Lieb-Schultz-Mattis theorem with long-range interactions

We prove the Lieb-Schultz-Mattis theorem in $d$-dimensional spin systems exhibiting $SO\left(3\right)$ 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 $SO\left(3\right)$ 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, i.e., 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.