Lieb-Schultz-Mattis theorem in higher dimensions from approximate magnetic translation symmetry

We prove the Lieb-Schultz-Mattis (LSM) theorem on the energy spectrum of a general two or three-dimensional quantum many-body system with the U(1) particle number conservation and translation symmetry. Especially, it is demonstrated that the theorem holds in a system with long-range interactions. To this end, we introduce approximate magnetic translation symmetry under the total magnetic flux $\mathrm{\Phi }=2\pi$ instead of the exact translation symmetry, and explicitly construct low energy variational states. The energy spectrum at $\mathrm{\Phi }=2\pi$ is shown to agree with that at $\mathrm{\Phi }=0$ in the thermodynamic limit, which concludes the LSM theorem.