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

Abstract

Motivated by the recent generalization of the Haldane conjecture to $\mathrm{SU}(3)$ chains [M. 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 ${\Delta }_{[300]}/J=0.040\pm 0.006$ 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.

6 pages, 4 figures, + Supplemental Material 12 pages, 15 figures

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• Exact quantization of topological order parameter in SU($N$) spin models, $N$-ality transformation and ingappabilities
• Density Matrix Renormalization Group simulations of the SU(N) Fermi-Hubbard chain implementing the full SU(N) symmetry via Semi-Standard Young Tableaux and Unitary Group Subduction Coefficients
• Edge states and universality class of the critical two-box symmetric SU(3) chain
• The dynamical structure factor of the SU(3) Heisenberg chain: The variational Monte Carlo approach
• Bosonic realization of SU(3) chiral Haldane phases