Field-induced triplon condensation ($\eta (H_c-H_g)=\nu n$)

A magnetic field can close the lowest gap of a dilute bosonic triplet branch. In a shifted-parabola model with actual chemical potential zero, ${ϵ}_k={\hslash }^2{k}^2/(2m)+\nu n-\eta (H-H_g)$ and repulsion $\nu >0$ prevent indefinite density growth. The normal minimum reaches zero when $\eta (H_c-H_g)=\nu n$. At that point the finite thermal excited population saturates and macroscopic ground-state occupation can develop.

If the phenomenological single-density shift is continued into the condensed phase and the thermal population remains saturated at $n_T(T)$, the zero minimum requires $\nu [n_0+n_T]=\eta (H-H_g)$. At fixed temperature, $n_0=(\eta /\nu )[H-H_c(T)]$ above onset and zero below it. This prefactor belongs to that closure; a microscopic interaction model can distinguish normal and condensed self-energies. Continuing the normal parabola into negative energies without changing the state is not a consistent Bose distribution.

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.