Spin dynamics of the bilinear-biquadratic $S=1$ Heisenberg model on the triangular lattice: A quantum Monte Carlo study

We study thermodynamic properties as well as the dynamical spin and quadrupolar structure factors of the O(3)-symmetric spin-1 Heisenberg model with bilinear-biquadratic exchange interactions on the triangular lattice. Based on a sign-problem-free quantum Monte Carlo approach, we access both the ferromagnetic and the ferroquadrupolar ordered spin nematic phase as well as the SU(3)-symmetric point that separates these phases. Signatures of Goldstone soft modes in the dynamical spin and the quadrupolar structure factors are identified, and the properties of the low-energy excitations are compared to the thermodynamic behavior observed at finite temperatures as well as to Schwinger-boson flavor-wave theory.

Figure 1

Thermodynamic limit results of the $\mathbf{q}=\mathbf{0}$ values of the spin and quadrupolar structure factors $S_S(\mathbf{q}=\mathbf{0})/N$ and $S_Q(\mathbf{q}=\mathbf{0})/N$ as functions of $\theta$ within the considered parameter range.

Figure 2

Finite-size extrapolations of the $\mathbf{q}=\mathbf{0}$ values of the spin and quadrupolar structure factors $S_S(\mathbf{q}=\mathbf{0})/N$ (left panel) and $S_Q(\mathbf{q}=\mathbf{0})/N$ (right panel) as functions of $1/L$ for various values of $\theta$.

Figure 3

Uniform susceptibility ${\chi }_{\mathrm{u}}$ and specific heat $C_V$ as functions of temperature $T$ taken for different values of $\theta$, as indicated. The dashed lines show the $S=1$ single-spin Curie behavior ${\chi }_{\mathrm{u}}^{\mathrm{C}}=S(S+1)/\left(3T\right)$ of the uniform susceptibility.

Figure 4

Rescaled specific heat $C_V/T$ ($C_V/T^2$) for the ferromagnetic (ferroquadrupolar) region as a function of temperature $T$ for different values of $\theta$, as indicated.

Figure 5

Dynamical spin and quadrupolar structure factors $S_S(\omega ,\mathbf{q})$ and $S_Q(\omega ,\mathbf{q})$ for different values of $\theta$ along the path $\Gamma \to K\to M\to \Gamma$ through the Brillouin zone [$\Gamma ={(0,0)}^{⊺}, K={(4\pi /3,0)}^{⊺}, M={(2\pi /3,\pi /\sqrt{3})}^{⊺}$]. The dashed lines indicate the quadrupolar wave dispersion relations obtained within flavor-wave theory.

Figure 6

Integrated spectral weights $I_S\left(\mathbf{q}\right)$ and $I_Q\left(\mathbf{q}\right)$ for different values of $\theta$ along the same Brillouin-zone path as in Fig. 5. Values of $I_S\left(\mathbf{q}\right)$ and $I_Q\left(\mathbf{q}\right)$ at $\mathbf{q}=\Gamma$ that extend beyond the range of the figure are not on display.

Figure 7

Dynamical quadrupolar structure factor $S_Q(\omega ,\mathbf{q})$ at $\theta =-0.5\pi$ for different temperatures along the same Brillouin-zone path as in Fig. 5.

Figure 8

Dynamical quadrupolar structure factor $S_Q(\omega ,\mathbf{q})$ for different values of $\theta$ along the same Brillouin-zone path as in Fig. 5.