Electromagnetic and thermal responses in topological matter: Topological terms, quantum anomalies and D-branes

Akira Furusaki; Naoto Nagaosa; Kentaro Nomura; Shinsei Ryu; Tadashi Takayanagi

doi:10.1016/j.crhy.2013.03.002

Topological insulators/Isolants topologiques

Electromagnetic and thermal responses in topological matter: Topological terms, quantum anomalies and D-branes
[Réponses électromagnétiques et thermiques dans la matière topologique : Termes topologiques, anomalies quantiques et D-branes]

Akira Furusaki ^1,² ; Naoto Nagaosa ^2,³ ; Kentaro Nomura ⁴ ; Shinsei Ryu ⁵ ; Tadashi Takayanagi ^6,⁷

¹ Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan
² RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
³ Department of Applied Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
⁴ Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
⁵ Department of Physics, University of Illinois, 1110 West Green St, Urbana, IL 61801, United States
⁶ Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan
⁷ Institute for the Physics and Mathematics of the Universe (IPMU), The University of Tokyo, Kashiwa, Chiba 277-8582, Japan

Comptes Rendus. Physique, Volume 14 (2013) no. 9-10, pp. 871-883.

Résumé
Abstract

Nous traitons des réponses thermiques (ou gravitationnelles) des supraconducteurs topologiques et, plus généralement, des phases topologiques. Ces réponses thermiques (tout comme les réponses électromagnétiques pour la charge électrique conservée) fournissent une définition des isolants et supraconducteurs topologiques, qui reste valable au-delà du modèle à une particule. Pour les phases bidimensionnelles, la formule de Streda de la conductivité de Hall de charge est généralisée à la conductivité de Hall thermique. Appliquée aux états de surface de Majorana des supraconducteurs topologiques, cette formule conduit à des fonctions de réponse croisées entre le moment angulaire et la polarisation thermique. Nous discutons également de la théorie des cordes (D-branes) comme outil systématique pour obtenir ces réponses topologiques. En particulier, nous relions lʼinvariant topologique $Z_{2}$ des isolants topologiques introduit par Kane et Mele (ainsi que ses généralisations à dʼautres classes de symétrie en dimensions arbitraires) à la charge de certaines D-branes (celles qui ne saturent pas la limite de Bogomolʼnyi–Prasad–Sommerfield), et vice versa. Nous établissons ainsi un lien entre la stabilité de ces D-branes et celle des isolants topologiques.

We discuss the thermal (or gravitational) responses in topological superconductors and in topological phases in general. Such thermal responses (as well as electromagnetic responses for conserved charge) provide a definition of topological insulators and superconductors beyond the single-particle picture. In two-dimensional topological phases, the Strěda formula for the electric Hall conductivity is generalized to the thermal Hall conductivity. Applying this formula to the Majorana surface states of three-dimensional topological superconductors predicts cross-correlated responses between the angular momentum and thermal polarization (entropy polarization). We also discuss a use of D-branes in string theory as a systematic tool to derive all such topological terms and topological responses. In particular, we relate the $Z_{2}$ index of topological insulators introduced by Kane and Mele (and its generalization to other symmetry classes and to arbitrary dimensions) to the K-theory charge of non-BPS D-branes, and vice versa. We thus establish a link between the stability of non-BPS D-branes and the topological stability of topological insulators.

Publié le : 2013-05-02

DOI : 10.1016/j.crhy.2013.03.002

Keywords: Electromagnetic response, Thermal response, Topological classification, Quantum anomaly, Brane theory
Mot clés : Réponse électromagnétique, Réponse thermique, Classification topologique, Anomalie quantique, Théorie des branes

Affiliations des auteurs :

Akira Furusaki ^{1, 2} ; Naoto Nagaosa ^{2, 3} ; Kentaro Nomura ⁴ ; Shinsei Ryu ⁵ ; Tadashi Takayanagi ^{6, 7}

¹ Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan
² RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
³ Department of Applied Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
⁴ Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
⁵ Department of Physics, University of Illinois, 1110 West Green St, Urbana, IL 61801, United States
⁶ Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan
⁷ Institute for the Physics and Mathematics of the Universe (IPMU), The University of Tokyo, Kashiwa, Chiba 277-8582, Japan

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     title = {Electromagnetic and thermal responses in topological matter: {Topological} terms, quantum anomalies and {D-branes}},
     journal = {Comptes Rendus. Physique},
     pages = {871--883},
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     year = {2013},
     doi = {10.1016/j.crhy.2013.03.002},
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Akira Furusaki; Naoto Nagaosa; Kentaro Nomura; Shinsei Ryu; Tadashi Takayanagi. Electromagnetic and thermal responses in topological matter: Topological terms, quantum anomalies and D-branes. Comptes Rendus. Physique, Volume 14 (2013) no. 9-10, pp. 871-883. doi : 10.1016/j.crhy.2013.03.002. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2013.03.002/

Bibliographie
Cité par

[1] M.V. Berry Proc. R. Soc. Lond. Ser. A, 392 (1984), p. 45

[2] R. Karplus; J.M. Luttinger Phys. Rev., 95 (1954), p. 1154

[3] N. Nagaosa; J. Sinova; S. Onoda; A.H. MacDonald; N.P. Ong Rev. Mod. Phys., 82 (2010), p. 1539

[4] S. Murakami; N. Nagaosa; S.-C. Zhang; J. Sinova; D. Culcer; Q. Niu; N.A. Sinitsyn; T. Jungwirth; A.H. MacDonald Phys. Rev. Lett., 301 (2003), p. 1348

[5] Di Xiao; M.-C. Chang; Q. Niu Rev. Mod. Phys., 82 (2010), p. 1959

[6] Y. Onose; T. Ideue; H. Katsura; Y. Shiomi; N. Nagaosa; Y. Tokura Science, 329 (2010), p. 297

[7] The Quantum Hall Effect (R.E. Prange; S.M. Girvin, eds.), Springer, New York, 1987

[8] D.J. Thouless; M. Kohmoto; P. Nightingale; M. den Nijs Phys. Rev. Lett., 49 (1982), p. 405

[9] M. Kohmoto Ann. Phys. (N.Y.), 160 (1985), p. 355

[10] M.Z. Hasan; C.L. Kane Rev. Mod. Phys., 82 (2010), p. 3045

[11] X.-L. Qi; S.-C. Zhang Rev. Mod. Phys., 83 (2011), p. 1057

[12] C.L. Kane; E.J. Mele; C.L. Kane; E.J. Mele Phys. Rev. Lett., 95 (2005), p. 146802

[13] R. Roy; R. Roy Phys. Rev. B, 79 (2009), p. 195321

[14] J.E. Moore; L. Balents Phys. Rev. B, 75 (2007), p. 121306(R)

[15] L. Fu; C.L. Kane; E.J. Mele Phys. Rev. Lett., 98 (2007), p. 106803

[16] B.A. Bernevig; T.L. Hughes; S.-C. Zhang Science, 314 (2006), p. 1757

[17] L. Fu; C.L. Kane Phys. Rev. B, 76 (2007), p. 045302

[18] X.-L. Qi; T. Hughes; S.-C. Zhang Phys. Rev. B, 78 (2008), p. 195424

[19] A.M. Essin; J.E. Moore; D. Vanderbilt Phys. Rev. Lett., 102 (2009), p. 146805

[20] F. Wilczek Phys. Rev. Lett., 58 (1987), p. 1799

[21] N. Read; D. Green Phys. Rev. B, 61 (2000), p. 10267

[22] A.P. Mackenzie; Y. Maeno Rev. Mod. Phys., 75 (2003), p. 657

[23] A.P. Schnyder; S. Ryu; A. Furusaki; A.W.W. Ludwig; A.P. Schnyder; S. Ryu; A. Furusaki; A.W.W. Ludwig AIP Conf. Proc., 78 (2008), p. 195125

[24] R. Roy | arXiv

[25] X.-L. Qi; T.L. Hughes; S. Raghu; S.-C. Zhang Phys. Rev. Lett., 102 (2009), p. 187001

[26] M.M. Salomaa; G.E. Volovik Phys. Rev. B, 37 (1988), p. 9298

[27]

³He–B should be called topological superfluid, rather than topological SC. Since we are interested in topological properties of fermionic quasi-particle wavefunctions, and since thermal responses we are after are generic and common to both of them, we do not distinguish topological superfluid and topological SCs in this paper.

[28] Y. Aoki; et al.; Y. Wada; et al.; S. Murakawa et al. Phys. Rev. Lett., 95 (2005), p. 075301

[29] S. Murakawa; Y. Wada; Y. Tamura; M. Wasai; M. Saitoh; Y. Aoki; R. Nomura; Y. Okuda; Y. Nagato; M. Yamamoto; S. Higashitani; K. Nagai J. Phys. Soc. Jpn., 80 (2011), p. 013602

[30] Y.S. Hor et al. Phys. Rev. Lett., 104 (2010), p. 057001

[31] L. Fu; E. Berg Phys. Rev. Lett., 105 (2010), p. 097001

[32] S. Sasaki; M. Kriener; K. Segawa; K. Yada; Y. Tanaka; M. Sato; Y. Ando Phys. Rev. Lett., 107 (2011), p. 217001

[33] A.Yu. Kitaev AIP Conf. Proc., 1134 (2009), p. 22

[34] S. Ryu; A.P. Schnyder; A. Furusaki; A.W.W. Ludwig New J. Phys., 12 (2010), p. 065010

[35] P. Strěda J. Phys. C, Solid State Phys., 15 (1982), p. 717

[36] L. Smrčka; P. Strěda J. Phys. C, Solid State Phys., 10 (1977), p. 2153

[37] J.M. Luttinger Phys. Rev., 135 (1964), p. A1505

[38] B. Mashhoon; B. Mashhoon (See for example) | arXiv

[39] K. Nomura; S. Ryu; A. Furusaki; N. Nagaosa Phys. Rev. Lett., 108 (2012), p. 026802

[40] S. Ryu; J.E. Moore; A.W.W. Ludwig Phys. Rev. B, 85 (2012), p. 045104 | arXiv

[41] Z. Wang; X.-L. Qi; S.-C. Zhang Phys. Rev. B, 84 (2011), p. 014527

[42] P. Hosur; S. Ryu; A. Vishwanath Phys. Rev. B, 81 (2010), p. 045120 (For a similar calculation for the electromagnetic θ-angle, see, for example)

[43] K. Fujikawa; H. Suzuki Path Integrals and Quantum Anomalies, Oxford Univ. Press, 2004

[44]

It is, however, not clear if we can reduce the gravitational instanton term, by taking a weak gravitational field limit, to $\propto θ E_{g} \cdot B_{g}$ , which we would expect from the discussion in Section 3.3. This seems related to the fact that the temperature gradient alone does not produce a non-trivial curvature background, which is a source for an anomalous current (topological current) to flow [61]. While the cross-correlation discussed in Section 3.3 is obtained from the application of the Kubo formula to a microscopic model, it may be the case that it has a different origin from that of the anomalous current captured by the gravitational instanton term (23). For our purposes in Sections 4.2 and 4.3, however, we regard the presence/absence of the gravitational instanton term (and of its analogues in other dimensions and in other symmetry classes) as a hallmark of the topologically non-trivial/trivial ground state, without making much contact with thermal/mechanical responses.

[45] G.E. Volovik JETP Lett., 51 (1990), p. 125

[46] E. Witten Nucl. Phys. B, 500 (1997), p. 3 | arXiv

[47] R. Minasian; G.W. Moore J. High Energy Phys., 9711 (1997), p. 002 | arXiv

[48] E. Witten J. High Energy Phys., 9812 (1998), p. 019 | arXiv

[49] P. Horava Adv. Theor. Math. Phys., 2 (1999), p. 1373 | arXiv

[50] S. Ryu; T. Takayanagi Phys. Lett. B, 693 (2010), p. 175

[51] S. Ryu; T. Takayanagi Phys. Rev. D, 82 (2010), p. 086014

[52]

$D p – D q$ systems have common $(d + 1)$ dimensions; this is where a d-dimensional topological insulator exists. From the viewpoint of open string, the $(d + 1)$ dimensions have the Neumann boundary condition at the both ends (called the NN direction). In the same way, $(p + q - 2 d)$ coordinates are along Neumann–Dirichlet (ND) directions, while the remaining $(9 - p - q + d)$ ones are in the DD direction.

[53] J. Polchinski String Theory, vols. 1 and 2, Cambridge Univ. Press, Cambridge, UK, 1998 (402 pp)

[54]

In this section, for illustration purposes, we have not discussed how discrete symmetries are implemented in the D-brane realization of topological phases. We found that particle-hole symmetry corresponds to the orientation reverse Ω of strings, while chiral (or sublattice) symmetry to the parity transformation in one of the DD directions, respectively. Time-reversal is nothing but the orientifold projection. With these implementations of discrete symmetries, our D-brane systems are characterized by the directions of $D p$ , $D q$ and the orientifold. Classifying these conditions (aided by an analysis on the stability of boundary modes of these systems), we found that the allowed D-brane systems are in one-to-one correspondence to the ten-fold classification of topological phases.

[55]

Their edge states are obtained by replacing one of DD directions by a ND direction. The chiral fermions typical in edge states appear due to the topological mechanism. This construction leads to the correct fermion spectra for the topological insulators and their edge states.

[56] J. Polchinski Phys. Rev. Lett., 75 (1995), p. 4724 | arXiv

[57] M.B. Green; J.A. Harvey; G.W. Moore Class. Quantum Gravity, 14 (1997), p. 47 | arXiv

[58] O. Bergman; M.R. Gaberdiel Phys. Lett. B, 441 (1998), p. 133 | arXiv

[59] A. Sen J. High Energy Phys., 9808 (1998), p. 012 | arXiv

[60] A. Sen Int. J. Mod. Phys. A, 20 (2005), p. 5513 | arXiv

[61] M. Stone Phys. Rev. B, 85 (2012), p. 184503

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