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Specify the spin counting first. Let $g_s=g_s(E_F)$ denote the unpolarized single-spin density of states per atom per energy, and put $\delta n=n_{\uparrow }-n_{\downarrow }$. At fixed total occupation $n$, the two Hubbard model mean-field energies can be writtenThe common shift $Un/2$ can be absorbed into the chemical potential. Lowering the up-spin band by $\Delta$ adds $g_s\Delta$ carriers, while raising the down-spin band removes the same number. ThusIf $n_a$ is the atom number density, the magnetization is $M=n_a{\mu }_B\delta n$. The Stoner-enhanced magnetic susceptibility in SI volume units is thereforeHere ${\chi }_P$ is the noninteracting Pauli paramagnetism result. If the paper's $g(E_F)$ counts both spins, $g=2g_s$ and the same result isA susceptibility per atom omits $n_a$. Stating the spin and volume conventions avoids an otherwise spurious factor of two or a dimensional mismatch.

The Stoner criterion is $U{g}_s(E_F)>1$, equivalently $Ug(E_F)>2$ with spin-summed $g$. To see the instability, transfer $\eta$ Electrons per atom from one spin band to the other. The band-filling cost is ${\eta }^2/g_s$, whereas the interaction energy changes by $-U{\eta }^2$. Their sum $(1/g_s-U){\eta }^2$ becomes negative at the threshold. The unpolarized state then ceases to be a local minimum, and nonlinear band filling determines the spontaneous ferromagnetism. The negative continuation of the displayed linear susceptibility beyond the threshold is not a stable paramagnetic response.