Abstract
In topological terms, we compute the spectral flow of an arbitrary family of self-adjoint Dirac-type operators with classical (local) boundary conditions on a compact Riemannian manifold with boundary under the assumption that the initial and terminal operators of the family are conjugate by an automorphism of the bundle in which the operators act. We use this result to study conditions for the existence of a nonzero spectral flow of a family of self-adjoint Dirac-type operators with local boundary conditions in a two-dimensional domain with a nontrivial topology and discuss possible physical realizations of a nonzero spectral flow.
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References
N. D. Mermin, Rev. Modern Phys., 51, 591–648 (1979).
D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett., 49, 405–408 (1982).
J. Bellisard, A. van Elst, and H. Schulz-Baldes, J. Math. Phys., 35, 5373–5451 (1994); arXiv:cond-mat/9411052v1 (1994).
F. Wilczek, Fractional Statistics and Anyon Superconductivity, World Scientific, Singapore (1990).
G. E. Volovik, The Universe in a Helium Droplet (Internat. Series Monogr. Phys., Vol. 117), Oxford Univ. Press, Oxford (2003).
M. A. H. Vozmediano, M. I. Katsnelson, and F. Guinea, Phys. Rep., 496, 109–148 (2010); arXiv:1003.5179v2 [cond-mat.mes-hall] (2010).
M. I. Katsnelson, Graphene: Carbon in Two Dimensions, Cambridge Univ. Press, Cambridge (2012).
M. Z. Hasan and C. L. Kane, Rev. Modern Phys., 82, 3045–3067 (2010); arXiv:1002.3895v2 [cond-mat.mes-hall] (2010).
X.-L. Qi and S.-C. Zhang, Rev. Modern Phys., 83, 1057–1110 (2011).
M. F. Atiyah and I. M. Singer, Bull. Amer. Math. Soc., 69, 422–433 (1963).
N. B. Kopnin, Rep. Prog. Phys., 65, 1633–1678 (2002).
Y. Aharonov and D. Bohm, Phys. Rev., 115, 485–491 (1959).
S. Olariu and I. I. Popescu, Rev. Modern Phys., 57, 339–436 (1985).
Y. Avishai, Y. Hatsugai, and M. Kohmoto, Phys. Rev. B, 47, 9501–9512 (1993).
P. Recher, B. Trauzettel, A. Rycerz, Ya. M. Blanter, C. W. J. Beenakker, and A. F. Morpurgo, Phys. Rev. B, 76, 235404 (2007).
R. Jackiw, A. I. Milstein, S.-Y. Pi, and I. S. Terekhov, Phys. Rev. B, 80, 033413 (2009); arXiv:0904.2046v3 [cond-mat.mes-hall] (2009).
M. I. Katsnelson, Europhys. Lett., 89, 17001 (2010); arXiv:0911.3263v1 [cond-mat.mes-hall] (2009).
M. F. Atiyah, V. K. Patodi, and I. M. Singer, Math. Proc. Cambridge Philos. Soc., 79, 71–99 (1976).
M. Berry and R. J. Mondragon, Proc. Roy. Soc. London A, 412, 53–74 (1987).
M. Prokhorova, “The spectral flow for Dirac operators on compact planar domains with local boundary conditions,” Preprint No. 76, Max Planck Inst. Math., Bonn (2011); arXiv:1108.0806v3 [math-ph] (2011).
M. F. Atiyah and R. Bott, “The index problem for manifolds with boundary,” in: Proc. Bombay Colloquium on Differential Analysis (Bombay, India, 7–14 January 1964), Oxford Univ. Press, Oxford (1964), pp. 175–186.
L. Hörmander, The Analysis of Linear Partial Differential Operators III (Grundlehren Math. Wiss., Vol. 274), Springer, Berlin (1985).
V. E. Nazaikinskii and B. Yu. Sternin, Dokl. Math., 61, 13–16.
V. E. Nazaikinskii and B. Yu. Sternin, Funct. Anal. Appl., 35, No. 2, 111–123 (2001).
V. Nazaikinskii and B. Sternin, Abstr. Appl. Anal., 2006, 98081 (2006).
V. E. Nazaikinskii, A. Savin, B.-W. Schulze, and B. Sternin, Elliptic Theory on Singular Manifolds (Differential Integral Equat. Their Appl., Vol. 7), CRC Press, Boca Raton, Fla. (2006).
B. Boo-Bavnbek, M. Lesch, and J. Phillips, Canad. J. Math., 57, 225–250 (2005).
B. BooElliptic Boundary Problems for Dirac Operators, Birkhäuser, Boston, Mass. (1993).
J. Brüning and M. Lesch, J. Funct. Anal., 185, 1–62 (2001).
B. Boo J. Geom. Phys., 59, 784–826 (2009); arXiv:0803.4160v3 [math.DG] (2008).
A. R. Akhmerov and C. W. J. Beenakker, Phys. Rev. B, 77, 085423 (2008); arXiv:0710.2723v3 [cond-mat.meshall] (2007).
D. W. Boukhvalov and M. I. Katsnelson, Nano Lett., 8, 4373–4379 (2008); arXiv:0807.3855v2 [cond-mat.mtrlsci] (2008).
B. Büttner, C. X. Liu, G. Tkachov, E. G. Novik, C. Brune, H. Buhmann, E. M. Hankiewicz, P. Recher, B. Trauzettel, S. C. Zhang, and L. W. Molenkamp, Nature Physics, 7, 418–422 (2011); arXiv:1009.2248v2 [cond-mat.mes-hall] (2010).
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Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 172, No. 3, pp. 437–453, September, 2012.
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Katsnelson, M.I., Nazaikinskii, V.E. The Aharonov-Bohm effect for massless Dirac fermions and the spectral flow of Dirac-type operators with classical boundary conditions. Theor Math Phys 172, 1263–1277 (2012). https://doi.org/10.1007/s11232-012-0112-8
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DOI: https://doi.org/10.1007/s11232-012-0112-8