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Communications in Mathematical Physics

, Volume 371, Issue 3, pp 1179–1230 | Cite as

Parseval Frames of Exponentially Localized Magnetic Wannier Functions

  • Horia D. Cornean
  • Domenico MonacoEmail author
  • Massimo Moscolari
Article

Abstract

Motivated by the analysis of gapped periodic quantum systems in presence of a uniform magnetic field in dimension \(d \le 3\), we study the possibility to construct spanning sets of exponentially localized (generalized) Wannier functions for the space of occupied states. When the magnetic flux per unit cell satisfies a certain rationality condition, by going to the momentum-space description one can model m occupied energy bands by a real-analytic and \({{\mathbb {Z}}}^{d}\)-periodic family \(\left\{ P(\mathbf{k}) \right\} _{\mathbf{k}\in {{\mathbb {R}}}^{d}}\) of orthogonal projections of rank m. A moving orthonormal basis of \({{\,\mathrm{Ran}\,}}P(\mathbf{k})\) consisting of real-analytic and \({{\mathbb {Z}}}^d\)-periodic Bloch vectors can be constructed if and only if the first Chern number(s) of P vanish(es). Here we are mainly interested in the topologically obstructed case. First, by dropping the generating condition, we show how to algorithmically construct a collection of \(m-1\)orthonormal, real-analytic, and \({{\mathbb {Z}}}^d\)-periodic Bloch vectors. Second, by dropping the linear independence condition, we construct a Parseval frame of \(m+1\) real-analytic and \({{\mathbb {Z}}}^d\)-periodic Bloch vectors which generate \({{\,\mathrm{Ran}\,}}P(\mathbf{k})\). Both algorithms are based on a two-step logarithm method which produces a moving orthonormal basis in the topologically trivial case. A moving Parseval frame of analytic, \({{\mathbb {Z}}}^d\)-periodic Bloch vectors corresponds to a Parseval frame of exponentially localized composite Wannier functions. We extend this construction to the case of magnetic Hamiltonians with an irrational magnetic flux per unit cell and show how to produce Parseval frames of exponentially localized generalized Wannier functions also in this setting. Our results are illustrated in crystalline insulators modelled by 2d discrete Hofstadter-like Hamiltonians, but apply to certain continuous models of magnetic Schrödinger operators as well.

Notes

Acknowledgements

The authors would like to thank G. Panati, G. Nenciu, G. De Nittis and P. Kuchment for inspiring discussions. Financial support from Grant 8021-00084B of the Danish Council for Independent Research | Natural Sciences, from the German Science Foundation (DFG) within the GRK 1838 “Spectral theory and dynamics of quantum systems”, and from the ERC Consolidator Grant 2016 “UNICoSM – Universality in Condensed Matter and Statistical Mechanics” is gratefully acknowledged.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesAalborg UniversityAalborgDenmark
  2. 2.Dipartimento di Matematica e FisicaUniversità degli Studi di Roma TreRomeItaly
  3. 3.Dipartimento di Matematica“La Sapienza” Università di RomaRomeItaly

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