### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 077, 25 pages      arXiv:1302.0349      https://doi.org/10.3842/SIGMA.2014.077
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

### Quantitative $K$-Theory Related to Spin Chern Numbers

Terry A. Loring
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA

Received January 15, 2014, in final form July 13, 2014; Published online July 19, 2014

Abstract
We examine the various indices defined on pairs of almost commuting unitary matrices that can detect pairs that are far from commuting pairs. We do this in two symmetry classes, that of general unitary matrices and that of self-dual matrices, with an emphasis on quantitative results. We determine which values of the norm of the commutator guarantee that the indices are defined, where they are equal, and what quantitative results on the distance to a pair with a different index are possible. We validate a method of computing spin Chern numbers that was developed with Hastings and only conjectured to be correct. Specifically, the Pfaffian-Bott index can be computed by the ''log method'' for commutator norms up to a specific constant.

Key words: $K$-theory; $C^{*}$-algebras; matrices.

pdf (975 kb)   tex (525 kb)

References

1. Eilers S., Exel R., Finite-dimensional representations of the soft torus, Proc. Amer. Math. Soc. 130 (2002), 727-731, math.OA/9810165.
2. Eilers S., Loring T.A., Computing contingencies for stable relations, Internat. J. Math. 10 (1999), 301-326.
3. Eilers S., Loring T.A., Pedersen G.K., Morphisms of extensions of $C^*$-algebras: pushing forward the Busby invariant, Adv. Math. 147 (1999), 74-109.
4. Exel R., The soft torus and applications to almost commuting matrices, Pacific J. Math. 160 (1993), 207-217.
5. Exel R., Loring T.A., Almost commuting unitary matrices, Proc. Amer. Math. Soc. 106 (1989), 913-915.
6. Exel R., Loring T.A., Invariants of almost commuting unitaries, J. Funct. Anal. 95 (1991), 364-376.
7. Fulga I.C., Hassler F., Akhmerov A.R., Scattering theory of topological insulators and superconductors, Phys. Rev. B 85 (2012), 165409, 12 pages, arXiv:1106.6351.
8. Glebsky L., Almost commuting matrices with respect to normalized Hilbert-Schmidt norm, arXiv:1002.3082.
9. Gygi F., Fattebert J., Schwegler E., Computation of maximally localized Wannier functions using a simultaneous diagonalization algorithm, Comput. Phys. Comm. 155 (2003), 1-6.
10. Halmos P.R., Some unsolved problems of unknown depth about operators on Hilbert space, Proc. Roy. Soc. Edinburgh Sect. A 76 (1976), 67-76.
11. Hastings M.B., Loring T.A., Topological insulators and $C^*$-algebras: theory and numerical practice, Ann. Physics 326 (2011), 1699-1759, arXiv:1012.1019.
12. Lin H., Almost commuting selfadjoint matrices and applications, in Operator Algebras and their Applications (Waterloo, ON, 1994/1995), Fields Inst. Commun., Vol. 13, Amer. Math. Soc., Providence, RI, 1997, 193-233.
13. Loring T.A., The torus and noncommutative topology, Ph.D. Thesis, University of California, Berkeley, 1986.
14. Loring T.A., $C^*$-algebra relations, Math. Scand. 107 (2010), 43-72, arXiv:0807.4988.
15. Loring T.A., Computing a logarithm of a unitary matrix with general spectrum, Numer. Linear Algebra Appl., to appear, arXiv:1203.6151.
16. Loring T.A., Hastings M.B., Disordered topological insulators via $C^*$-algebras, Europhys. Lett. 92 (2010), 67004, 6 pages, arXiv:1005.4883.
17. Loring T.A., Sørensen A.P.W., Almost commuting unitary matrices related to time reversal, Comm. Math. Phys. 323 (2013), 859-887, arXiv:1107.4187.
18. Loring T.A., Vides F., Estimating norms of commutators, arXiv:1301.4252.
19. Marzari N., Souza I., Vanderbilt D., An introduction to maximally-localized Wannier functions, Psi-K Newsletter 57 (2003), 129-168, available at http://www.psi-k.org/newsletters/News_57/Highlight_57.pdf.
20. Rieffel M.A., $C^{\ast}$-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), 415-429.
21. Ruhe A., Closest normal matrix finally found!, BIT 27 (1987), 585-598.
22. Sørensen A.P.W., Semiprojectivity and the geometry of graphs, Ph.D. Thesis, University of Copenhagen, 2012, available at http://www.math.ku.dk/noter/filer/phd12apws.pdf.