= Spin-projection-protected crossings in an exchange spectrum
In the <two-spin exchange spectrum with unequal signed magnetic moments>, the axial <Hamiltonian> conserves $M=S_{1z}+S_{2z}$. Parallel-spin states have $M=\pm\hbar$, while the two mixed states lie in the $M=0$ block. Levels in different $M$ sectors may cross as the axial <magnetic field> changes because the conserved projection forces their coupling <matrix element> to vanish.
This is a symmetry qualification of the <non-crossing rule>. A nonzero isotropic <exchange interaction> does not by itself turn every field-dependent crossing into an <avoided crossing>. A transverse perturbation that breaks conservation of $M$ can mix the sectors and open a gap, provided its relevant <matrix element> is nonzero. Plotting the <energy eigenvalues> by their conserved projection therefore distinguishes protected crossings from the avoided splitting already present within the $M=0$ block.
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