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.

arXiv comment6 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