Probable absence of a quadrupolar spin-nematic phase in the bilinear-biquadratic spin-1 chain

We study numerically the ground-state phase diagram of the bilinear-biquadratic spin-1 chain near the ferromagnetic instability point, where the existence of a gapped or gapless nondimerized quantum nematic phase has been suggested. Our results, obtained by a highly accurate density-matrix renormalization-group calculation, are consistent with the view that the order parameter characterizing the dimer phase vanishes only at the point where the system becomes ferromagnetic, although the existence of a gapped or gapless nondimerized phase in a very narrow parameter range between the ferromagnetic and the dimerized regimes cannot be ruled out.

Figure 1

Schematic plot of the phase diagram of the bilinear-biquadratic $S=1$ model as a function of $\theta$.

Figure 2

Schematic plot of the excitation gap $\left(\Delta \right)$ and the dimer order parameter $D$ as a function of $\theta$ around the phase boundary of the ferromagnetic and dimerized phases. (a) Chubukov’s suggestion; (b) the result of earlier numerical work; (c) the case of a critical nondimerized phase.

Figure 3

The excitation gap $\left(\Delta \right)$ and dimer order parameter $\left(D\right)$ at $\theta =-0.5\pi$ as a function of $1∕N^2$. The symbol × denotes the exact value. The solid lines are least-square fits using the scaling forms discussed in the text.

Figure 4

Typical valence-bond configurations for chains with even lattice sites.

Figure 5

Finite-size scaling of the dimer order parameter at $\theta =-0.67\pi$ and $-0.70\pi$ for $100⩽N⩽1000$.

Figure 6

The extrapolated value of the dimer order parameter as a function of $\theta ∕\pi$. The solid curve is a least-square fit using the form in Eq. (7) with $a=2.4\pm 0.6$, $c=0.11\pm 0.05$, and $\sigma =1.3\pm 0.3$.