Abstract
Deformations in piezoelectric materials lead to conduction effects, which are due to two contributions: the relative displacements of the ionic cores, and the so-called orbital polarization. This work is devoted to the rigorous derivation of the celebrated King-Smith and Vanderbilt formula for orbital polarization in a generalized setting which includes continuous random systems among others.
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Notes
- 1.
In the sense of the resolvent.
- 2.
We will assume that \(\varOmega \) is also metrizable, and in turn separable. This assumption implies that \(L^2(\varOmega )\) is a separable Hilbert space.
- 3.
In the sense of Bellissard [2].
- 4.
In the interesting examples \(\mathcal {G}\) is also separable and metrizable (e.g. \(\mathcal {G}=\mathbb {R}^d,\mathbb {Z}^d,\mathbb {T}^d\)) and this implies that \(L^2(\mathcal {G})\) is a separable Hilbert space.
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Acknowledgements
GD’s research is supported by the grant Fondecyt Regular - 1230032. DP’s research is supported by ANID-Subdirección de Capital Humano/ Doctorado Nacional/ 2022-21220144. GD’s would like to thank the Alexander von Humboldt Foundation for supporting his stay at the University of Erlangen-Nürnberg during July 2022 where the large part of this work was completed. He is also grateful to Camping due barche (Scanzano, Italy) where the peaceful atmosphere of this place provided an invaluable help for the preparation of the final version of this manuscript. The authors are indebted to M. Lein and S. Teufel for many stimulating discussions.
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De Nittis, G., Polo Ojito, D. (2023). Topological Polarization in Disordered Systems. In: Correggi, M., Falconi, M. (eds) Quantum Mathematics I. INdAM 2022. Springer INdAM Series, vol 57. Springer, Singapore. https://doi.org/10.1007/978-981-99-5894-8_6
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