Lieb-Schultz-Mattis Theorem in Higher Dimensions from Approximate Magnetic Translation Symmetry

We prove the Lieb-Schultz-Mattis 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 Φ=2π instead of the exact translation symmetry, and explicitly construct low energy variational states. The energy spectrum at Φ=2π is shown to agree with that at Φ=0 in the thermodynamic limit, which concludes the Lieb-Schultz-Mattis theorem.