Luttinger's Theorem Violation and Green's Function Topological Invariants in a Fractional Chern Insulator

TL;DR

This paper investigates the violation of Luttinger's theorem in fractional Chern insulators, demonstrating how Green's function topological invariants encode fractionalized quasiparticle properties and proposing experimental methods to measure these invariants.

Contribution

It provides the first direct numerical evidence of Luttinger's theorem violation in fractional Chern insulators and links Green's function invariants to measurable topological responses.

Findings

Luttinger's theorem is violated in fractional Chern insulators.

The fractional Chern number is reflected in the Středa response of the Luttinger integral.

An experimental protocol is proposed to measure Green's function topological invariants.

Abstract

Luttinger's theorem constrains the particle density of interacting fermions through global properties of the single-particle Green's function, and its violation signals a breakdown of the identification between the quantized Hall response and the Green-function-based Ishikawa-Matsuyama invariant. This phenomenon becomes especially compelling in strongly correlated topological phases, such as fractional Chern insulators, where fractionalized quasiparticles lack an adiabatic connection to electrons, raising the question of how Green's-function-based topological invariants manifest in such phases. Using exact diagonalization of the fermionic Harper-Hofstadter-Hubbard model, we compute bulk single-particle Green's functions deep inside a fractional Chern insulating phase and directly evaluate the Luttinger count, its possible correction (the Luttinger integral), and the Ishikawa-Matsuyama invariant $N_3[\mathrm{G}]$. We demonstrate a clear violation of Luttinger's theorem and show that the fractional nature of the many-body Chern number is encoded in the St\v{r}eda response of the Luttinger integral, while the integer invariant $N_3[\mathrm{G}]$ arises from the St\v{r}eda response of the Luttinger count. We also analytically prove that $N_3[\mathrm{G}]$ is fully determined by the Luttinger count together with the Chern number of the occupied Bloch band, upon neglecting Bloch-band mixing. Finally, we propose an experimental protocol to extract all Green-function-based topological invariants from local density-of-states measurements, experimentally accessible in fractional quantum Hall systems.