Long-range order in gapped magnetic systems induced by Bose-Einstein condensation

We study the Heisenberg antiferromagnet with single-ion anisotropy in two and three dimensions and present self-consistent intuitive theory to show the Bose-Einstein condensation-induced long-range order in the gapped magnetic systems, when the energy gap is tuned to zero by changing the physical parameters or by applying an external magnetic field. The recent experimental results on ${\mathrm{NiCl}}_2\bullet 4\mathrm{SC}{\left({\mathrm{NH}}_2\right)}_2$ are interpreted by the theory. Many other gapped magnetic systems share the same physical picture. The theory is also helpful in understanding the superfluid-Mott insulator transition observed in the system of ultracold atoms in the optical lattice.

Figure 1

Changes of the energy gap $\Delta$ (right axis) and the staggered magnetization $M_x$ (left axis) with $D$ in two dimensions (a) and three dimensions (b) with $R=0$ (triangles), 0.5 (circles), 1 (squares), 1.5 (diamonds), 2 (hexagons).

Figure 2

Critical temperature $T_c\left(D\right)$ for various $D$ with $R=0$ (squares), 0.5 (circles), 1 (triangles), 1.5 (diamonds), 2 (hexagons) in three dimensions.

Figure 3

Critical temperature $T_c\left(h\right)$ with $D=8$, $R=1$ in two dimensions (a), and with $D=12$ (triangles), 16 (squares), 20 (circles) and $R=1$ in three dimensions (b).

Figure 4

Changes of the field-induced staggered magnetization $M_x$ with temperature in three dimensions, with $D=16$ and $R=1$. The corresponding applied field is $h=10$ (squares), 13 (circles), 15 (triangles), 18 (hexagons), and 20 (diamonds).