Abstract
Topological phases of matter can be classified by using Clifford algebras through Bott periodicity. We consider effective topological field theories of quantum Hall systems and topological insulators that are Chern–Simons and BF field theories. The edge states of these systems are related to the gauge invariance of the effective actions. For the edge states at the interface of two topological insulators, transgression field theory is proposed as a gauge invariant effective action. Transgression actions of Chern–Simons theories for (2+1)D and (4+1)D and BF theories for (3+1)D are constructed. By using transgression actions, the edge states are written in terms of the bulk connections of effective Chern–Simons and BF theories.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bernevig, B.A., Hughes, T.L., Zhang, S.C.: Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006)
Bernevig, B.A., Chern, C.H., Hu, J.P., Toumbas, N., Zhang, S.C.: Effective field theory description of the higher dimensional quantum Hall liquid. Ann. Phys. 300, 185–207 (2002)
Bertlmann, R.: Anomalies in Quantum Field Theory. Oxford University Press, Oxford (1996)
Borowiec, A., Ferraris, M., Francaviglia, M.: A covariant formalism for Chern-Simons gravity. J. Phys. A Math. Theor. 36, 2589 (2003)
Borowiec, A., Fatibene, L., Ferraris, M., Francaviglia, M.: Covariant Lagrangian formulation of Chern-Simons and BF theories. Int. J. Geom. Methods Mod. Phys. 3, 755–774 (2006)
Cho, G.Y., Moore, J.E.: Topological BF field theory description of topological insulators. Ann. Phys. 326, 1515–1535 (2011)
Fradkin, E.: Field Theories of Condensed Matter Physics, 2nd edn. Cambridge University Press, Cambridge (2013)
Hasan, M.Z., Kane, C.L.: Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010)
Huerta, L., Zanelli, J.: Optical properties of a \(\theta \) vacuum. Phys. Rev. D 85, 085024 (2012)
Izaurieta, F., Rodriguez, E., Salgado, P.: On transgression forms and Chern–Simons (super)gravity (2005). arXiv:hep-th/0512014
Kane, C.L., Mele, E.J.: \(\mathbb{Z}_2\) topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005)
Kitaev, A.: Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22–30 (2009)
Lawson, B.H., Michelsohn, M.L.: Spin geometry. Princeton University Press, Princeton (1989)
Moore, J.E.: The birth of topological insulators. Nature 464, 194–198 (2010)
Mora, P., Olea, R., Troncoso, R., Zanelli, J.: Transgression forms and extensions of Chern-Simons gauge theories. J. High Energy Phys. 0602, 067 (2006)
Nakahara, M.: Geometry, Topology and Physics, 2nd edn. Taylor Francis, London (2003)
Qi, X.L., Zhang, S.C.: Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011)
Qi, X.L., Hughes, T.L., Zhang, S.C.: Topological field theory of time reversal invariant insulators. Phys. Rev. B 78, 195424 (2008)
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)
Stone, M., Chiu, C.K., Roy, A.: Symmetries, dimensions and topological insulators: the mechanism behind the face of the Bott clock. J. Phys. A Math. Theor. 44, 0450001 (2011)
Thouless, D.J., Kohmoto, M., Nightingale, M.P., den Nijs, M.: Quantized Hall conductance in a two dimensional perodic potential. Phys. Rev. Lett. 49, 405–408 (1982)
Wang, Z., Qi, X.L., Zhang, S.C.: Equivalent topological invariants of topological insulators. New J. Phys. 12, 065007 (2010)
Wen, X.G.: Theory of the edge states in fractional quantum Hall effects. Int. J. Mod. Phys. B 6, 1711–1762 (1992)
Wen, X.G.: Topological orders and edge excitations in fractional quantum Hall states. Adv. Phys. 44, 405–473 (1995)
Willison, S., Zanelli, J.: Chern–Simons foam (2008). arXiv:0810.3041
Zhang, S.C., Hu, J.P.: A four dimensional generalization of the quantum Hall effect. Science 294, 823–828 (2001)
Zhang, S.C., Hansson, T.H., Kivelson, S.: Effective field theory model for the fractional quantum Hall effect. Phys. Rev. Lett. 62, 82–85 (1989)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Rafał Abłamowicz
Rights and permissions
About this article
Cite this article
Açık, Ö., Ertem, Ü. Transgression Field Theory at the Interface of Topological Insulators. Adv. Appl. Clifford Algebras 27, 2235–2245 (2017). https://doi.org/10.1007/s00006-017-0761-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00006-017-0761-7