Localization properties of Chern insulators

TL;DR

This paper investigates how the electron Green's function behaves in Chern insulators across various dimensions and band structures, revealing fundamental limits on its decay related to the quantum Hall response.

Contribution

It provides rigorous proofs that connect the decay properties of the Green's function with the topological quantum Hall response in Chern insulators.

Findings

Green's function cannot decay super-exponentially in finite-range Hamiltonians with nonzero Hall response.

For general Hamiltonians, the Green's function cannot be finite-range if the Hall response is nonzero.

Uses algebraic geometry methods to establish these fundamental decay constraints.

Abstract

We study the localization properties of the equal-time electron Green's function in a Chern insulator in an arbitrary dimension and with an arbitrary number of bands. We prove that the Green's function cannot decay super-exponentially if the Hamiltonian is finite-range and the quantum Hall response is nonzero. For a general band Hamiltonian (possibly infinite-range), we prove that the Green's function cannot be finite-range if the quantum Hall response is nonzero. The proofs use methods of algebraic geometry.