zbMATH — the first resource for mathematics

Twisted equivariant matter. (English) Zbl 1286.81109
In quantum mechanics the pure states form the projective space \(PH\) of a complex Hibert space \(H\). The well-konw Wigner’s theorem states that every symmetry that acts on pure states can be lifted to \(H\) as a unitary or antiunitary operator. Hence every symmetry group \(G\) is equipped with a homomorphism \(\varphi: G\rightarrow \pm 1\), which encodes whether a symmetry is unitary or antiunitary. Also with \(G\) a group extension us associated
\[ 0\rightarrow T\rightarrow G^{\tau}\rightarrow G\rightarrow 0, \] where \(T\) is the group of scalar unitary transformations of \(H\).
Typically a quantum symmetry acts on the space-time. Thus \(G\) is equipped with a homomorphism \(t: G\rightarrow \pm 1\), which encodes whether a symmetry preserves or reverses time-orientation. Denote \(c:=\varphi t\).
One says that a quantum system is gapped if it’s Hamiltoniam is invertible. Two gapped system are called to be in the same topological phase if there is a continious family of quantum systems that connect the considered systems. The set \(TF\) of topological phases has two algebraic structures - tensor product an direct sum. The associated abelian group is denoted as \(RTP\).
The main result of the paper under review is an identification of the group \(RTF\) with the a topological \(K\)-theory group.
Also in the paper some examples are discussed in details. In particular the Kane-Mele invariant and the orbital magnetolelectric polarizability are considered in this context.

81R20 Covariant wave equations in quantum theory, relativistic quantum mechanics
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
20G45 Applications of linear algebraic groups to the sciences
20C25 Projective representations and multipliers
20C35 Applications of group representations to physics and other areas of science
Full Text: DOI arXiv
[1] Atiyah M.F.: \(K\)-Theory. 2nd edn., Advanced Book Classics. Addison-Wesley, Redwood City (1989)
[2] Atiyah, M.F., \(K\)-theory and reality, Q. J. Math. Oxford Ser., 17, 367-386, (1966) · Zbl 0146.19101
[3] Atiyah, M.F.; Bott, R.; Shapiro, A.A., Clifford modules, Topology, 3, 3-38, (1964) · Zbl 0146.19001
[4] Atiyah, M.F.; Hirzebruch, F., Vector bundles and homogeneous spaces, Proc. Symp. Pure Math., 3, 7-38, (1961) · Zbl 0108.17705
[5] Abramovici, G., Kalugin, P.: Clifford modules and symmetries of topological insulators. Int. J. Geomet. Methods Mod. Phys. 9(03) (2012). http://arxiv.org/abs/arXiv:1101.1054v2 · Zbl 1252.15030
[6] Atiyah, M., Segal, G.: Twisted \(K\)-theory. Ukr. Mat. Visn. 1(3), 287-330 (2004). http://arxiv.org/abs/arXiv:math/0407054 · Zbl 1151.55301
[7] Altland, A., Zirnbauer, M.R.: Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B 55, 1142-1161 (1997). doi:10.1103/PhysRevB.55.1142
[8] Bouwknegt, P., Carey, A.L., Mathai, V., Murray, M.K., Stevenson, D.: Twisted \(K\)-theory and \(K\)-theory of bundle gerbes. Comm. Math. Phys. 228(1), 17-45 (2002). doi:10.1007/s002200200646. http://arxiv.org/abs/arXiv:hep-th/0106194 · Zbl 1036.19005
[9] Bassani F., Pastori Parravicini G.: Electronic States and Optical Transitions in Solids. Pergamon Press Ltd., New York (1975)
[10] Bouckaert, L.P., Smoluchowski, R., Wigner, E.: Theory of Brillouin Zones and Symmetry Properties of Wave Functions in Crystals. Phys. Rev. 50, 58-67 (1936). doi:10.1103/PhysRev.50.58 · Zbl 0014.37407
[11] Deligne, P.: Notes on spinors, Quantum Fields and Strings: a course for mathematicians, vol. 1, 2, pp. 99-135. (Princeton, NJ, 1996/1997), Am. Math. Soc., Providence, RI (1999)
[12] Deligne, P., Etingof, P., Freed, D.S., Jeffrey, L.C., Kazhdan, D., Morgan, J.W., Morrison, D.R., Witten, E. (eds.): Quantum fields and strings: a course for mathematicians, vol. 1, 2, American Mathematical Society, Providence, RI (1999). Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton (1996-1997) · Zbl 0984.00503
[13] Deligne, P., Freed, D.S.: Classical field theory. Quantum fields and strings: a course for mathematicians, vol. 1, 2, pp. 137-225. (Princeton, NJ, 1996/1997), Am. Math. Soc., Providence, RI (1999) · Zbl 1170.53315
[14] Deligne, P., Freed, D.S.: Sign manifesto. Quantum fields and strings: a course for mathematicians, vol. 1, 2, pp. 357-363. (Princeton, NJ, 1996/1997), Am. Math. Soc., Providence, RI (1999) · Zbl 1170.58301
[15] Donovan, P.; Karoubi, M., Graded Brauer groups and \(K\)-theory with local coefficients, Inst. Hautes Études Sci. Publ. Math., 38, 5-25, (1970) · Zbl 0207.22003
[16] Deligne, P., Morgan, J.W.: Notes on supersymmetry (following Joseph Bernstein). Quantum fields and strings: a course for mathematicians, vol. 1, 2, pp. 41-97. (Princeton, NJ, 1996/1997), Am. Math. Soc., Providence (1999) · Zbl 1170.58302
[17] Dyson, Freeman J., The threefold way. algebraic structure of symmetry groups and ensembles in quantum mechanics, J. Math. Phys., 3, 1199-1215, (1962) · Zbl 0134.45703
[18] Essin, A.M., Moore, J.E., Vanderbilt, D.: Magnetoelectric polarizability and axion electrodynamics in crystalline insulators. Phys. Rev. Lett. 102, 146805 (2009). http://arxiv.org/abs/arXiv:0810.2998 · Zbl 0145.43002
[19] Essin, A.M., Turner, A.M., Moore, J.E., Vanderbilt, D.: Orbital magnetoelectric coupling in band insulators. Phys. Rev. B 81, 205104 (2010). http://arxiv.org/abs/arXiv:1002.0290
[20] Fidkowski, L., Kitaev, A.: Topological phases of fermions in one dimension. Phys. Rev. B 83, 075103 (2011). doi:10.1103/PhysRevB.83.075103, http://arxiv.org/abs/arXiv:1008.4138
[21] Fidkowski, L., Kitaev, A.: Effects of interactions on the topological classification of free fermion systems. Phys. Rev. B 81, 134509 (2010). doi:10.1103/PhysRevB.81.134509, http://arxiv.org/abs/arXiv:0904.2197
[22] Fredenhagen, K., Rejzner, K.: Perturbative algebraic quantum field theory. http://arxiv.org/abs/arXiv:1208.1428 · Zbl 1315.81070
[23] Freed, D.S.: On Wigner’s theorem. Proceedings of the Freedman Fest. In: Vyacheslav, K., Rob, K., Zhenghan, W. (eds.), Geometry & Topology Monographs, vol. 18, pp. 83-89. Mathematical Sciences Publishers (2012). http://arxiv.org/abs/arXiv:1112.2133 · Zbl 1259.81033
[24] Freed, D.S.: Classical Chern-Simons theory. I. Adv. Math. 113(2), 237-303 (1995). http://arxiv.org/abs/arXiv:hep-th/9206021 · Zbl 0844.58039
[25] Freed, D.S.: Lectures on twisted \(K\)-theory and orientifolds. June. http://www.ma.utexas.edu/users/dafr/ESI.pdf. Notes from graduate workshop “\(K\)-theory and quantum fields” (2012)
[26] Freed, D.S., Hopkins, M.J., Teleman, C.: Loop groups and twisted \(K\)-theory I. J. Topol. 4, 737-798 (2011). http://arxiv.org/abs/arXiv:0711.1906
[27] Freed, D.S., Hopkins, M.J., Teleman, C.: Loop groups and twisted \(K\)-theory III. Ann. Math. 174(2), 974-1007 (2011). http://arxiv.org/abs/arXiv:math/0511232 · Zbl 1239.19002
[28] Freedman, M., Hastings, M.B., Nayak, C., Qi, X.-L., Walker, K., Wang, Z.: Projective ribbon permutation statistics: A remnant of non-Abelian braiding in higher dimensions. Phys. Rev. B 83, 115132 (2011). doi:10.1103/PhysRevB.83.115132, http://arxiv.org/abs/arXiv:1005.0583v4
[29] Fu, L., Kane, C.L.: Topological insulators with inversion symmetry. Phys. Rev. B 76, 045302 (2007). doi:10.1103/PhysRevB.76.045302, http://arxiv.org/abs/arXiv:cond-mat/0611341
[30] Fu, L., Kane, C.L., Mele, E.J.: Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007). doi:10.1103/PhysRevLett.98.106803, http://arxiv.org/abs/arXiv:cond-mat/0607699
[31] Gawe¸dzki, K., Suszek, R.R., Waldorf, K.: Bundle gerbes for orientifold sigma models. http://arxiv.org/abs/0809.5125 · Zbl 1280.81089
[32] Hořava, P.: Stability of Fermi Surfaces and \(K\) Theory. Phys. Rev. Lett. 95, 016405 (2005). doi:10.1103/PhysRevLett.95.016405, http://arxiv.org/abs/arXiv:hep-th/0503006
[33] Haldane, F.D.M.: Model for a quantum hall effect without landau levels: condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett. 61, 2015-2018 (1988). doi:10.1103/PhysRevLett.61.2015
[34] Heinzner, P., Huckleberry, A., Zirnbauer, M.R.: Symmetry classes of disordered fermions. Comm. in Math. Phys. 257(3), 725-771 (2005). doi:10.1007/s00220-005-1330-9, http://arxiv.org/abs/arXiv:math-ph/0411040 · Zbl 1092.82020
[35] Hasan, M.Z., Kane, C.L.: Colloquium : Topological insulators. Rev. Mod. Phys. 82, 3045-3067 (2010). doi:10.1103/RevModPhys.82.3045, http://arxiv.org/abs/arXiv:1002.3895
[36] Hasan, M.Z, Moore, J.E.: Three-Dimensional Topological Insulators. http://arxiv.org/abs/arXiv:1011.5462v1
[37] Kitaev, A.: Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22-30 (2009). doi:10.1063/1.3149495, http://arxiv.org/abs/0901.2686 · Zbl 1180.82221
[38] Konig, M., Buhmann, H., Molenkamp, L.W., Hughes, T.L., Liu, C.-X., et al.: The quantum spin hall effect: theory and experiment. J. Phys. Soc. Jpn. 77, 031007 (2008). doi:10.1143/JPSJ.77.031007, http://arxiv.org/abs/arXiv:0801.0901
[39] Kane, C.L., Mele, E.J.: Quantum spin hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005). doi:10.1103/PhysRevLett.95.226801
[40] Kane, C.L., Mele, E.J.: \(Z\)_{2} Topological order and the quantum spin hall effect. Phys. Rev. Lett. 95, 146802 (2005). doi:10.1103/PhysRevLett.95.146802
[41] Liu, Z.X., Wen, X.G., Chen, X., Gu, Z.C.: Symmetry protected topological orders and the cohomology class of their symmetry group. http://arxiv.org/abs/1106.4772
[42] Mermin, N.D.: The topological theory of defects in ordered media. Rev. Mod. Phys. 51, 591-648 (1979). doi:10.1103/RevModPhys.51.591 · Zbl 0207.22003
[43] Moore, J. E., Balents, L.: Topological invariants of time-reversal-invariant band structures. Phys. Rev. B 75, 121306 (2007). doi:10.1103/PhysRevB.75.121306, http://arxiv.org/abs/arXiv:cond-mat/0607314
[44] Milnor, J.: Morse theory. Based on lecture notes. Spivak, M., Wells, R. Annals of Mathematics Studies, No 51. Princeton University Press, Princeton (1963) · Zbl 0108.10401
[45] Moore, G.W.: Quantum symmetries and \(K\)-theory. http://www.physics.rutgers.edu/users/gmoore/QuantumSymmetryKTheory-Part1.pdf. Notes from St. Ottilien lectures (2012)
[46] Murray, M.K.: Bundle gerbes. J. Lond. Math. Soc. (2) 54(2), 403-416 (1996). http://arxiv.org/abs/arXiv:dg-ga/9407015 · Zbl 0867.55019
[47] Qi, X.-L., Hughes, T.L., Zhang, S.-C.: Topological field theory of time-reversal invariant insulators. Phys. Rev. B 78, 195424 (2008). doi:10.1103/PhysRevB.78.195424, http://arxiv.org/abs/arXiv:0802.3537
[48] Qi, X.-L., Zhang, S.-C.: Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057-1110 (2011). doi:10.1103/RevModPhys.83.1057, http://arxiv.org/abs/arXiv:1008.2026
[49] Roy, R.: \(Z\)_{2} classification of quantum spin Hall systems: An approach using time-reversal invariance. Phys. Rev. B 79, 195321 (2009). doi:10.1103/PhysRevB.79.195321
[50] Roy, R.: Topological phases and the quantum spin Hall effect in three dimensions. Phys. Rev. B 79, 195322 (2009). doi:10.1103/PhysRevB.79.195322
[51] Roy, R.: Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect. http://arxiv.org/abs/arXiv:cond-mat/0607531v3
[52] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I. Second ed., Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York,. Functional analysis (1980) · Zbl 0459.46001
[53] Ryu, S., Schnyder, A.P., Furusaki, A., Ludwig, A.W.W.: Topological insulators and superconductors: Tenfold way and dimensional hierarchy. New J. Phys. 12, 065010 (2010). http://arxiv.org/abs/arXiv:0912.2157
[54] Stone, M., Chiu, C.-K., Roy, A.: Symmetries, dimensions and topological insulators: the mechanism behind the face of the Bott clock, J. Phys. A 44(4), 045001,18 (2011). doi:10.1088/1751-8113/44/4/045001, http://arxiv.org/abs/arXiv:1005.3213 · Zbl 1209.82049
[55] Segal, G., Equivariant \(K\)-theory, Inst. Hautes Études Sci. Publ. Math., 34, 129-151, (1968) · Zbl 0199.26202
[56] Schnyder, A.P., Ryu, S., Furusaki, A., Ludwig, A.W.W.: Classification of Topological Insulators and Superconductors. http://arxiv.org/abs/arXiv:0905.2029v1 · Zbl 1180.82228
[57] Schnyder, A.P., Ryu, S., Furusaki, A., Ludwig, A.W.W.: Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008). doi:10.1103/PhysRevB.78.195125, http://arxiv.org/abs/arXiv:0803.2786 · Zbl 1180.82228
[58] Steenrod, N.E., A convenient category of topological spaces, Mich. Math. J., 14, 133-152, (1967) · Zbl 0145.43002
[59] Teo, J.C.Y., Kane, C.L.: Topological defects and gapless modes in insulators and superconductors. Phys. Rev. B 82, 115120 (2010). doi:10.1103/PhysRevB.82.115120
[60] Thouless, D.J., Kohmoto, M., Nightingale, M.P., den Nijs, M.: Quantized hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405-408 (1982). doi:10.1103/PhysRevLett.49.405 · Zbl 0199.26202
[61] Tu, J.-L., Xu, P., Laurent-Gengoux, C.: Twisted \(K\)-theory of differentiable stacks. Ann. Sci. École Norm. Sup. (4) 37(6), 841-910 (2004). doi:10.1016/j.ansens.2004.10.002, http://arxiv.org/abs/arXiv:math/0306138 · Zbl 1069.19006
[62] Turner, A.M., Zhang, Y., Mong, R.S.K., Vishwanath, A.: Quantized response and topology of insulators with inversion symmetry. http://arxiv.org/abs/arXiv:1010.4335v2
[63] Wall, C.T.C.: Graded Brauer groups. J. Reine Angew. Math. 213, 187-199 (1963/1964) · Zbl 0125.01904
[64] Weinberg, S.: The quantum theory of fields. Vol. I. Cambridge University Press, Cambridge, Foundations (2005) · Zbl 1069.81501
[65] Wen, X.-G.: Symmetry-protected topological phases in noninteracting fermion systems. Phys. Rev. B 85, 085103 (2012). doi:10.1103/PhysRevB.85.085103, http://arxiv.org/abs/arXiv:1111.6341 [cond-mat.mes-hall] · Zbl 0145.43002
[66] Wigner, E.P.: Group theory: and its application to the quantum mechanics of atomic spectra. Expanded and improved ed. Translated from the German by J.J. Griffin. Pure and Applied Physics, vol. 5, Academic Press, New York (1959)
[67] Wang, Z., Qi, X.-L., Zhang, S.-C.: Equivalent topological invariants of topological insulators. New J. Phys. 12, 065007 (2010). http://arxiv.org/abs/arXiv:0910.5954
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.