 Ginzburg reduced-temperature window

For dimensionless scalar Landau-Ginzburg theory with quadratic coefficient $r=a_{t}t$, gradient coefficient $c>0$ and quartic term $u{\varphi }^4/4$, the correlation-volume derivation of the scalar Ginzburg ratio gives $\mathcal{G}\sim u{c}^{-d/2}{a}_t^{(d-4)/2}|t{|}^{-(4-d)/2}$. For $d<4$, solving $\mathcal{G}\sim 1$ gives the displayed nonuniversal window. Mean-field self-consistency requires $|t|\gg \mathrm{Gi}$ while still staying in the domain of the local expansion. In three dimensions $\mathrm{Gi}\sim u^2/(c^3{a}_t)$. Numerical prefactors depend on the averaging convention and microscopic normalization; no universal material value follows from dimensional power counting. At four dimensions use the marginal Ginzburg criterion instead of this singular exponent formula.