Phases of a Triangular-Lattice Antiferromagnet Near Saturation

We consider 2D Heisenberg antiferromagnets on a triangular lattice with spatially anisotropic interactions in a high magnetic field close to the saturation. We show that this system possesses a rich phase diagram in a field or anisotropy plane due to competition between classical and quantum orders: an incommensurate noncoplanar spiral state, which is favored classically, and a commensurate coplanar state, which is stabilized by quantum fluctuations. We show that the transformation between these two states is highly nontrivial and involves two intermediate phases—the phase with coplanar incommensurate spin order and the one with noncoplanar double-$\mathbf{Q}$ spiral order. The transition between the two coplanar states is of commensurate-incommensurate type, not accompanied by softening of spin-wave excitations. We show that a different sequence of transitions holds in triangular antiferromagnets with exchange anisotropy, such as ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$.

Figure 1

Phase diagram of the spatially anisotropic triangular lattice antiferromagnet with large $S$ near the saturation field, as a function of spatial anisotropy of the interactions. The phases at small and large anisotropy are the commensurate coplanar $V$ phase, whose order parameter manifold is $U(1)\times Z_3$, and incommensurate noncoplanar chiral cone phase, which lives in the $U(1)\times Z_2$ manifold. In between, there are two incommensurate phases: a coplanar phase, with $U(1)\times U(1)$ symmetry, and a noncoplanar double cone phase, which is characterized by the $U(1)\times U(1)\times Z_2$ manifold. Line $AC$ denotes the CI transition from the $V$ phase to the incommensurate planar phase. The inset shows the geometry of the lattice: exchange is $J$ on horizontal bonds (bold) and $J^{\prime }$ on diagonal bonds (thin).

Figure 2

The phase diagram of the $XXZ$ model in a magnetic field near a saturation value, $\mathrm{\Delta }=(J-J_z)/J$. The cone and $V$ states are the same as in Fig. 1, but the transformation from one phase to the other with increasing spin exchange anisotropy proceeds differently from the case of spatial exchange anisotropy and involves one intermediate coplanar commensurate phase with a $\mathrm{\Psi }$-like spin pattern.