Localization of Generalized Wannier Bases Implies Chern Triviality in Non-periodic Insulators

Abstract

We investigate the relation between the localization of generalized Wannier bases and the topological properties of two-dimensional gapped quantum systems of independent electrons in a disordered background, including magnetic fields, as in the case of Chern insulators and quantum Hall systems. We prove that the existence of a well-localized generalized Wannier basis for the Fermi projection implies the vanishing of the Chern character, which is proportional to the Hall conductivity in the linear response regime. Moreover, we state a localization dichotomy conjecture for general non-periodic gapped quantum systems.

Finding spectral gaps in quasicrystals

“…Proof. Without loss of generality, we can assume that m > l. To compute the sum in ( 14), we use the Cauchy-Maclaurin criterion adapted to our setting in same spirit as in [80]. The main idea is to estimate the sum from above by using an integral.…”

Ultra-generalized Wannier bases: Are they relevant to topological transport?

“…In [22] it is shown that GWBs can be used to investigate topological and transport properties of non-periodic gapped quantum systems. In particular, the Fermi projections that admit an exponentially localized (or just a well-localized [22]) GWB with a uniformly discrete set D, are Chern trivial in the sense that their Chern character is zero.…”

“…In [22] it is shown that GWBs can be used to investigate topological and transport properties of non-periodic gapped quantum systems. In particular, the Fermi projections that admit an exponentially localized (or just a well-localized [22]) GWB with a uniformly discrete set D, are Chern trivial in the sense that their Chern character is zero. As well known, the Chern character is proportional to the Hall conductivity, thus showing that topological quantum transport and well-localized generalized Wannier bases cannot coexist.…”

“…This point of view has been pushed forward and generalized in several directions. Ludewig and Thiang [18] considered systems which are periodic with respect to a suitable non-abelian discrete group, while Bourne and Mesland generalized [22] to a broader C * -algebraic setting [6]. Lu and Stubbs enlarged the class of welllocalized GWB for which the Chern character of the Fermi projection vanishes [19,20].…”

Compact Wannier Functions in One Dimension

“…The localization properties of Wannier functions also have a direct bearing on the topological properties of electronic bands. [5][6][7][8][9]. Exponentially localized Wannier functions can be constructed if and only if the Chern number and Hall conductivity of the corresponding filled band(s) are zero [5,10,11].…”