Lieb-Schultz-Mattis Theorem with Long-Range Interactions

We prove the Lieb-Schultz-Mattis theorem in $d$-dimensional spin systems exhibiting $SO(3)$ spin rotation and lattice translation symmetries in the presence of $k-$local interactions decaying as $\sim 1/r^\alpha$ with distance $r$. Two types of Hamiltonians are considered: Type I comprises long-range spin-spin couplings, while Type II features long-range couplings between $SO(3)$ symmetric local operators. For spin-$\frac{1}{2}$ systems, it is shown that Type I cannot have a unique symmetric ground state with a nonzero excitation gap when the interaction decays sufficiently fast, \ie when $\alpha>\max(3d,4d-2)$. For Type II, the condition becomes $\alpha>\max(3d-1,4d-3)$. In $1d$, this ingappability condition is improved to $\alpha>2$ for Type I and $\alpha>0$ for Type II by examining the energy of a state with a uniform $2\pi$ twist. Notably, in $2d$, a Type II Hamiltonian with van der Waals interaction is subject to the constraint of the theorem.