-theory and operator algebras (including cyclic theory)
##### Keywords:
$$C^*$$-algebra; topological insulator; topological phase
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##### References:
 [1] Bellissard, J.: K-theory of $$C^*$$-Algebras in Solid State Physics. Statistical Mechanics and Field Theory: Mathematical Aspects. Springer, Berlin (1986) · Zbl 0612.46066 [2] Bellissard, J; Elst, A; Schulz-Baldes, H, The noncommutative geometry of the quantum Hall effect, J. Math. Phys., 35, 5373-5451, (1994) · Zbl 0824.46086 [3] Bellissard, J; Herrmann, D; Zarrouati, M, Hull of aperiodic solids and gap labeling theorems, Dir. Math. Quasicryst., 13, 207-258, (2000) · Zbl 0972.52014 [4] Blackadar, B.: K-theory for Operator Algebras, 2nd edn. Mathematical Sciences Research Institute Publications, vol. 5. Cambridge University Press, Cambridge (1998) · Zbl 0913.46054 [5] Boersema, JL; Loring, TA, K-theory for real $$C^*$$-algebra s via unitary elements with symmetries, N. Y. J. Math., 22, 1139-1220, (2016) · Zbl 1358.46068 [6] de la Harpe, P.: Classical groups and classical Lie algebras of operators. In: Proceedings of Symposia in Pure Mathematics Operator Algebras and Applications, Part I (Kingston, Ont., 1980), vol. 38, pp. 477-513. American Mathematical Society, Providence (1982) [7] Nittis, G; Gomi, K, Classification of “real” Bloch-bundles: topological quantum systems of type AI, J. Geom. Phys., 86, 303-338, (2014) · Zbl 1316.57019 [8] Nittis, G; Gomi, K, Classification of “quaternionic” Bloch-bundles: topological insulators of type AII, Commun. Math. Phys., 339, 1-55, (2015) · Zbl 1326.57047 [9] Echterhoff, S; Williams, D, Inducing primitive ideals, Trans. Am. Math. Soc., 360, 6113-6129, (2008) · Zbl 1159.46037 [10] De Nittis, G., Gomi, K.: Chiral vector bundles: a geometric model for class AIII topological quantum systems. arXiv:1504.04863 (2015) · Zbl 1326.57047 [11] Elliott, GA; Natsume, T; Nest, R, Cyclic cohomology for one-parameter smooth crossed products, Acta Math., 160, 285-305, (1988) · Zbl 0655.46054 [12] Freed, DS; Moore, GW, Twisted equivariant matter, Ann. Henri Poincaré, 14, 1927-2023, (2013) · Zbl 1286.81109 [13] Grossmann, J; Schulz-Baldes, H, Index pairings in presence of symmetries with applications to topological insulators, Commun. Math. Phys., 343, 477-513, (2016) · Zbl 1348.82083 [14] Kellendonk, J., Richard, S.: Topological boundary maps in physics. In: Perspectives in Operator Algebras and Mathematical Physics. Theta Series in Advanced Mathematics, vol. 8, pp. 105-121. Theta, Bucharest (2008) · Zbl 1199.81029 [15] Kellendonk, J, Noncommutative geometry of tilings and gap labelling, Rev. Math. Phys., 7, 1133-1180, (1995) · Zbl 0847.52022 [16] Kennedy, R; Zirnbauer, M, Bott periodicity for $${\mathbb{Z}}_2$$ symmetric ground states of gapped free-fermion systems, Commun. Math. Phys., 342, 909-963, (2016) · Zbl 1346.81159 [17] Kitaev, A.: Periodic table for topological insulators and superconductors. In: Advances in Theoretical Physics: Landau Memorial Conference. AIP Conference Proceedings, vol. 1134, pp. 22-30 (2009) · Zbl 1180.82221 [18] Mantoiu, M., Purice, R., Richard, S.: Twisted crossed products and magnetic pseudodifferential operators. In: Advances in Operator Algebras and Mathematical Physics, Theta Series in Advanced Mathematics, vol. 5, pp. 137-172. Theta, Bucharest (2005) · Zbl 1199.46158 [19] Packer, J., Raeburn, I.: Twisted crossed products of C*-algebras. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 106. no. 02. Cambridge University Press, Cambridge (1989) · Zbl 0757.46056 [20] Pimsner, M; Voiculescu, D, Exact sequences for $$K$$-groups of certain cross products of $$C^*$$-algebras, J. Oper. Theory, 4, 93-118, (1980) · Zbl 0474.46059 [21] Roe, J, Paschke duality for real and graded $$C^*$$-algebras, Q. J. Math., 55, 325-331, (2004) · Zbl 1072.46047 [22] Rordam, M., Larsen, F., Laustsen, N.J.: An Introduction to K-theory of $$C^*$$-Algebras. Cambridge University Press, Cambridge (2000) · Zbl 0967.19001 [23] Sadun, L; Williams, RF, Tiling spaces are Cantor set fiber bundles, Ergod. Theory Dyn. Syst., 23, 307-316, (2003) · Zbl 1038.37014 [24] Schnyder, AP; Ryu, S; Furusaki, A; Ludwig, AWW, Classification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B, 78, 195125, (2008) [25] Schröder, H.: $$K$$-Theory for Real $$C^*$$-Algebras and Applications. Volume 290 of Pitman Research Notes in Mathematics Series (1993) · Zbl 0655.46054 [26] Schulz-Baldes, H, $${\mathbb{Z}}_2$$-indices and factorization properties of odd symmetric Fredholm operators, Doc. Math., 20, 1481-1500, (2015) · Zbl 1341.47014 [27] Thiang, GC, On the $$K$$-theoretic classification of topological phases of matter, Ann. Henri Poincaré, 17, 757-794, (2016) · Zbl 1344.81144 [28] Thouless, DJ; Kohmoto, M; Nightingale, MP; Nijs, M, Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett., 49, 405-408, (1982) [29] Daele, A, A note on the K-group of a graded Banach algebra, Bull. Soc. Math. Belg. Sér. B, 40, 353-359, (1988) · Zbl 0668.46037 [30] Daele, A, K-theory for graded Banach algebras I, Q. J. Math., 39, 185-199, (1988) · Zbl 0657.46050
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