Quantum Phase Diagram of the Triangular-Lattice $XXZ$ Model in a Magnetic Field

The triangular lattice of $S=1/2$ spins with $XXZ$ anisotropy is a ubiquitous model for various frustrated systems in different contexts. We determine the quantum phase diagram of the model in the plane of the anisotropy parameter and the magnetic field by means of a large-size cluster mean-field method with a scaling scheme. We find that quantum fluctuations break up the nontrivial continuous degeneracy into two first-order phase transitions. In between the two transition boundaries, the degeneracy-lifting results in the emergence of a new coplanar phase not predicted in the classical counterpart of the model. We suggest that the quantum phase transition to the nonclassical coplanar state can be observed in triangular-lattice antiferromagnets with large easy-plane anisotropy or in the corresponding optical-lattice systems.

Figure 1

Ground-state phase diagram of the spin-$1/2$ triangular-lattice $XXZ$ model from (a) the classical and (b) $\mathrm{CMF}+\mathrm{S}$ analyses ($J_z>0$). The thick blue (thin black) solid curves correspond to first- (second-)order transitions. The latest QMC data [45, 46] are shown by the red dashed (first-order) and dotted (second-order) curves. The symbol ($\times$) is the value from the dilute Bose-gas expansion.

Figure 2

Five types of spin configurations for $J/J_z>0$. The sets of three arrows represent each spin angle on the three sublattices. The lower illustrations in (a)–(c) depict the corresponding distributions of magnon BECs at the corners of the hexagonal first Brillouin zone.

Figure 3

(a) Coplanar-umbrella phase boundary $(J/J_z{)}_{c2}$ just below the saturation field as a function of the interlayer coupling strength obtained from ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_2$. We display the cases of the isotropic (circles) and $XY$-type (squares) antiferromagnetic interlayer couplings as examples. (b) Cluster-size scaling of the CMF data for the phase boundaries $(J/J_z{)}_{c1}$ between 0- and $\pi$-coplanar phases as well as $(J/J_z{)}_{c2}$ just below the saturation field.

Figure 4

(a) Classical solution of $\chi$ as a function of $J/J_z$ for $H/J_z=3$. (b) Quantum degeneracy lifting obtained by the CMF analysis ($H/J_z=4.5$). The vertical dashed lines mark the first-order transition points determined by the Maxwell construction. The inset is the enlarged view around the weak first-order $0–\pi$ transition. The phase diagrams in (c) show the quantum breakup of the continuous degeneracy into two first-order transitions for $H/H_s\gtrsim 0.84$.