Phase diagram of the antiferromagnetic XXZ model on the triangular lattice

We determine the quantum phase diagram of the antiferromagnetic spin-1/2 XXZ model on the triangular lattice as a function of magnetic field and anisotropic coupling $J_z$. Using the density matrix renormalization group algorithm in two dimensions, we establish the locations of the phase boundaries between a plateau phase with 1/3 Néel order and two distinct coplanar phases. The two coplanar phases are characterized by a simultaneous breaking of both translational and U(1) symmetries, which is reminiscent of supersolidity. A translationally invariant umbrella phase is entered via a first-order phase transition at relatively small values of $J_z$ compared to the corresponding case of ferromagnetic hopping and the classical model. The phase transition lines meet at two tricritical points on the tip of the lobe of the plateau state, so that the two coplanar states are completely disconnected. Interestingly, the phase transition between the plateau state and the upper coplanar state changes from second order to first order for large values of $J_z\gtrsim 2.5J$.

Figure 1

The phase diagram of the XXZ model on the triangular lattice with two-dimensional DMRG and exact diagonalization [38] for different sizes. The solid (dashed) line represents first- (second-) order phase transitions, respectively. Arrows indicate the classical spin configurations in the different phases. The black circles indicate the regions analyzed in Fig. 4. Linear finite size scaling with $1/N$ of the interpolated data predicts the black solid line as the phase boundary in the thermodynamic limit (TD). The two black arrows show the finite size scaling of the tricritical points (see the Supplemental Material [39] for details).

Figure 2

The structure factors $S^z\left(\mathbf{Q}\right)/N$ and $S^{\pm }\left(\mathbf{Q}\right)/N$ at $J_z=J$ as a function of magnetization for different sizes. The dotted line at $M=1/6$ indicates the location of the 1/3 Néel phase. Inset: The magnetization as a function of magnetic field.

Figure 3

The magnetization as a function of field at (a) $J_z=2.5J$ (inset: larger range) and (b) $J_z=5J$ for different sizes. The dotted vertical line (red) indicates the Maxwell construction.

Figure 4

The structure factor $S^z\left(\mathbf{Q}\right)/N$, $S^{\pm }\left(\mathbf{Q}\right)/N$, concurrence, entanglement entropy, quantum discord, and the variational angle $\theta$ in the $6\times 9$ lattice in the regions shown by red circles in Fig. 1 indicate a phase transition to the umbrella state from (a) the antiferromagnetic ($B=1.398J_z$) and (b) the ferrimagnetic coplanar state ($B=2.161J_z$).

Figure 5

The order parameter $\Delta S^3$ in Eq. (6) as a function of $J/J_z$ close to saturation $M=1/2-3/N.$ Inset: Finite size scaling of the $\pi$-coplanar plateau width.