$S=1$ bilinear biquadratic spin model on the square lattice: A series expansion study

We use extensive series expansions at $T=0$ to investigate the phase diagram of a spin-1 Hamiltonian on the square lattice. The model includes bilinear and biquadratic interactions (the “$J$-$K$” model) and has been studied recently using a variety of other methods. We find a clear indication of the three-sublattice order conjectured recently by Toth et al. [Phys. Rev. B 85, 140403(R) (2012)]. We also compute the energy and order parameter in the quadrupolar phases.

Figure 1

Phase diagram as a function of angle $\theta$. Key: FM: ferromagnetic; AF: antiferromagnetic; FQ: ferroquadrupolar; AFQ3: antiferroquadrupolar, with 3-sublattice order; AFM3: antiferromagnetic, with 3-sublattice order.

Figure 2

(a) Ground state energy per site from expansions in the Néel phase (squares) and the AFM3 phase (circles). Dashed line: flavor-wave prediction for the Néel phase; (b) estimates of the magnetization in the Néel phase. Dashed line: flavor-wave prediction; solid line: a fit to the data.

Figure 3

The figures show (a) the ground state energy per site and (b) the quadrupolar order parameter $Q$ in the FQ phase. The circular points are our series estimates, and the dashed lines are flavor-wave calculations. The squares are estimates of the ground state energy extracted from Harada and Kawashima.[9]

Figure 4

(a) Ground state energy per site from expansions in the AFQ3 phase (squares) and the AFM3 phase (circles). Dashed line: flavor-wave prediction.[11] (b) Estimates of the quadrupole order parameter Q in the AFQ3 phase.