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Article title

The Non-Commutative Geometry of the Complex Classes of Topological Insulators

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Content
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EN
Abstracts
EN
Alain Connes’ Non-Commutative Geometry program
[1] has been recently carried out [2, 3] for the entire
A- and AIII-symmetry classes of topological insulators,
in the regime of strong disorder where the insulating
gap is completely filled with dense localized spectrum.
This is a short overview of these results, whose goal is to
highlight the methods of Non-Commutative Geometry involved
in these studies. The exposition proceeds gradually
through the cyclic cohomology, quantized calculus with
Fredholm-modules, local formulas for the odd and even
Chern characters and index theorems for the odd and even
Chern numbers. The characterization of the A- and AIIIsymmetry
classes in the presence of strong disorder and
magnetic fields emerges as a natural application of these
tools.
Publisher

Year
Volume
1
Issue
1
Physical description
Dates
received
23 - 2 - 2014
online
30 - 6 - 2014
accepted
6 - 6 - 2014
Contributors
author
  • Department of Physics,
    Yeshiva University, 245 Lexington Av, 10016 New York, USA, prodan@yu.edu
References
  • [1] A. Connes, Noncommutative Geometry (Academic Press, SanDiego, CA, 1994).
  • [2] E. Prodan, B. Leung, and J. Bellissard, J. Phys. A: Math. Theor.46, 485202 (2013).
  • [3] E. Prodan and H. Schulz-Baldes,http://arxiv.org/abs/1402.5002.
  • [4] R. E. Prange and S. M. Girvin, eds., The Quantum Hall Effect,Graduate Texts in Contemporary Physics (Springer-Verlag,1987), 1st ed.
  • [5] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. Ludwig, New J.Phys. 12, 065010 (2010).
  • [6] I. C. Fulga, F. Hassler, and A. R. Akhmerov, Phys. Rev. B 85,165409 (2012).
  • [7] B. Leung and E. Prodan, Phys. Rev. B 85, 205136 (2012).
  • [8] B. Sbierski and P. W. Brouwer, arXiv:1401.7461 (2014).
  • [9] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Phys.Rev. B 78, 195125 (2008).
  • [10] A. Kitaev, in Adv. Theor. Phys.: Landau Memorial Conference,edited by V. Lebedev and M. Feigel’man (AIP, 2009), vol. 1134,pp. 22–30.
  • [11] J. Bellissard, A. van Elst, and H. Schulz-Baldes, J. Math. Phys.35, 5373 (1994).
  • [12] M. Stone, C.-K. Chiu, and A. Roy, J. Phys. A: Math. Theor. 44,045001 (2011).
  • [13] E. Park, Complex Topological K-Theory (Cambridge UniversityPress, Cambridge, UK, 2008).
  • [14] A. P. Schnyder, S. Ryu, and A. W. W. Ludwig, Phys. Rev. Lett.102, 196804 (2009).
  • [15] N. E. Wegge-Olsen, K-Theory and C*-Algebras (Oxford UniversityPress, Oxford, 1993).
  • [16] M. Rordam, F. Larsen, and N. J. Laustsen, An Introduction toK-Theory for C*-algebras, vol. 49 of London MathematicalSociety Student Texts (Cambridge University Press, New York,2000).
  • [17] A. Connes, Tech. Rep. M/82/53, I.H.E.S. (1982).
  • [18] A. Connes, Tech. Rep. M/83/19, I.H.E.S. (1983).
  • [19] A. Connes and H. Moscovici, Geom. Funct. Anal. 5, 174 (1995).
  • [20] J. Bellissard, in Lecture Notes in Physics, edited by T. Dorlas,M. Hugenholtz, and M. Winnink (Springer-Verlag, 1986), vol.257, pp. 99–156.
  • [21] J. Bellissard, in Geometric and Topological Methods for QuantumField Theory (World Sci. Publ., River Edge, NJ, 2003), pp.86–156.
  • [22] H. Schulz-Baldes and J. Bellissard, Rev. Math. Phys. 10, 1(1998).
  • [23] H. Schulz-Baldes and J. Bellissard, J. Stat. Phys. 91, 991 (1998).
  • [24] H. Schulz-Baldes and S. Teufel, Commun. Math. Phys. 319,649 (2013).
  • [25] G. D. Nittis and M. Lein, J. Phys. A: Math. Theor. 46, 385001(2013).
  • [26] E. Prodan, Phys. Rev. B 80, 125327 (2009).
  • [27] B. Leung and E. Prodan, J. Phys. A: Math. and Theor. 46,085205 (2012).
  • [28] I. Mondragon-Shem, J. Song, T. L. Hughes, and E. Prodan,arXiv:1311.5233 (2013).
  • [29] E. Prodan, Appl. Math. Res. eXpress 2013, 176 (2013).
  • [30] E. Prodan, J. Phys. A: Math. Theor. 44, 113001 (2011).
  • [31] J. Song and E. Prodan (2014), http://arxiv.org/abs/1402.7116.
  • [32] S.-C. Zhang and J. Hu, Science 294, 823 (2001).
  • [33] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78, 195424(2008).
Document Type
Publication order reference
Identifiers
YADDA identifier
bwmeta1.element.-psjd-doi-10_2478_topor-2014-0001
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