Phase diagrams of the $XXZ$ model on a depleted square lattice

Using quantum Monte Carlo (QMC) simulations and a mean field theory, we investigate the spin-1/2 $XXZ$ model with nearest-neighbor interactions on a periodic depleted square lattice. In particular, we present results for 1/4 depleted lattice in an applied magnetic field and investigate the effect of depletion on the ground state. The ground state phase diagram is found to include an antiferromagnetic (AF) phase of magnetization $m_z=\pm 1/6$ and an in-plane ferromagnetic (FM) phase with finite spin stiffness. The agreement between the QMC simulations and the mean-field theory based on resonating trimers suggests the AF phase and in-plane FM phase can be interpreted as a Mott insulator and superfluid of trimer states, respectively. While the thermal transitions of the in-plane FM phase are well described by the Kosterlitz-Thouless transition, the quantum-phase transition from the AF phase to in-plane FM phase undergo a direct second-order insulator-superfluid transition upon increasing magnetic field.

Figure 1

The unit cell and site indices of the 1/4 depleted square lattice. The spin pattern represents the $m_z=1/6$ AF phase.

Figure 2

(Color online) The ground-state phase diagram obtained by the QMC (denoted by black dots) simulations of lattice size $28\times 28$. AF phase, characterized by the magnetization plateau $m_z=\pm 1/6$, extends further into the in-plane FM phase as explained in the text. Results of a trimer-basis MF theory (blue dashed line) agree very well to the QMC data. For $\Delta <1$, phase separation is observed at $h=0$ while the ground state is a uniform superfluid for $\Delta >1$. FP denotes the fully polarized states for large fields. The vertical red line shows the scan for the order parameters in Fig. 3.

Figure 3

(Color online) QMC data of order parameters of the (a) total and sublattice magnetizations, (b) spin stiffness ${\rho }_s$, and (c) structure factors as a function of field $h$ for $\Delta =0.8$. The dashed lines in (a) denote the corresponding mean-field magnetizations.

Figure 4

(Color online) Finite-size scaling of ${\rho }_s$ at $h_{c1}$, the first critical field of in-plane FM to AF phase transition in Fig. 3 for $\Delta =0.8$. (a) Data of ${\rho }_s{L}^2$ for different system sizes intersect at a critical field $h_{c1}=0.2568\left(1\right)$. (b) Data of ${\rho }_s{L}^2$ collapse well into a universal scaling curve when plotting against the $\left(h-h_{c1}\right)L^{1/\nu }$. The dynamical critical exponent is set to $z=0$ while correlation exponent $\nu =0$. The temperature used is $\beta =L^z/12$. The same for Fig. 5, 6.

Figure 5

(Color online) Finite-size scaling ${\rho }_s$ at $h_{c2}$, the second critical field of AF to in-plane FM phase transition in Fig. 3 for $\Delta =0.8$. Data also appear to collapse for $z=2$ and $\nu =0.5$, though not as well as in Fig. 4 for $h_{c1}$.

Figure 6

(Color online) Finite-size scaling of ${\rho }_s$ crossing the tip of the lobe in Fig. 2 (red horizontal line in Fig. 2) with $m_z=1/6$. Using $z=1$ and $\nu =0.67$, (a) the critical ${\Delta }_c$ is found to be ${\Delta }_c=0.5685\left(5\right)$ and (b) data collapse well onto the universal scaling curve.

Figure 7

(Color online) ${\rho }_s$ is plotted as a function of $T$ for different system sizes $L$ near the KT transition temperature $T_{\text{KT}}$ for $\Delta =0.8$ at $h=0$. The dashed line ${\rho }_s\left(T\right)=\frac{2}{\pi }T$ intersects ${\rho }_s$ curves of different $L$ at temperatures $T^{\ast }\left(L\right)$, which follows the logarithmic behavior given by Eq. (9) as shown in the inset.

Figure 8

(Color online) Same as Fig. 7 but now $h=1.8$.