On Parseval frames of exponentially decaying composite Wannier functions
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2018 American Mathematical Society. Let L be a periodic self-adjoint linear elliptic operator in Rn with coefficients periodic with respect to a lattice , e.g. Schrdinger operator (i1 /x A(x))2 + V (x) with periodic magnetic and electric potentials A, V, or a Maxwell operator (x)1 in a periodic medium. Let also S be a finite part of its spectrum separated by gaps from the rest of the spectrum. We address here the question of existence of a finite set of exponentially decaying Wannier functions wj (x) such that their -shifts wj, (x) = wj (x ) for span the whole spectral subspace corresponding to S. It was shown by D. Thouless in 1984 that a topological obstruction sometimes exists to finding exponentially decaying wj, that form an orthonormal (or any) basis of the spectral subspace. This obstruction has the form of non-triviality of certain finite dimensional (with the dimension equal to the number of spectral bands in S) analytic vector bundle (Bloch bundle), which we denote S. It was shown by G. Nenciu in 1983 that in the presence of time reversal symmetry (which implies absence of magnetic fields), and if S is a single band, the bundle is trivial and thus the desired Wannier functions do exist. In 2007, G. Panati proved that in dimensions n 3, even if S consists of several spectral bands, the time reversal symmetry removes the obstruction as well. If the bundle is non-trivial, it was shown in 2009 by one of the authors that it is always possible to find a finite number l (estimated there as m l 2n m) of exponentially decaying Wannier functions wj such that their -shifts form a tight (Parseval) frame in the spectral subspace. A Parseval frame is the next best thing after an orthonormal basis (unavailable in the presence of the topological obstacle). This appears to be the best one can do when the topological obstruction is present. Here we significantly improve the estimate on the number of extra Wannier functions needed, showing that in physical dimensions the number l can be chosen equal to m+1, i.e. only one extra family of Wannier functions is required. This is the lowest number possible in the presence of the topological obstacle. The result for dimension four is also stated (without a proof), in which case m + 2 functions are needed.