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 $\mathrm{\Phi }=2\pi$ 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 Lieb-Schultz-Mattis theorem.

中文翻译：

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来自近似磁平移对称性的高维 Lieb-Schultz-Mattis 定理

我们在具有 U(1) 粒子数守恒和平移对称性的一般二维或三维量子多体系统的能谱上证明了 Lieb-Schultz-Mattis 定理。特别是，证明了该定理在具有长程相互作用的系统中成立。为此，我们引入了总磁通量下的近似磁平移对称性$\mathrm{\Phi }=2\pi$而不是精确的平移对称性，并明确构建低能量变分状态。能谱在$\mathrm{\Phi }=2\pi$ 显示同意在 $\mathrm{\Phi }=0$ 在热力学极限中，得出 Lieb-Schultz-Mattis 定理。