Top

Published in:

2017 | OriginalPaper | Chapter

Chern and Fu–Kane–Mele Invariants as Topological Obstructions

Author : Domenico Monaco

Published in: Advances in Quantum Mechanics

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

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.

Dont have a licence yet? Then find out more about our products and how to get one now:

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!

inform now

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!

inform now

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!

inform now

previous chapter Dispersive Estimates for Schrödinger Operators with Point Interactions in ℝ3

next chapter Norm Approximation for Many-Body Quantum Dynamics and Bogoliubov Theory

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].

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.

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

$$\displaystyle{U(k)A_{\mu }^{\varPhi }(k) = A_{\mu }^{\varPsi }(k)U(k) -\mathrm{ i}\partial _{\mu }U(k).}$$

The action of any (anti)unitary operator on $\mathbb{H}_{\text{f}}$ is lifted to $\mathbb{H}_{\text{f}}^{m}$ componentwise.

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).

J.E. Avron, R. Seiler, B. Simon, Charge deficiency, charge transport and comparison of dimensions. Commun. Math. Phys. 159, 399–422 (1994)MathSciNetCrossRefMATH

J. Bellissard, A. van Elst, H. Schulz-Baldes, The noncommutative geometry of the quantum Hall effect. J. Math. Phys. 35, 5373–5451 (1994)MathSciNetCrossRefMATH

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

Ch. Brouder, G. Panati, M. Calandra, Ch. Mourougane, N. Marzari, Exponential localization of Wannier functions in insulators. Phys. Rev. Lett. 98, 046402 (2007)CrossRef

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

H.D. Cornean, I. Herbst, G. Nenciu, On the construction of composite Wannier functions. Ann. Henri Poincaré 17, 3361–3398 (2016)MathSciNetCrossRefMATH

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

G. De Nittis, K. Gomi, Classification of “Quaternionic” Bloch bundles. Commun. Math. Phys. 339, 1–55 (2015)CrossRefMATH

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.

T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, 1966)CrossRefMATH

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.

G. Panati, Triviality of Bloch and Bloch-Dirac bundles. Ann. Henri Poincaré 8, 995–1011 (2007)MathSciNetCrossRefMATH

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

Title: Chern and Fu–Kane–Mele Invariants as Topological Obstructions
Author: Domenico Monaco
Publisher: Springer International Publishing
Book: Advances in Quantum Mechanics
Print ISBN: 978-3-319-58903-9

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

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-3-319-58904-6_12

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner