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.
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Notes
- 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).
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Acknowledgements
The author is indebted with D. Fiorenza and G. Panati for valuable discussions, as well as with G. dell’Antonio and A. Michelangeli for the invitation to the INdAM meeting “Contemporary Trends in the Mathematics of Quantum Mechanics”. Financial support from INdAM and from the German Science Foundation (DFG) within the GRK 1838 “Spectral theory and dynamics of quantum systems” is gratefully acknowledged.
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Monaco, D. (2017). Chern and Fu–Kane–Mele Invariants as Topological Obstructions. In: Michelangeli, A., Dell'Antonio, G. (eds) Advances in Quantum Mechanics. Springer INdAM Series, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-319-58904-6_12
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