Abstract
We review recent results concerning the localization of gapped periodic systems of independent fermions, as, e.g., electrons in Chern and Quantum Hall insulators. We show that there is a “localization dichotomy” which shows some analogies with phase transitions in Statistical Mechanics: either there exists a system of exponentially localized composite Wannier functions for the Fermi projector, or any possible system of composite Wannier functions yields a diverging expectation value for the squared position operator. This fact is largely model-independent, covering both tight-binding and continuous models. The results are discussed with emphasis on the main ideas and the broader context, avoiding most of the technical details.
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Notes
- 1.
The analogous claim for spin transport does not hold in general, since the vanishing of spin torque response is a necessary condition to obtain the equality of spin conductance and spin conductivity for two-dimensional systems, see [14] and references therein.
- 2.
An exception is provided by the well-known Landau Hamiltonian. Notice, however, that if a periodic background potential is included in the model, one is generically back to the absolutely continuous setting.
- 3.
We use Hartree atomic units, and moreover we reabsorb the reciprocal of the speed of light 1 / c and the sign of the charge carriers in the definition of the function \(A_{\Gamma }\).
- 4.
We carefully avoid the use of the word “lattice”, since the latter has a different meaning in physics and in mathematics. In particular, a honeycomb or a triangular structure is not a lattice in the mathematical sense (i.e., a discrete subgroup of \((\mathbb {R}^d, +)\) with maximal rank). As far as the Bravais lattice \(\Gamma \) is concerned, no ambiguity raises, since it is a lattice in both senses.
- 5.
It is easy to check that, for every lattice \(\Gamma \subset \mathbb {R}^d\), one has \(|Y| |\mathbb {B}| = (2\pi )^d\).
- 6.
To be precise, one should introduce the Hilbert space
$$ \mathcal {H}_{\tau } = \left\{ \Phi \in L^2{}_{\mathrm {loc}}(\mathbb {R}^d, \ell ^2(F)) : \Phi _{\mathbf {k}+ \lambda } = \tau _{\lambda } \Phi _{\mathbf {k}} \text { for a.e. } \mathbf {k}\in \mathbb {R}^d, \forall \lambda \in \Gamma ^* \right\} . $$Then the transform \(\mathcal {U}\), defined by (4.2), extends to a unitary operator from \(\ell ^2(\mathcal {C})\) to \(\mathcal {H}_{\tau }\), and formula (4.4) holds true for every \(\Phi \in \mathcal {H}_{\tau }\). The subtle point is that the components of an element of \( \ell ^2(F) \simeq \mathbb {C}^N\) are intrinsic, while the components of an element of \( \ell ^2(Y) \simeq \mathbb {C}^N\) change according to the choice of the periodicity cell. Not all the \(\mathbb {C}^N\)s are equal, some are more equal than others.
- 7.
To agree with the physics literature, we use the notation \(H_{\mathbf {k}}\), \(P_{\mathbf {k}}\) and \(u_{n,\mathbf {k}}\) although in mathematics the subscript usually refers to labels varying in a discrete set, and clearly \(\mathbf {k}\in \mathbb {R}^d/\Gamma ^* \simeq \mathbb {T}^d\) varies in a continuum set.
- 8.
The normalization here differs from the one used in [28] but agrees with the one used in [24], which is also the most common convention among solid-state and computational physicists. The latter is more convenient when a numerical grid in \(\mathbf {k}\)-space is considered, which becomes finer and finer in the thermodynamic limit.
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Acknowledgements
I am indebted to Herbert Spohn for his precious guidance in my early scientific steps. The opportunity to work with him has been invaluable to me. This survey would not have been possible without the fruitful collaboration with many colleagues, including (in chronological order) S. Teufel, Ch. Brouder, N. Marzari, A. Pisante, D. Fiorenza, D. Monaco, É. Cancès, A. Levitt, and G. Stolz.
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Panati, G. (2018). The Localization Dichotomy for Gapped Periodic Systems and Its Relevance for Macroscopic Transport. In: Cadamuro, D., Duell, M., Dybalski, W., Simonella, S. (eds) Macroscopic Limits of Quantum Systems. MaLiQS 2017. Springer Proceedings in Mathematics & Statistics, vol 270. Springer, Cham. https://doi.org/10.1007/978-3-030-01602-9_11
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