Connecting the Many-Body Chern Number to Luttinger's Theorem through St\v{r}eda's Formula

TL;DR

This paper derives a relation between the many-body Chern number and single-particle Green's functions using Streda's formula, linking topological invariants to Luttinger's theorem and clarifying conditions for quantized Hall responses in correlated insulators.

Contribution

It introduces a decomposition of the many-body topological invariant into known and new terms, connecting it to Luttinger's theorem and providing insights into fractionalization in correlated topological phases.

Findings

Decomposition of the many-body Chern number into $N_3[G]$ and $\Delta N_3[G]$

Expression of the invariant in terms of Green's functions and magnetic derivatives

Conditions under which the Hall conductivity is determined solely by $N_3[G]$

Abstract

Relating the quantized Hall response of correlated insulators to many-body topological invariants is a key challenge in topological quantum matter. Here, we use Streda's formula to derive an expression for the many-body Chern number in terms of the single-particle interacting Green's function and its derivative with respect to a magnetic field. In this approach, we find that this many-body topological invariant can be decomposed in terms of two contributions, $N_3[G] + \Delta N_3[G]$, where $N_3[G]$ is known as the Ishikawa-Matsuyama invariant and where the second term involves derivatives of Green's function and the self energy with respect to the magnetic perturbation. As a by-product, the invariant $N_3[G]$ is shown to stem from the derivative of Luttinger's theorem with respect to the probe magnetic field. These results reveal under which conditions the quantized Hall conductivity of correlated topological insulators is solely dictated by the invariant $N_3[G]$, providing new insight on the origin of fractionalization in strongly-correlated topological phases.