Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Bücher
Zeitschriften
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
MTZ-Fachkongress

Antriebe und Energiesysteme von morgen


Austausch von Automobil- und Energiewirtschaft

Am 14./15. Mai geht es in Chemnitz um die Mobilitätswende durch Elektro- und Wasserstofffahrzeuge.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Shell Interactions for Dirac Operators Kapitel lesen Erstes Kapitel lesen

2017 | OriginalPaper | Buchkapitel

Chern and Fu–Kane–Mele Invariants as Topological Obstructions

verfasst von : Domenico Monaco

Erschienen in: Advances in Quantum Mechanics

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

The use of topological invariants to describe geometric phases of quantum matter has become an essential tool in modern solid state physics. The first instance of this paradigmatic trend can be traced to the study of the quantum Hall effect, in which the Chern number underlies the quantization of the transverse Hall conductivity. More recently, in the framework of time-reversal symmetric topological insulators and quantum spin Hall systems, a new topological classification has been proposed by Fu, Kane and Mele, where the label takes value in \(\mathcal{Z}_{2}\).
We illustrate how both the Chern number \(c \in \mathbb{Z}\) and the Fu–Kane–Mele invariant \(\delta \in \mathbb{Z}_{2}\) of 2-dimensional topological insulators can be characterized as topological obstructions. Indeed, c quantifies the obstruction to the existence of a frame of Bloch states for the crystal which is both continuous and periodic with respect to the crystal momentum. Instead, δ measures the possibility to impose a further time-reversal symmetry constraint on the Bloch frame.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel Dispersive Estimates for Schrödinger Operators with Point Interactions in ℝ3
Nächstes Kapitel Norm Approximation for Many-Body Quantum Dynamics and Bogoliubov Theory
Fußnoten
1
In continuous models, where H is a Schrödinger operator, these assumptions usually amount to asking that the electromagnetic potentials be infinitesimally Kato-small (possibly in the sense of quadratic forms) with respect to the kinetic part [26].
 
2
In order for (P2) and (P3) to be compatible with each other, one should also require that τ λ  Θ = τ λ −1 Θ for all λ ∈ Λ. We will assume this in the following.
 
3
An easy way to realize this is the following. The connection matrices A μ Ψ (k) and A μ Φ (k) satisfy
$$\displaystyle{\varPsi (k) \vartriangleleft A_{\mu }^{\varPsi }(k) = -\mathrm{i}\partial _{\mu }\varPsi (k),\quad \varPhi (k) \vartriangleleft A_{\mu }^{\varPhi }(k) = -\mathrm{i}\partial _{\mu }\varPhi (k).}$$
As by definition we have Φ(k) = Ψ(k) ⊲ U(k), we obtain
$$\displaystyle\begin{array}{rcl} \varPsi (k) \vartriangleleft (U(k)A_{\mu }^{\varPhi }(k))& =& \left (\varPsi (k) \vartriangleleft U(k)\right ) \vartriangleleft A_{\mu }^{\varPhi }(k) =\varPhi (k) \vartriangleleft A_{\mu }^{\varPhi }(k) = -\mathrm{i}\partial _{\mu }\varPhi (k) {}\\ & =& -\mathrm{i}\partial _{\mu }\left (\varPsi (k) \vartriangleleft U(k)\right ) = \left (-\mathrm{i}\partial _{\mu }\varPsi (k)\right ) \vartriangleleft U(k) +\varPsi (k) \vartriangleleft \left (-\mathrm{i}\partial _{\mu }U(k)\right ) {}\\ & =& \left (\varPsi (k) \vartriangleleft A_{\mu }^{\varPsi }(k)\right ) \vartriangleleft U(k) +\varPsi (k) \vartriangleleft \left (-\mathrm{i}\partial _{\mu }U(k)\right ) {}\\ & =& \varPsi (k) \vartriangleleft \left (A_{\mu }^{\varPsi }(k)U(k) -\mathrm{ i}\partial _{\mu }U(k)\right ) {}\\ \end{array}$$
by which we deduce that
$$\displaystyle{U(k)A_{\mu }^{\varPhi }(k) = A_{\mu }^{\varPsi }(k)U(k) -\mathrm{ i}\partial _{\mu }U(k).}$$
 
4
The action of any (anti)unitary operator on \(\mathbb{H}_{\text{f}}\) is lifted to \(\mathbb{H}_{\text{f}}^{m}\) componentwise.
 
5
The presence of the reshuffling matrix ɛ is needed to make the time-reversal symmetry condition self-consistent. This follows essentially from the fact that the antiunitary operator Θ defines by restriction a symplectic structure on the invariant subspace \(\mathop{\mathrm{Ran}}\nolimits P(k_{\sharp }) \subset \mathbb{ H}_{\text{f}}\) if \(k_{\sharp } \equiv -k_{\sharp }\bmod \varLambda\). Notice that in particular the rank m of P(k) must be even under (P3).
 
Literatur
1.
J.E. Avron, R. Seiler, B. Simon, Charge deficiency, charge transport and comparison of dimensions. Commun. Math. Phys. 159, 399–422 (1994)MathSciNetCrossRefMATH
2.
J. Bellissard, A. van Elst, H. Schulz-Baldes, The noncommutative geometry of the quantum Hall effect. J. Math. Phys. 35, 5373–5451 (1994)MathSciNetCrossRefMATH
3.
A.J. Bestwick, E.J. Fox, X. Kou, L. Pan, K.L. Wang, D. Goldhaber-Gordon, Precise quantization of the anomalous Hall effect near zero magnetic field. Phys. Rev. Lett. 114, 187201 (2015)CrossRef
4.
Ch. Brouder, G. Panati, M. Calandra, Ch. Mourougane, N. Marzari, Exponential localization of Wannier functions in insulators. Phys. Rev. Lett. 98, 046402 (2007)CrossRef
5.
C.Z. Chang et al., High-precision realization of robust quantum anomalous Hall state in a hard ferromagnetic topological insulator. Nat. Mater. 14, 473 (2015)CrossRef
6.
H.D. Cornean, I. Herbst, G. Nenciu, On the construction of composite Wannier functions. Ann. Henri Poincaré 17, 3361–3398 (2016)MathSciNetCrossRefMATH
7.
H.D. Cornean, D. Monaco, S. Teufel, Wannier functions and \(\mathbb{Z}_{2}\) invariants in time-reversal symmetric topological insulators. Rev. Math. Phys. 29, 1730001 (2017)MathSciNetCrossRefMATH
8.
G. De Nittis, K. Gomi, Classification of “Quaternionic” Bloch bundles. Commun. Math. Phys. 339, 1–55 (2015)CrossRefMATH
9.
B.A. Dubrovin, S.P. Novikov, A.T. Fomenko, Modern Geometry – Methods and Applications, Part II: The Geometry and Topology of Manifolds. Graduate Texts in Mathematics, vol. 93 (Springer, New York, 1985)
10.
D. Fiorenza, D. Monaco, G. Panati, \(\mathbb{Z}_{2}\) invariants of topological insulators as geometric obstructions. Commun. Math. Phys. 343, 1115–1157 (2016)
11.
L. Fu, C.L. Kane, Time reversal polarization and a Z 2 adiabatic spin pump. Phys. Rev. B 74, 195312 (2006)CrossRef
12.
K. Fu, C.L. Kane, E.J. Mele, Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007)CrossRef
13.
G.M. Graf, Aspects of the integer quantum hall effect, in Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday, ed. by F. Gesztesy, P. Deift, C. Galvez, P. Perry, W. Schlag. Proceedings of Symposia in Pure Mathematics, vol. 76 (American Mathematical Society, Providence, RI, 2007), pp. 429–442
14.
F.D.M. Haldane, Model for a quantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett. 61, 2017 (1988)
15.
M.Z. Hasan, C.L. Kane, Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010)CrossRef
16.
L.-K. Hua, On the theory of automorphic functions of a matrix variable I – Geometrical basis. Am. J. Math. 66, 470–488 (1944)CrossRefMATH
17.
D. Husemoller, Fibre Bundles, 3rd edn. Graduate Texts in Mathematics, vol. 20 (Springer, New York, 1994)
18.
C.L. Kane, E.J. Mele, Z 2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005)CrossRef
19.
20.
M. Kohmoto, Topological invariant and the quantization of the Hall conductance. Ann. Phys. 160, 343–354 (1985)MathSciNetCrossRef
21.
N. Marzari, A.A. Mostofi, J.R. Yates, I. Souza, D. Vanderbilt, Maximally localized Wannier functions: theory and applications. Rev. Mod. Phys. 84, 1419 (2012)CrossRef
22.
D. Monaco, G. Panati, Symmetry and localization in periodic crystals: triviality of Bloch bundles with a Fermionic time-reversal symmetry. Acta Appl. Math. 137, 185–203 (2015)MathSciNetCrossRefMATH
23.
24.
G. Panati, A. Pisante, Bloch bundles, Marzari-Vanderbilt functional and maximally localized Wannier functions. Commun. Math. Phys. 322, 835–875 (2013)MATH
25.
E. Prodan, H. Schulz-Baldes, Bulk and Boundary Invariants for Complex Topological Insulators: From K-theory to Physics. Mathematical Physics Studies (Springer, Basel, 2016)CrossRefMATH
26.
M. Reed, B. Simon, Methods of Modern Mathematical Physics. Volume IV: Analysis of Operators (Academic, New York, 1978)
27.
D.J. Thouless, M. Kohmoto, M.P. Nightingale, M. den Nijs, Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982)CrossRef
28.
K. von Klitzing, G. Dorda, M. Pepper, New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 45, 494 (1980)CrossRef
Metadaten
Titel
Chern and Fu–Kane–Mele Invariants as Topological Obstructions
verfasst von
Domenico Monaco
Copyright-Jahr
2017
Buch
Advances in Quantum Mechanics

Print ISBN: 978-3-319-58903-9

Electronic ISBN: 978-3-319-58904-6

Copyright-Jahr: 2017

DOI
https://doi.org/10.1007/978-3-319-58904-6_12

Premium Partner

    Bildnachweise