Parseval frames of localized Wannier functions

Horia D. Cornean, Domenico Monaco

Published 2017-04-04Version 1

Let $d \le 3$ and consider a real analytic and $\mathbb{Z}^d$-periodic family $\{P(\mathbf{k})\}_{\mathbf{k} \in \mathbb{R}^d}$ 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 can construct a collection of $m-1$ orthonormal, real analytic, and $\mathbb{Z}^d$-periodic Bloch vectors. Second, by dropping the orthonormality condition, we can construct a Parseval frame of $m+d-1$ real analytic and $\mathbb{Z}^d$-periodic Bloch vectors which generate $\mathrm{Ran}\, P(\mathbf{k})$. Both constructions are based on a two-step logarithm method which produces a moving orthonormal basis in the topologically trivial case. In applications to condensed matter systems, a moving Parseval frame of analytic, $\mathbb{Z}^d$-periodic Bloch vectors generates a Parseval frame of exponentially localized composite Wannier functions for the occupied states of a gapped periodic Hamiltonian.