Abstract.
In this paper, we stress the importance of momentum-space geometry in the understanding of two-dimensional topological phases of matter. We focus, for simplicity, on the gapped boundary of three-dimensional topological insulators in class AII, which are described by a massive Dirac Hamiltonian and characterized by an half-integer Chern number. The gap is induced by introducing a magnetic perturbation, such as an external Zeeman field or a ferromagnet on the surface. The quantum Bures metric acquires a central role in our discussion and identifies a cigar geometry. We first derive the Chern number from the cigar geometry and we then show that the quantum metric can be seen as a solution of two-dimensional non-Abelian BF theory in momentum space. The gauge connection for this model is associated to the Maxwell algebra, which takes into account the Lorentz symmetries related to the Dirac theory and the momentum-space magnetic translations connected to the magnetic perturbation. The Witten black-hole metric is a solution of this gauge theory and coincides with the Bures metric. This allows us to calculate the corresponding momentum-space entanglement entropy that surprisingly carries information about the real-space conformal field theory describing the defect lines that can be created on the gapped boundary.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Fradkin, Field Theories of Condensed Matter Physics (Cambridge University Press, 2013)
J.E. Avron, R. Seiler, P.G. Zograf, Phys. Rev. Lett. 75, 697 (1995)
N. Read, Phys. Rev. B 79, 045308 (2009)
C.L. Kane, M.P.A. Fisher, Phys. Rev. B 55, 15832 (1997)
N. Read, D. Green, Phys. Rev. B 61, 10267 (2000)
A. Cappelli, M. Huerta, G.R. Zemba, Nucl. Phys. B 636, 568 (2002)
T.L. Hughes, R.G. Leigh, E. Fradkin, Phys. Rev. Lett. 107, 075502 (2011)
A. Mesaros, D. Sadri, J. Zaanen, Phys. Rev. B 82, 073405 (2010)
F.D.M. Haldane, Phys. Rev. Lett. 107, 116801 (2011)
X.G. Wen, A. Zee, Phys. Rev. Lett. 69, 953 (1992)
G. Palumbo, Ann. Phys. 386, 15 (2017)
G. Palumbo, J.K. Pachos, Ann. Phys. 372, 175 (2016)
S. Ryu, J.E. Moore, A.W.W. Ludwig, Phys. Rev. B 85, 045104 (2012)
Z. Wang, X.-L. Qi, S.-C. Zhang, Phys. Rev. B 84, 014527 (2011)
M. Stone, Phys. Rev. B 85, 184503 (2012)
P. Gentile, M. Cuoco, C. Ortix, Phys. Rev. Lett. 115, 256801 (2015)
D. Cangemi, Phys. Lett. B 297, 261 (1992)
P. Salgado, R.J. Szabo, O. Valdivia, Phys. Rev. D 89, 084077 (2014)
X.-L. Qi, T.L. Hughes, S.-C. Zhang, Phys. Rev. B 78, 195424 (2008)
S.M. Girvin, A.H. MacDonald, P.M. Platzman, Phys. Rev. B 33, 2481 (1986)
A. Cappelli, C.A. Trugenberger, G.R. Zemba, Nucl. Phys. B 396, 465 (1993)
J. Wang, B. Lian, S.-C. Zhang, Phys. Scr. 2015, 014003 (2015)
G.P. Provost, G. Vallee, Comm. Math. Phys. 76, 289 (1980)
R. Resta, Eur. Phys. J. B 79, 121 (2011)
R. Roy, Phys. Rev. B 90, 165139 (2014)
S.A. Parameswaran, R. Roy, S.L. Sondhi, Phys. Rev. B 85, 241308(R) (2012)
T.S. Jackson, G. Moller, R. Roy, Nat. Commun. 6, 8629 (2015)
M. Claassen, C.H. Lee, R. Thomale, X.-L. Qi, T.P. Devereaux, Phys. Rev. Lett. 114, 236802 (2015)
A.G. Grushin, T. Neupert, C. Chamon, C. Mudry, Phys. Rev. B 86, 205125 (2012)
M. Legner, T. Neupert, Phys. Rev. B 88, 115114 (2013)
E. Dobardzic, M.V. Milovanovic, N. Regnault, Phys. Rev. B 88, 115117 (2013)
Y.-Q. Ma, S.-J. Gu, S. Chen, H. Fan, W.-M. Liu, EPL 103, 10008 (2013)
S. Matsuura, S. Ryu, Phys. Rev. B 82, 245113 (2010)
Yu.A. Golfand, J. Exp. Theor. Phys. 43, 256 (1962)
V.G. Kadyshevsky, Nucl. Phys. B 141, 477 (1978)
R.M. Mir-Kasimov, Phys. Lett. B 378, 181 (1996)
G.W. Gibbons, S. Gielen, Class. Quantum Grav. 26, 135005 (2009)
L.N. Chang, D. Minic, T. Takeuchi, Mod. Phys. Lett. A 25, 2947 (2010)
M. Arzano, J. Kowalski-Glikman, Class. Quantum Grav. 28, 105009 (2011)
G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman, L. Smolin, Phys. Rev. D 84, 084010 (2011)
D. Cangemi, R. Jackiw, Phys. Rev. Lett. 69, 233 (1992)
E. Witten, Phys. Rev. D 44, 314 (1991)
G. Mandal, A.M. Sengupta, S.R. Wadia, Mod. Phys. Lett. A 6, 1685 (1991)
C.R. Nappi, E. Witten, Phys. Rev. Lett. 71, 3751 (1993)
D. Birmingham, M. Blau, M. Rakowski, G. Thompson, Phys. Rep. 209, 129 (1991)
G.Y. Cho, J.E. Moore, Ann. Phys. 326, 1515 (2011)
T.H. Hansson, V. Oganesyan, S.L. Sondhi, Ann. Phys. 313, 497 (2004)
A. Marzuoli, G. Palumbo, EPL 99, 10002 (2012)
G. Palumbo, M. Cirio, Sci. Rep. 5, 10824 (2015)
A. Blasi, A. Braggio, M. Carrega, D. Ferraro, N. Maggiore, N. Magnoli, New J. Phys. 14, 013060 (2012)
A.M. Chan, T.L. Hughes, S. Ryu, E. Fradkin, Phys. Rev. B 87, 085132 (2013)
S.N. Solodukhin, Phys. Lett. B 646, 268 (2007)
S.N. Solodukhin, Living Rev. Relativ. 14, 8 (2011)
D.J. Thouless, M. Kohmoto, M.P. Nightingale, M. den Nijs, Phys. Rev. Lett. 49, 405 (1982)
B. Lian, C. Vafa, F. Vafa, S.-C. Zhang, Phys. Rev. B 95, 094512 (2017)
P. Zanardi, P. Giorda, M. Cozzini, Phys. Rev. Lett. 99, 100603 (2007)
V.A. Mousolou, C.M. Canali, E. Sjoqvist, Phys. Rev. A 88, 012310 (2013)
M. Kolodrubetz, V. Gritsev, A. Polkovnikov, Phys. Rev. B 88, 064304 (2013)
A. Julku, S. Peotta, T.I. Vanhala, D.-H. Kim, P. Torma, Phys. Rev. Lett. 117, 045303 (2016)
Y. Gao, S.A. Yang, Q. Niu, Phys. Rev. B 91, 214405 (2015)
F. Piechon, A. Raoux, J.-N. Fuchs, G. Montambaux, Phys. Rev. B 94, 134423 (2016)
M. Nakahara, Geometry, Topology, and Physics (Institute of Physics Publishing, Bristol, 2003)
B. Chow, D. Knopf, The Ricci Flow: An Introduction (American Mathematical Society, Providence, RI, 2004)
Y. Hatsugai, Phys. Rev. Lett. 71, 3697 (1993)
I. Mandal, EPL 110, 67005 (2015)
V. Dwivedi, V. Chua, Phys. Rev. B 93, 134304 (2016)
D.N. Page, Phys. Rev. A 36, 3479(R) (1987)
T. Ozawa, H.M. Price, I. Carusotto, Phys. Rev. B 93, 195201 (2016)
C.G. Callan jr., S.B. Giddings, J.A. Harvey, A. Strominger, Phys. Rev. D 45, R1005(R) (1992)
J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, Phys. Rev. D 94, 025007 (2016)
L.-J. Lang, S.-L. Zhang, Q. Zhou, Phys. Rev. A 95, 053615 (2017)
H. Li, F.D.M. Haldane, Phys. Rev. Lett. 101, 010504 (2008)
V. Balasubramanian, M.B. McDermott, M. Van Raamsdonk, Phys. Rev. D 86, 045014 (2012)
I. Mondragon-Shem, M. Khan, T.L. Hughes, Phys. Rev. Lett. 110, 046806 (2013)
R. Lundgren, J. Blair, M. Greiter, A. Lauchli, G.A. Fiete, R. Thomale, Phys. Rev. Lett. 113, 256404 (2014)
C.H. Lee, P. Ye, X.-L. Qi, J. Stat. Mech. 2014, P10023 (2014)
A. Chandran, M. Hermanns, N. Regnault, B.A. Bernevig, Phys. Rev. B 84, 205136 (2011)
P. Calabrese, J. Cardy, J. Stat. Mech. 06, P002 (2004)
I. Affleck, Phys. Rev. Lett. 56, 746 (1986)
H.W.J. Blote, J.L. Cardy, M.P. Nightingale, Phys. Rev. Lett. 56, 742 (1986)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Palumbo, G. Momentum-space cigar geometry in topological phases. Eur. Phys. J. Plus 133, 23 (2018). https://doi.org/10.1140/epjp/i2018-11856-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/i2018-11856-8