Bose-Einstein condensation and the magnetically ordered state of ${\text{TlCuCl}}_3$

The dimerized $S=\frac{1}{2}$ spins of the ${\text{Cu}}^{2+}$ ions in ${\text{TlCuCl}}_3$ are ordered antiferromagnetically in the presence of a field larger than about 54 kOe in the zero-temperature limit. Within the mean-field approximation all thermal effects are frozen out below 6 K. Nevertheless, experiments show significant changes in the critical field and the magnetization below this temperature, which reflect the presence of low-energetic dimer-spin excitations. We calculate the dimer-spin correlation functions within a self-consistent random-phase approximation, using as input the effective exchange-coupling parameters obtained from the measured excitation spectra. The calculated critical field and magnetization curves exhibit the main features of those measured experimentally but differ in important respects from the predictions of simplified boson models.

Figure 1

(Color online) The susceptibility of ${\text{TlCuCl}}_3$ determined experimentally with a field of 10 kOe along the $b$ axis. The pluses show the data of Oosawa et al. (Ref. 3) and the open circles are the results of Dell’Amore et al. (Ref. 4). The experimental results are compared with the calculated ones obtained by assuming $g=1.97$ or $g=2.33$. In all other calculations we use $g=2.06$.

Figure 2

The solid line is the theoretical result for the critical field as a function of temperature using the exchange parameters introduced by Eq. (36). The dashed line is the result obtained if using instead the exchange parameters of Oosawa et al. (Ref. 6). The critical field is here defined to be the one at which the paramagnetic phase becomes unstable. The experimental points are those obtained when the field is applied in the $b$ direction by Oosawa et al.(Ref. 23) and Shindo and Tanaka (Ref. 24).

Figure 3

(Color online) The parallel magnetic moment per Cu ion, $m_z$, as a function of temperature calculated at various values of the field applied along the $b$ direction. In the case of $H=53\text{kOe}$ the system is predicted to stay disordered all the way to zero temperature. The experimental points are a selection of those obtained by Oosawa et al. (Refs. 3, 8).

Figure 4

(Color online) The minimum energies of the three different dimer excitations as functions of field at 1.5 K. The calculated results (the solid lines) are compared with the experimental results of Rüegg et al. (Refs. 25, 26).

Figure 5

(Color online) The squares are the experimental results for the field dependence of the parallel moment $m_z$ at 1.3 K and are the data of Tatani et al. (Ref. 27) presented in Ref. 11. The open circles are the neutron-diffraction results for the square of the ordered antiferromagnetic moment $m_{xy}^2$ obtained by Tanaka et al. (Ref. 14) with the field along the $b$ direction at 0.2 K. The solid lines show the corresponding theoretical predictions and the dashed ones are the results of using the MF approximation.

Figure 6

(Color online) The renormalization parameters $a_{xy}$ and $a_z$ (left scale) and the population numbers (right scale) calculated as functions of temperature at an applied field of 70 kOe, where the transition occurs at 3.34 K.