Competition between two- and three-sublattice ordering for $S=1$ spins on the square lattice

We provide strong evidence that the $S=1$ bilinear-biquadratic Heisenberg model with nearest-neighbor interactions on the square lattice possesses an extended three-sublattice phase induced by quantum fluctuations for sufficiently large biquadratic interactions, in spite of the bipartite nature of the lattice. The argumentation relies on exact diagonalizations of finite clusters and on a semiclassical treatment of quantum fluctuations within linear flavor-wave theory. In zero field, this three-sublattice phase is purely quadrupolar, and upon increasing the field it replaces most of the plateau at 1/2 that is predicted by the classical theory.

Figure 1

Schematic phase diagram of the $S=1$ bilinear-biquadratic model on the square lattice. The inner circle shows the variational, the outer the numerical results. FM and AFM are ferro- and antiferromagnetic, FQ is ferroquadrupolar, while SO stands for semiordered. In the outer circle, we have shown the three-sublattice ordered antiferromagnetic and antiferroquadrupolar phases between $\vartheta \approx 0.2\pi$ and $\pi /2$.

Figure 2

(a) The variational phase diagram in a magnetic field. AFM2 denotes a two-sublattice ordered Néel phase, while FM stands for ferromagnet. Note the presence of a magnetization plateau at 1/2 which is separated from AFM2 by a supersolid phase. (b) Schematic plot of the anticipated phase diagram based on exact diagonalization calculations. AFM3 (AFQ3) is a three-sublattice ordered antiferromagnetic (antiferroquadrupolar) phase. The upper boundary of the 1/2 plateau is a result of finite-size scaling. The dashed line is estimated within flavor-wave theory (see text).

Figure 3

Spin and quadrupole structure factor from exact diagonalizations of square clusters with periodic boundary conditions. The geometry of the clusters can be found in Fig. 72 of Ref. 10. Open (closed) symbols stand for the spin (quadrupole) structure factor.

Figure 4

Anderson tower for the 18-site cluster showing the two-sublattice (a) and the three-sublattice order (b). The dashed line is a guide to the eye. The energy levels are characterized by their momenta and irreducible representations.

Figure 5

Magnetization curves of finite clusters for $\vartheta =3\pi /8$. Dashed lines represent variational magnetization curves calculated for two-sublattice and three-sublattice ordered states. The 3D plot shows the magnetization process of the 18-site cluster as a function of $\vartheta$.

Figure 6

Dynamical structure factors for the plateau state in the reduced Brillouin zone for three different values of $\vartheta$ at the upper critical magnetic field where the gap closes. The color code goes from yellow for spin waves to red for quadrupolar waves according to the dominant character of the mode. The dispersing bands have finite $S^x=S^y$ and $Q^{yz}=Q^{zx}$ matrix elements. The flat dispersion at $\omega =2h$ is a pure quadrupolar wave with $Q^{xy}=Q^{x^2-y^2}$ fluctuations. The spectra are displayed at the critical field to show that it is a partially magnetic mode that softens at the transition.