= Stoner-enhanced magnetic susceptibility
{c}
{title2=$\chi=\chi_P/[1-Ug_s(E_F)]$}
Let $g_s(E_F)$ be the <single-spin density of states> per site per energy, and let the on-site <Hubbard model> repulsion have <energy> $U$. For a weak uniform magnetic field $H$, the half-splitting of the two spin bands is $\Delta=U\delta n/2+\mu_0\mu_BH$, where $\delta n=n_\uparrow-n_\downarrow$. Constant density of states gives $\delta n=2g_s\Delta$. Therefore
$$
\frac{\mu_B\delta n}{H}=\frac{2\mu_0\mu_B^2g_s}{1-Ug_s}.
$$
This is the response per site; multiplying by the site <number density> gives the dimensionless SI volume <magnetic susceptibility>. With <spin-summed density of states> $g=2g_s$, the denominator is $1-Ug/2$. Its vanishing is the mean-field <Stoner criterion>. The unpolarized linear-response expression is stable only for $Ug_s<1$.
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