Spin nematics correlations in bilinear-biquadratic $S=1$ spin chains

We present an extensive numerical study of spin quadrupolar correlations in single and coupled bilinear-biquadratic spin one chains, using several methods such as exact diagonalization, density matrix renormalization group, and strong coupling series expansions. For the single chain, we clarify the dominant correlation function in the enigmatic gapless period-three phase for $\theta ∊(\pi ∕4,\pi ∕2)$, which is of spin quadrupolar nature with a period three spatial structure. Then we revisit the open problem of the possible existence of a ferroquadrupolar phase between the dimerized and the ferromagnetic phases. Although an extended critical region is in principle compatible with the numerical results, a scenario with a huge crossover scale is more plausible. Finally we study the fate of the dimerized phase upon coupling two chains in a ladder geometry. The dimerized phase rapidly vanishes and an extended gapped phase takes over. This gapped phase presumably has dominant short-ranged ferroquadrupolar correlations for $\theta ∊(-3\pi ,4,-\pi ∕2)$ and—surprisingly—seems to be adiabatically connected to the plaquette single solid phase of the Heisenberg $S=1$ ladder and therefore also with the Haldane phase of isolated chains.

Figure 1

Phase diagram of the bilinear-biquadratic spin-one chain. The firmly established phases are the Haldane, the ferromagnetic and the dimerized phase. We characterize the extended gapless phase $\pi ∕4\le \theta <\pi ∕2$ by having dominant $k=\pm 2\pi ∕3$ spin quadrupolar correlations. The possible occurrence of a spin nematic like phase close to $-3\pi ∕4$ is investigated and critically discussed.

Figure 2

(Color online) Upper plot: Static structure factors for spin and quadrupolar correlations at momentum $k=2\pi ∕3$ as a function of $\theta$ for different system sizes. At the SU(3) point $\theta =+\pi ∕4$ the two correlation functions are related by symmetry. By going deeper into the critical region the spin correlations get weaker, while the quadrupolar correlations are enhanced. Lower plot: the central charge, the excitation velocity and the scaling dimension of the quintuplet field in the critical region. The filled red square is the exact result (Ref. 33) for the velocity at $\theta =\pi ∕4$.

Figure 3

(Color online) Extrapolated DMRG $S=2$ gaps as a function of $\theta$. The gray region denotes the estimated onset of the critical behavior. The filled diamond is the exactly known gap result at $\theta =-\pi ∕2$. Inset: finite size extrapolation of the $S=2$ gaps (32 to 512 sites). $\theta ∕\pi$ varies from $-0.5$ to $-0.7$ with decrements of $0.025$ from top to bottom.

Figure 4

Strong coupling dimer expansion of the $S=2$ gap. Left panel: $S=2$ gap as a function of $\theta$. The gap closes at ${\theta }_c\approx -0.67\pi$. Upper right panel: Critical value ${\lambda }_c$ where the gap vanishes. The uniformly coupled chain of interest here corresponds to $\lambda =1$ (dashed line). Lower right panel: Critical exponent of the $S=2$ gap opening as a function of the dimerization $(\mathrm{\Delta }\sim \mid \lambda -{\lambda }_c{\mid }^{\gamma })$ calculated by various DLog Padé approximants.

Figure 5

Finite size scaling of the crossing points between the singlet gap at momentum $\pi$ and the quintuplet $(S=2)$ gap at momentum 0. The extrapolation yields a critical value ${\theta }_c\approx -0.67\pi$ in the thermodynamic limit.

Figure 6

Static and generalized susceptibilities for the bilinear dimerization (left panel) and the ferroquadrupolar correlations (right panel), obtained by ED on systems of 8 to 16 sites. Solid symbols: Static structure factors. Open symbols: Corresponding generalized susceptibilities.

Figure 7

ED dynamical correlation results on a $L=16$ chain. Upper panel: dynamical staggered dimer correlation function $\mathcal{D}(q=\pi ,\omega )$ plotted for a range of values of $\theta$. There is an important transfer of spectral weight upon lowering $\theta$ from a low energy level—collapsing to the ground state as a function of system size—to a level which will converge to the energy $\sqrt{2}$ at $\theta =-3\pi ∕4$. Lower panel: dynamical ferroquadrupolar correlations $\mathcal{Q}(q=0,\omega )$ for $\theta ∊(0.75,0.60]\pi$.

Figure 8

(Color online) Hypothetical conformal field theory parameters calculated in ED on chains of 8 up to 18 sites. The scaling dimensions of the singlet $(k=\pi )$, and the quintuplet $(k=0)$ field are shown. Inset: effective central charge. The value at the boundary of the critical region $(\theta \approx -0.67\pi )$ is $c\approx 3∕2$.

Figure 9

(Color online) Phase diagram of two coupled bilinear-biquadratic $S=1$ chains. The dimerization along the chains is rapidly suppressed by a finite interchain coupling. The dominating phase is a unique gapped phase which is adiabatically connected to the Haldane phase of two isolated Heisenberg chains.

Figure 10

(Color online) Finite size scaling of the dimerization measured in the middle of a ladder as a function of $K∕J$. It can be clearly seen that the dimerization extrapolates to a finite value for $K∕J\le 0.125$, while it tends to zero for $K∕J\ge 0.15$.

Figure 11

(Color online) Evolution of the spin gap for constant $K∕J=1$ as a function of $\theta ∊[-3\pi ∕4,0]$. The nature of the gap changes at the $\mathrm{SU}\left(3\right)$ point $\theta =-\pi ∕2$, from a $S=1$ state for $\theta >-\pi ∕2$ to a $S=2$ state for $\theta <-\pi ∕2$. The QMC value of the gap at $\theta =0$ is from Ref. 25.