Many-body Chern number without integration

TL;DR

This paper demonstrates that the many-body Chern number can be computed without integration over twisted boundary conditions, simplifying calculations by showing the integrand is effectively quantized and errors decay exponentially with system size.

Contribution

The work proves that the integration over twisted angles is unnecessary for calculating the many-body Chern number, reducing computational complexity.

Findings

Integrand of the many-body Chern number is effectively quantized.

Error in calculation decays exponentially with system size.

Computing the Berry connection at a single boundary condition suffices.

Abstract

The celebrated work of Niu, Thouless, and Wu demonstrated the quantization of Hall conductance in the presence of many-body interactions by revealing the many-body counterpart of the Chern number. The generalized Chern number is formulated in terms of the twisted angles of the boundary condition, instead of the single particle momentum, and involves an integration over all possible twisted angles. However, this formulation is physically unnatural, since topological invariants directly related to observables should be defined for each Hamiltonian under a fixed boundary condition. In this work, we show via numerical calculations that the integration is indeed unnecessary - the integrand itself is effectively quantized and the error decays exponentially with the system size. This implies that the numerical cost in computing the many-body Chern number could, in principle, be significantly reduced as it suffices to compute the Berry connection for a single value of the twisted boundary condition if the system size is sufficiently large.