A spin-energy operator inequality for Heisenberg-coupled qubits

We slightly strengthen an operator inequality identified by Correggi et al. that lower bounds the energy of a Heisenberg-coupled graph of $s=1/2$ spins using the total spin. In particular, $\Delta H \ge C \Delta\vec{S}^2$ for a graph-dependent constant $C$, where $\Delta H$ is the energy above the ground state and $\Delta\vec{S}^2$ is the amount by which the square of the total spin $\vec{S} = \sum_i \vec{\sigma}_i/2$ falls below its maximum possible value. We obtain explicit constants in the special case of a cubic lattice. We briefly discuss the interpretation of this bound in terms of low-energy, approximately non-interacting magnons in spin wave theory and contrast it with another inequality found by B\"arwinkel et al.