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 $s=1\mathrm{/}2$ spins using the total spin. In particular, $\mathrm{\Delta }H\ge C\mathrm{\Delta }{\vec{S}}^2$ for a graph-dependent constant $C$, where $\mathrm{\Delta }H$ is the energy above the ground state and $\mathrm{\Delta }{\vec{S}}^2$ is the amount by which the square of the total spin $\vec{S}={\sum }_i{\vec{\sigma }}_i\mathrm{/}2$ 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 Bärwinkel et al.