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Motivated by the analysis of gapped periodic quantum systems in presence of auniform magnetic field in dimension $d \le 3$, we study the possibility toconstruct spanning sets of exponentially localized (generalized) Wannierfunctions for the space of occupied states. When the magnetic flux per unitcell satisfies a certain rationality condition, by going to the momentum-spacedescription one can model $m$ occupied energy bands by a real-analytic and$\mathbb Z^{d}$-periodic family $\{P({\bf k})\}_{{\bf k} \in \mathbb R^{d}}$ oforthogonal projections of rank $m$. A moving orthonormal basis of $\mathrm{Ran}P({\bf k})$ consisting of real-analytic and $\mathbb Z^d$-periodic Blochvectors 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 algorithmicallyconstruct a collection of $m-1$ orthonormal, real-analytic, and periodic Blochvectors. Second, by dropping the linear independence condition, we construct aParseval frame of $m+1$ real-analytic and periodic Bloch vectors which generate$\mathrm{Ran} P({\bf k})$. Both algorithms are based on a two-step logarithmmethod which produces a moving orthonormal basis in the topologically trivialcase. A moving Parseval frame of analytic, periodic Bloch vectors correspondsto a Parseval frame of exponentially localized composite Wannier functions. Weextend this construction to the case of magnetic Hamiltonians with anirrational magnetic flux per unit cell and show how to produce Parseval framesof exponentially localized generalized Wannier functions also in this setting.Our results are illustrated in crystalline insulators modelled by $2d$ discreteHofstadter-like Hamiltonians, but apply to certain continuous models ofmagnetic Schr\"{o}dinger operators as well.

Parseval frames of exponentially localized magnetic Wannier functions