Triplon condensation field scaling ($H_c-H_g\propto T^{3/2}$)

For a three-dimensional quadratic branch, the critical thermal density is $n_T=\zeta (3/2)[m{k}_{B}T/(2\pi {\hslash }^2){]}^{3/2}$. Inserting this into the mean-field gap-closing condition gives $H_c(T)-H_g=(\nu /\eta )n_T\propto T^{3/2}$. This is a dilute low-temperature model; finite bandwidth, anisotropy and changes in interactions can alter its applicability outside that regime.