Bose–Einstein condensation of triplons with a weakly broken U(1) symmetry

Abstract

The low-temperature properties of certain quantum magnets can be described in terms of a Bose-Einstein condensation (BEC) of magnetic quasiparticles (triplons). Some mean-field approaches (MFA) to describe these systems, based on the standard grand canonical ensemble, do not take the anomalous density into account and leads to an internal inconsistency, as it has been shown by Hohenberg and Martin, and may therefore produce unphysical results. Moreover, an explicit breaking of the U(1) symmetry as observed, for example, in TlCuCl 3 makes the application of MFA more complicated. In the present work, we develop a self-consistent MFA approach, similar to the Hartree-Fock-Bogolyubov approximation in the notion of representative statistical ensembles, including the effect of a weakly broken U(1) symmetry. We apply our results on experimental data of the quantum magnet TlCuCl 3 and show that magnetization curves and the energy dispersion can be well described within this approximation assuming that the BEC scenario is still valid. We predict that the shift of the critical temperature T c due to a finite exchange anisotropy is rather substantial even when the anisotropy parameter γ is small, e.g., ∆T c ≈ 10% of T c in H= 6 T and for γ ≈ 4µeV.

Critical behavior of Tan's contact for bosonic systems with a fixed chemical potential

“…Since the triplon BEC occurs in solids, the integration is performed over the unit cell of the crystal with the corresponding momenta defined in the first Brillouin zone [38]. The parameter µ characterizes an additional direct contribution to the triplon energy due to the external field…”

Magnetocaloric effect and Grüneisen parameter of quantum magnets with a spin gap

“…Although the critical temperature T c or the density of triplons of the BEC may be obtained within Hamiltonian formalism [12][13][14][15], the thermodynamic potential and, in particular, the entropy can be also derived by using a Gaussian functional approximation [16], which is in fact, equivalent to the Hartree-Fock-Bogoliubov (HFB) approach.…”

“…where we have used Eq. (15). Due to Hugenholtz-Pines theorem, the excitation energy becomes gapless,…”

Effects of exchange and weak Dzyaloshinsky-Moriya anisotropies on thermodynamic characteristics of spin-gapped magnets

“…Below we bring explicit expressions for E k,T and ρ T for normal (T > T c ) and BEC phases (T ≤ T c ), where the critical temperature is defined at the point ρ 0 (T = T c ) = 0. a. Critical temperature and densityThe condition ρ 0 (T = T c ) = 0 leads the following coupled equations with respect to T c and σ c[53]:µ 2U + σ c + 3 γ 2 − k f b (E c k ) E c k [ε k + U (σ c + 3 γ)] = 0 σ c + U (σ c + γ) k f b (E c k ) k = E k (T → T c ) = ε k + X c ε k + 2γ , X c 1 = 2U (σ c + γ) , f b (x) =1/(exp(x/T c )−1) and γ = γ/U . The critical density is given by ρ(T = T c ) = µ…”