Lieb-Schultz-Mattis Theorem with Long-Range Interactions

TL;DR

This paper extends the Lieb-Schultz-Mattis theorem to higher-dimensional spin systems with long-range interactions, establishing conditions under which a symmetric gapped ground state cannot exist.

Contribution

It proves the Lieb-Schultz-Mattis theorem for $d$-dimensional systems with long-range interactions decaying as $1/r^eta$, covering both spin-spin and operator-based couplings.

Findings

For spin-1/2 systems, no unique gapped symmetric ground state exists if $eta$ exceeds certain bounds.

In 1D, the decay rate condition is improved to $eta>2$ for spin-spin interactions.

In 2D, the theorem constrains systems with van der Waals 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 $\sim 1/r^\alpha$ 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.