Two-spin exchange spectrum with unequal signed magnetic moments

For $H=(\Delta /4){\mathbf{\sigma }}_1\cdot {\mathbf{\sigma }}_2-({\mu }_{0}B/2)(g_1{\sigma }_{1z}+g_2{\sigma }_{2z})$, the parallel-spin energies are $\Delta /4\mp {\mu }_{0}B(g_1+g_2)/2$. The zero-projection block has energies $-\Delta /4\pm \sqrt{{\Delta }^2+[{\mu }_{0}B(g_1-g_2){]}^2}/2$. Signed moment conventions must be fixed before assigning a physical opposite-charge case.

In the two-spin exchange spectrum with unequal signed magnetic moments, the axial Hamiltonian conserves $M=S_{1z}+S_{2z}$. Parallel-spin states have $M=\pm \hslash$, 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.

If $g_2=-g_1$ in a shared magneton convention, the Zeeman energy of both parallel-spin states cancels. Their energy stays $\Delta /4$, with exact degeneracy two for every field. Opposite charges alone do not enforce this equality because the masses and magnetic factors may differ.

If the differential Zeeman energy $|w|=|{\mu }_{0}B(g_1-g_2)|$ greatly exceeds $|\Delta |$, the two mixed levels are $-\Delta /4\pm |w|/2+O({\Delta }^2/|w|)$ and approach opposite-spin product states. Four energies are generically distinct. Equal signed moments have $w=0$, so that block remains singlet/triplet rather than being unmixed by a stronger field.