Localization anisotropy and complex geometry in two-dimensional insulators

The localization tensor is a measure of distinguishability between insulators and metals. This tensor is related to the quantum metric tensor associated with the occupied bands in momentum space. In two dimensions and in the thermodynamic limit, it defines a flat Riemannian metric over the twist-angle space, topologically a torus, which endows this space with a complex structure, described by a complex parameter $\tau$. It is shown that the latter is a physical observable related to the anisotropy of the system. The quantity $\tau$ and the Riemannian volume of the twist-angle space provide an invariant way to parametrize the flat quantum metric obtained in the thermodynamic limit. Moreover, if by changing the couplings of the theory, the system undergoes quantum phase transitions in which the gap closes, the complex structure $\tau$ is still well defined, although the metric diverges (metallic state), and it is fixed by the form of the Hamiltonian near the gap closing points. The Riemannian volume is responsible for the divergence of the metric at the phase transition.