A spin-energy operator inequality for Heisenberg-coupled qubits

TL;DR

This paper refines an inequality relating the energy of Heisenberg-coupled qubits to their total spin, providing explicit constants for cubic lattices and discussing implications for low-energy magnon states.

Contribution

It improves a known operator inequality by providing explicit constants and extends the understanding of energy bounds in Heisenberg spin systems.

Findings

Established a stronger lower bound on energy using total spin.

Derived explicit constants for cubic lattice configurations.

Connected the inequality to spin wave theory and magnon states.

Abstract

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.