Half-magnetization plateau in a Heisenberg antiferromagnet on a triangular lattice

We present the phase diagram of a 2D isotropic triangular Heisenberg antiferromagnet in a magnetic field. We consider spin-$S$ model with nearest-neighbor ($J_1$) and next-nearest-neighbor ($J_2$) interactions. We focus on the range of $1/8<J_2/J_1<1$, where the ordered states are different from those in the model with only nearest-neighbor exchange. A classical ground state in this range is infinitely degenerate in any field. The actual order is then determined by quantum fluctuations via “order from disorder” phenomenon. We argue that the phase diagram is rich due to competition between competing quantum states which break either orientational or sublattice symmetry. At small and high fields, the ground state is a canted stripe state, which breaks orientational symmetry, but at intermediate fields the ordered states break sublattice symmetry. The most noticeable of such states is “three up, one down” state in which spins in three sublattices are directed along the field and in one sublattice opposite to the field. In such a state, magnetization is quantized at exactly one half of the saturation value. We identify gapless states, which border the “three up, one down” state and discuss the transitions between these states and the canted stripe state.

Figure 1

Schematic semiclassical phase diagram of a spin-$S{J}_1-J_2$ antiferromagnet on a triangular lattice at $1/8<J_2/J_1<1$. Solid (dotted) lines are second-order (first-order) phase transitions, which we identified and analyzed in this work. Dashed line is a first-order transition, which we expect to hold, but did not analyze. Arrows indicate magnetic order in the four-sublattice representation, and symbols like $\text{U}\left(1\right)\times Z_3$ indicate the broken symmetry in each state. The physics in a narrow range (at order 1/S) of $J_2/J_1$ near $J_2/J_1=1/8$ and $J_2/J_1=1$ is not analyzed in this work.

Figure 2

(a) and (b) Two candidate quantum four-sublattice ground states upon decreasing of the magnetic field $h$ towards a half of saturation value. (a) A $Z_3$ breaking canted stripe state. As field goes down, the angle between two pairs of parallel spins increases. (b) $\overline{\text{V}}$ and UUUD states. Both break $Z_4$ sublattice symmetry by selecting one sublattice with a different spin orientation compared to the other three. (c) Evolution from the UUUD state to the canted stripe state as $h$ decreases below $h_{\mathrm{sat}}/2$.

Figure 3

(a) The nearest-neighbor (${\mathbf{\delta }}_i$) and next-nearest-neighbor (${\mathit{l}}_i$) bonds on a triangular lattice. (b) Solid (dashed) line: single-sublattice (four-sublattice) Brillouin zone. The points labeled as $M_i$ are relevant to our discussion of spin-wave excitations near $h_{\text{sat}}$. The spin-wave excitations of the four-sublattice UUUD state soften at $\mathrm{\Gamma }$ point.

Figure 4

Evolution of the magnetic order below the UUUD state, depending on sign of the $K$ term in Eq. (4).