Lieb-Schultz-Mattis Theorem with Long-Range Interactions

Abstract

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 $\sim 1/r^{\hat{I}}\pm$ 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 $\hat{I}\pm >\max (3d,4d-2)$. For Type II, the condition becomes $\hat{I}\pm >\max (3d-1,4d-3)$. In $1d$, this ingappability condition is improved to $\hat{I}\pm >2$ for Type I and $\hat{I}\pm >0$ for Type II by examining the energy of a state with a uniform $2\ddot{I}€$ twist. Notably, in $2d$, a Type II Hamiltonian with van der Waals interaction is subject to the constraint of the theorem.

12 pages, 2 figures, published version