Supersolid Phase in Spin Dimer $XXZ$ Systems under a Magnetic Field

Using the quantum Monte Carlo method, we study, under external magnetic fields, the ground state phase diagram of the two-dimensional spin $S=1/2$ dimer model with an anisotropic intraplane antiferromagnetic coupling. With the anisotropy $4\gtrsim \Delta \gtrsim 3$, a supersolid phase characterized by a nonuniform Bose condensate density that breaks translational symmetry is found. The rich phase diagram also contains a checkerboard solid, an antiferromagnet in the $z$ axis, and a superfluid phase formed by $S_z=+1$ spin triplets which has a finite staggered magnetization in the in-plane direction. As we show, the model can be realized as a consequence of including the next nearest neighbor coupling among dimers and our results suggest that spin dimer systems may be an ideal model system to study the supersolid phase.

Figure 1

(a) The number densities $n_s$, $n_{t+}$, and $n_{t0}$, (b) the SF density $n_0$ and structure factor $S(\pi ,\pi )/N$, and (c) the uniform magnetization $m_z/2$ and staggered magnetization $m_z^s/2$ as a function of $h/J$. The upper inset shows the details of $n_{t0}$ around the SS phase with enlarged scale. The lower inset compares the QMC (symbols) and the MF (solid lines) results of $m_z^s/2$ as a function of $\Delta$.

Figure 2

Finite size scaling of (a) $S(\pi ,\pi )/N$ and (b) $n_0$. Fields increase from $h/J=2.05$ to 2.2 with steps 0.025 from bottom to top in (a) and the reverse in (b). In (c), results of $S(\pi ,\pi )/N$ (open symbols) and $n_0$ (filled symbols) for $L=12$, 16, and 20.

Figure 3

Histograms of the averaged $m_z/2$ of the whole lattice [(a)–(c)] and of a small area [(d)–(f)] of size $4\times 4$ (unshaded) and $6\times 6$ (shaded). $\Delta =3.5$. For comparison, histograms of SFII and QS are also presented.

Figure 4

Sublattice condensates $n_{AA}$ and $n_{BB}$ in the SS phase extrapolated to $T\to 0$ and $L\to \infty$.

Figure 5

(a) Order parameters as a function of $h/J$ for different $\Delta$ with lattice size $20\times 20\times 2$. The inset shows the isotropic case ($\Delta =1$). (b) Ground state phase diagram (extrapolated to $L\to \infty$) of $\Delta$ vs $h/J$. Solid (dotted) lines denote the first (second) order phase boundaries.