Numerical study of magnetic field induced ordering in $\mathrm{Ba}\mathrm{Cu}{\mathrm{Si}}_2{\mathrm{O}}_6$ and related systems

Thermodynamics of spin dimer system $\mathrm{Ba}\mathrm{Cu}{\mathrm{Si}}_2{\mathrm{O}}_6$ is studied using a quantum Monte Carlo calculation (QMC) and a bond-operator mean field theory. We propose that a new type of boson, which, rather than being hard core, allows up to two occupancies at each site, is responsible for the Bose-Einstein condensation of field induced ordering. Its superfluid density is identified as the square of the in-plane staggered magnetization $m_{xy}$ in the ordered phase. We also compare our QMC result of the spin Heisenberg model to those predicted by the mean field theory as well as by the simple hard-core boson model for both large and small intradimer coupling $J$. The asymmetry of the phase diagram of $m_{xy}\left(h\right)$ of small coupling $J$ in related systems such as $\mathrm{Ni}{\mathrm{Cl}}_2\text{−}4\mathrm{S}\mathrm{C}\left(\mathrm{N}{\mathrm{H}}_2\right)$ is explained with our new boson operator.

Figure 1

(Color online) QMC result of (a) specific heat $c_v$ and (b) spin susceptibility $\chi$ vs temperature for different magnetic fields. Data for different $h$ are shifted for clarity.

Figure 2

(Color online) (a) Low temperature regime of uniform magnetization per dimer $m_z$ for different magnetic fields that exhibit BEC. (b) The whole temperature range of $m_z$ for field $h=37\mathrm{T}$. (c) Magnetic field dependence of $m_z$ obtained from both QMC $(T=1\mathrm{K})$ and MFT together with the number density of $\mid s⟩$ and $\mid t_{+}⟩$ states.

Figure 3

(Color online) The slope of $m_z\left(h\right)$, i.e., the susceptibility, at $T=1\mathrm{K}$.

Figure 4

(Color online) Superfluid density $n_0$, which is equivalent to $m_{xy}^2$ in the MFT, as (a) a function of temperature and (b) a function of magnetic fields.

Figure 5

(Color online) Results of reduced interplane coupling $J=2.436\mathrm{meV}$. (a) Temperature dependence of $m_z$ and number density of $\mid s⟩$, $\mid t_{+}⟩$, and $\mid t_{-}⟩$ states. (b) Phase diagram of $n_0\left(h\right)$.