Quadrupolar Phases of the $S=1$ Bilinear-Biquadratic Heisenberg Model on the Triangular Lattice

Using mean-field theory, exact diagonalizations, and $SU(3)$ flavor theory, we have precisely mapped out the phase diagram of the $S=1$ bilinear-biquadratic Heisenberg model on the triangular lattice in a magnetic field, with emphasis on the quadrupolar phases and their excitations. In particular, we show that ferroquadrupolar order can coexist with short-range helical magnetic order, and that the antiferroquadrupolar phase is characterized by a remarkable $2/3$ magnetization plateau, in which one site per triangle retains quadrupolar order while the other two are polarized along the field. Implications for actual $S=1$ magnets are discussed.

Figure 1

(a) Zero-field phase diagram. The inner circle is the variational result; the outer circle the exact-diagonalization one. The magnetic phases are shaded in gray. (b) Probabilities of spin fluctuations $|⟨S(\widehat{\mathbf{n}})|\psi ⟩{|}^2$ in the pure state $\psi =|y⟩$ (left) and in a state with finite magnetization (right). $|S(\widehat{\mathbf{n}})⟩$ is the coherent spin state pointing in direction $\widehat{\mathbf{n}}$.

Figure 2

Spin and quadrupolar structure factors at the $\Gamma$ and $K$ points for different finite clusters. The system sizes are labeled by the symbol type.

Figure 3

Spin correlation function $S(\mathbf{k},\omega )$ from flavor-wave calculation for $\vartheta =-3\pi /4$, $-5\pi /8$, $-\pi /2$, $-3\pi /8$, and ${\Theta }_c^{MF}$ from top to bottom (shifted by $J$) in the FQ phase. The dispersion $\omega (q)$ is the solid curve; the matrix element is the dashed line measured from the solid line. Left upper corner: the Brillouin zone.

Figure 4

Magnetic phase diagram. Solid (dashed) lines denote 1st (2nd) order phase boundaries in the variational approach. The dotted line shows the exact boundary of the FQ phase [two magnon bound state formation, see Ref. 11 ]. Along the dashed-dotted lines the variational solution is highly degenerate. The plateaux are shaded in gray. Filled arrows represent fully polarized magnetic moments, empty arrows partially polarized ones.

Figure 5

Magnetization curves obtained by exact diagonalization for $\vartheta =3\pi /8$, where the presence of a plateau at $m=2/3$ is confirmed. The inset shows a magnetization curve at the Heisenberg point with a small plateau stabilized at $m=1/3$.