Generalized Boundary Condition Applied to Lieb-Schultz-Mattis-Type Ingappabilities and Many-Body Chern Numbers

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 in 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 π flux, utilizing a time-reversal, magnetic translations and an on-site internal U(N) symmetries. This demonstrates the role of the time-reversal anomaly on the ingappabilities of a lattice model. Moreover, by our new boundary condition, we show that the many-body Chern number of this lattice model is nonvanishing as N mod 2N in the presence of U(N) and magnetic translations. This can be a general mechanism of anomaly-based constraints on quantized Hall conductance, which generally depends on high-energy physics, from field theory.