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We consider a real periodic Schr\"odinger operator and a physically relevantfamily of $m \geq 1$ Bloch bands, separated by a gap from the rest of thespectrum, and we investigate the localization properties of the correspondingcomposite Wannier functions. To this aim, we show that in dimension $d \leq 3$there exists a global frame consisting of smooth quasi-Bloch functions whichare both periodic and time-reversal symmetric. Aiming to applications incomputational physics, we provide a constructive algorithm to obtain such aBloch frame. The construction yields the existence of a basis of compositeWannier functions which are real-valued and almost-exponentially localized.  The proof of the main result exploits only the fundamental symmetries of theprojector on the relevant bands, allowing applications, beyond the modelspecified above, to a broad range of gapped periodic quantum systems with atime-reversal symmetry of bosonic type.

Construction of real-valued localized composite Wannier functions for insulators