Haldane Gap of the Three-Box Symmetric SU(3) Chain

Motivated by the recent generalization of the Haldane conjecture to $\mathrm{SU}(3)$ chains [Lajkó et al., Nucl. Phys. B924, 508 (2017)] according to which a Haldane gap should be present for symmetric representations if the number of boxes in the Young diagram is a multiple of three, we develop a density matrix renormalization group algorithm based on standard Young tableaus to study the model with three boxes directly in the representations of the global $\mathrm{SU}(3)$ symmetry. We show that there is a finite gap between the singlet and the symmetric $[300]$ sector ${\mathrm{\Delta }}_{[300]}/J=0.040\pm 0.006$ where $J$ is the antiferromagnetic Heisenberg coupling, and we argue on the basis of the structure of the low energy states that this is sufficient to conclude that the spectrum is gapped.

Figure 1

(a) $\mathrm{SU}(3)$ AKLT state for the physical three-box symmetric irrep at each site. An ellipse denotes the projection of two adjoint representations onto the physical three-box symmetric irrep. The thick lines joining neighboring sites represent singlets made out of two adjoint irreps. (b) Decomposition of the tensor product of two adjoint representations.

Figure 2

Gap from the singlet sector to the symmetric $[300]$ sector versus inverse chain length for different values of the number $m$ of states kept. The inset shows a power-law fit with exponent $\gamma =0.47$ of the minimum of each finite-$m$ curve, denoted by a black diamond in the main plot, as a function of the total discarded weight ${\eta }_{\mathrm{tot}}$, which is dominated by the discarded weight in the $[300]$ sector [40]. The extrapolated value, shown with a blue diamond, falls approximately in the middle between our lower and upper bounds of the gap, denoted by red squares.

Figure 3

Gap from the singlet sector to the symmetric $[300]$ sector of the interpolation Hamiltonian ${\mathcal{H}}_{\lambda }=(1-\lambda ){\mathcal{H}}_{\mathrm{AKLT}}+\lambda \mathcal{H},\lambda \in [0,1]$.

Figure 4

Entanglement entropy of the ground state for a chain with $N_s=300$ sites. Over 839 000 states would be needed to reproduce the curve $m=12000$ with a code which does not have the $\mathrm{SU}(3)$ symmetry. The inset shows the scaling of the extrapolated central charge in the middle of the chain for different values of the chain length, and comparison with the central charges for the ${\mathrm{SU}(3)}_k$ WZW conformal field theory with $k=1$, 2, 3. There are three sets of data $c_q(m=\infty ,N_s),q=0$, 1, 2 because of the oscillation of the entanglement entropy along the chain for $x=q\text{mod}3$ where $x$ is the position of the cut.