Papers Published
Abstract:
Exponentially-localized Wannier functions are a basis of the Fermi projection
of a Hamiltonian consisting of functions which decay exponentially fast in
space. In two and three spatial dimensions, it is well understood for periodic
insulators that exponentially-localized Wannier functions exist if and only if
there exists an orthonormal basis for the Fermi projection with finite second
moment (i.e. all basis elements satisfy $\int |\boldsymbol{x}|^2
|w(\boldsymbol{x})|^2 \,\text{d}{\boldsymbol{x}} < \infty$). In this work, we
establish a similar result for non-periodic insulators in two spatial
dimensions. In particular, we prove that if there exists an orthonormal basis
for the Fermi projection which satisfies $\int |\boldsymbol{x}|^{5 + \epsilon}
|w(\boldsymbol{x})|^2 \,\text{d}{\boldsymbol{x}} < \infty$ for some $\epsilon >
0$ then there also exists an orthonormal basis for the Fermi projection which
decays exponentially fast in space. This result lends support to the
Localization Dichotomy Conjecture for non-periodic systems recently proposed by
Marcelli, Monaco, Moscolari, and Panati