@article {Fiorenza2016,
	title = {Construction of Real-Valued Localized Composite Wannier Functions for Insulators},
	journal = {Annales Henri Poincar{\'e}},
	volume = {17},
	number = {1},
	year = {2016},
	month = {Jan},
	pages = {63{\textendash}97},
	abstract = {<p>We consider a real periodic Schr{\"o}dinger operator and a physically relevant family of $m \geq 1$ Bloch bands, separated by a gap from the rest of the spectrum, and we investigate the localization properties of the corresponding composite Wannier functions. To this aim, we show that in dimension $d\leq 3$, there exists a global frame consisting of smooth quasi-Bloch functions which are both periodic and time-reversal symmetric. Aiming to applications in computational physics, we provide a constructive algorithm to obtain such a Bloch frame. The construction yields the existence of a basis of composite Wannier functions which are real-valued and almost-exponentially localized. The proof of the main result exploits only the fundamental symmetries of the projector on the relevant bands, allowing applications, beyond the model specified above, to a broad range of gapped periodic quantum systems with a time-reversal symmetry of bosonic type.</p>},
	issn = {1424-0661},
	doi = {10.1007/s00023-015-0400-6},
	url = {https://doi.org/10.1007/s00023-015-0400-6},
	author = {Domenico Fiorenza and Domenico Monaco and Gianluca Panati}
}
