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 interactions decaying as ~ 1/r^α 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-1/2 systems, it is shown that Type I cannot have a unique ground state with a nonzero excitation gap when the interaction decays sufficiently fast, i.e., when α>max(3d,4d-2). For Type II, the condition becomes α>max(3d-1,4d-3). In $1d$, this ingappability condition is improved to α>2 for Type I and α>0 for Type II by examining the energy of a state with a uniform 2π twist. Notably, in $2d$, a Type II Hamiltonian with van der Waals interaction is subject to the constraint of the theorem.