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

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.