Triviality of Bloch and Bloch–Dirac Bundles

By Gianluca Panati. 2006

202 citations recorded by the source corpus.

Abstract.In the framework of the theory of an electron in a periodic potential, we reconsider the longstanding problem of the existence of smooth and periodic quasi-Bloch functions, which is shown to be equivalent to the triviality of the Bloch bundle. By exploiting the time-reversal symmetry of the Hamiltonian and some bundle-theoretic methods, we show that the problem has a positive answer in any dimension d ≤ 3, thus generalizing a previous result by G. Nenciu. We provide a general formulation of the result, aiming at the application to the Dirac equation with a periodic potential and to piezoelectricity.

Subfields: Atomic and Molecular Physics, and Optics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/84077e397b34112ce8635e3fa7e5dc5b98963515

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In Atomic and Molecular Physics, and Optics

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Citations in the corpus

Superconductivity, superfluidity and quantum geometry in twisted multilayer systems (2021, background): “This is explained by the fact that if the Chern number is nonzero one cannot find exponentially localized Wannier functions [37, 38]; at best the Wannier states are algebraically decaying with a known exponent 1/r(2) [39].”
Maximally-localized Wannier Functions: Theory and Applications (2011, background): “The question of exponential decay in 2D and 3D was finally settled by Brouder et al. (2007) who showed, as a corollary to a theorem by Panati (2007), that a necessary and sufficient condition for the existence of exponentially localized WFs in 2D and 3D is that the so-called “Chern invariants” for…”
An overview of periodic elliptic operators (2015, background): “[176, 228] and references therein (see also [292] for the survey of techniques of Banach bundles).”
Z2Pack: Numerical Implementation of Hybrid Wannier Centers for Identifying Topological Materials (2016, background): “[84] Gianluca Panati, “Triviality of Bloch and Bloch–Dirac bundles,” in Annales Henri Poincaré, Vol.”