# A generalized boundary condition applied to Lieb-Schultz-Mattis type ingappabilities and many-body Chern numbers

## TL;DR

This paper introduces a new boundary condition that generalizes flux-insertion arguments for Lieb-Schultz-Mattis theorems, enabling their application in higher dimensions and linking them to anomaly field theories, with implications for lattice models.

## Contribution

It presents a boundary condition that removes size restrictions in LSM theorems and connects them to anomaly field theories in arbitrary dimensions.

## Key findings

- New boundary condition applicable in higher dimensions
- Formulation of LSM theorems via anomaly in field theories
- Demonstration of time-reversal anomaly affecting lattice model ingappabilities

## Abstract

We introduce a new boundary condition which renders the flux-insertion argument for the Lieb-Schultz-Mattis type theorems in two or higher dimensions free from the specific choice of system sizes. It also enables a formulation of the Lieb-Schultz-Mattis type theorems in arbitrary dimensions in terms of the anomaly in field theories of $1+1$ dimensions with a bulk correspondence as a BF-theory in 2+1 dimensions. Furthermore, we apply the anomaly-based formulation to the constraints on a half-filled spinless fermion on a square lattice with $\pi$ flux, utilizing time-reversal, the magnetic translation and on-site internal $U(N)$ symmetries. This demonstrates the role of time-reversal anomaly on the ingappabilities of a lattice model.