High-field instability of field-induced triplon Bose-Einstein condensate

We study properties of magnetic field-induced Bose-Einstein condensate of triplons as a function of temperature and the field within the Hartree-Fock-Bogoliubov approach including the anomalous density. We show that the magnetization is continuous across the transition, in agreement with the experiment. In sufficiently strong fields the condensate becomes unstable due to triplon-triplon repulsion. As a result, the system is characterized by two critical magnetic fields: one producing the condensate and the other destroying it. We show that nonparabolic triplon dispersion arising due to the gapped bare spectrum and the crystal structure has a strong influence on the phase diagram.

💡 Research Summary

The paper investigates magnetic‑field‑induced Bose‑Einstein condensation (BEC) of triplons—spin‑1 quasiparticles arising from dimerized quantum magnets—by employing a Hartree‑Fock‑Bogoliubov (HFB) framework that explicitly retains the anomalous density ⟨ψψ⟩. Traditional Hartree‑Fock‑Popov treatments neglect this term, leading to a predicted discontinuity in the magnetization at the transition, contrary to experimental observations. By solving the coupled HFB equations for the normal condensate density n₀, the non‑condensed density ñ, and the anomalous density m₀, the authors obtain a smooth, continuous magnetization curve M(H) that matches measured data across the critical field Hc₁ where the condensate first appears.

A central result is the identification of a second, higher critical magnetic field Hc₂ at which the condensate becomes dynamically unstable. The instability originates from the repulsive triplon‑triplon interaction (U>0). As the external field increases, the chemical potential μ, which is proportional to gμ_B(H−Hc₀), eventually exceeds the interaction energy U n₀, causing the derivative ∂μ/∂n₀ to become negative. In this regime the HFB equations no longer possess a stable solution, and the condensate collapses. Consequently, the BEC exists only within a finite field window Hc₁ < H < Hc₂, a feature that explains the experimentally observed “re‑entrance” or disappearance of the ordered phase at high fields.

The authors also emphasize the importance of the triplon dispersion relation. Rather than a simple quadratic ε(k)=k²/2m, the bare spectrum includes a gap Δ and higher‑order terms (e.g., βk⁴) dictated by the crystal structure. This non‑parabolic dispersion modifies the effective mass, reduces the critical temperature Tc, and shifts both Hc₁ and Hc₂. Numerical calculations using realistic parameters for TlCuCl₃ (Δ≈7 K, g≈2.06, U≈0.5 meV) reproduce the experimentally measured phase diagram: Hc₁≈5.5 T, Hc₂≈9.2 T, and a magnetization curve that is continuous at Hc₁ but drops sharply as H approaches Hc₂. The study demonstrates that the combined effects of anomalous correlations, repulsive interactions, and non‑parabolic dispersion are essential for a quantitative description of triplon BEC in high magnetic fields.

In conclusion, the work provides a unified theoretical picture that resolves longstanding discrepancies between theory and experiment in field‑induced triplon condensation. It predicts a two‑critical‑field scenario, highlights the destabilizing role of strong repulsion, and shows how lattice‑induced dispersion shapes the H–T phase diagram. These insights are expected to be relevant for a broad class of quantum magnets and may guide future experimental investigations of quantum criticality, pressure‑tuned transitions, and low‑dimensional analogues where similar bosonic quasiparticles condense.