Haldane Gap of the Three-Box Symmetric $\mathrm{SU}(3)$ Chain

Motivated by the recent generalization of the Haldane conjecture to $\mathrm{SU}(3)$ chains [M. Lajk\'o 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 $[3\,0\,0]$ sector $\Delta_{[3\,0\,0]}/J = 0.040\pm0.006$ 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.