 Field-induced triplon condensation

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.