9 problems

Localization dichotomy conjecture. The following statements are equivalent: (a) $P$ admits a generalized Wannier basis that is exponentially localized; (b) $P$ admits a generalized…

Marzari–Vanderbilt conjecture. Global minimizers of the MV functional are exponentially localized.

A periodic system in two dimensions is described by a Bloch bundle, and a Wannier function is obtained from a Bloch frame by the inverse Bloch transform. For a non-trivial Bloch bu…

Chern marker conjecture. The Chern marker characterizes the existence of a localized generalized Wannier basis for gapped non-periodic systems: vanishing of $C(P)$ should be equiva…

Localization Dichotomy Conjecture. The following statements are equivalent: $P$ admits a generalized Wannier basis that is exponentially localized; $P$ admits a generalized Wannier…

Let $P$ be a Fermi projection and let $\tilde{X}$ be a modified position operator. Say that $P\tilde{X}P$ has uniform spectral gaps when its spectrum can be decomposed into a disjo…

Let $P$ be a Fermi projection for a non-periodic insulating system, particularly one close to periodic, and let an exponentially localized basis mean a basis of the range of $P$ wh…

Exponentially localized generalized Wannier bases are conjectured to exist more generally when time-reversal symmetry holds. In the setting of non-periodic insulating systems, and…

Let $b$ be a magnetic flux in the interval $(b_1,b_2)$, let $c_1$ be the coefficient appearing in the paper's gap-labelling formula, and let $\Pi_b$ be the corresponding Fermi proj…