We apply field theory methods to SU(3) chains in the symmetric representation, with p boxes in the Young tableau, mapping them into a flag manifold nonlinear σ-model with a topological angle θ = 2πp/3. Generalizing the Haldane conjecture, we argue that the models are gapped for p = 3m but gapless for p = 3m ± 1 (for integer m), corresponding to a massless phase of the σ-model at θ = ±2π/3. We confirm this with Monte Carlo calculations on the σ-model. Figure 4: Breaking translational symmetry of the system by weakening the nearest neighbour bonds between sublattices 1 and 3. The R 13 mirror symmetry and a three site translational symmetry still remains.

Recently, SU(3) chains in the symmetric and self-conjugate representations have been studied using field theory techniques. For certain representations, namely rank-p symmetric ones with p not a multiple of 3, it was argued that the ground state exhibits gapless excitations. For the remaining representations considered, a finite energy gap exists above the ground state. In this paper, we extend these results to SU(n) chains in the symmetric representation. For a rank-p symmetric representation with n and p coprime, we predict gapless excitations above the ground state. If p is a multiple of n, we predict a unique ground state with a finite energy gap. Finally, if p and n have a greatest common divisor 1 < q < n, we predict a ground state degeneracy of n/q, with a finite energy gap. To arrive at these results, we derive a non-Lorentz invariant flag manifold sigma model description of the SU(n) chains, and use the renormalization group to show that Lorentz invariance is restored at low energies. We then make use of recently developed anomaly matching conditions for these Lorentz-invariant models. We also review the Lieb-Shultz-Mattis-Affleck theorem, and extend it to SU(n) models with longer range interactions. arXiv:1910.08196v2 [cond-mat.str-el]

It was recently argued that SU(3) chains in the p-box symmetric irreducible representation (irrep) exhibit a "Haldane gap" when p is a multiple of 3 and are otherwise gapless [Nucl. Phys. B 924, 508 (2017)]. We extend this argument to the self-conjugate irreps of SU(3) with p columns of length 2 and p columns of length 1 in the Young tableau (p = 1 corresponding to the adjoint irrep), arguing that they are always gapped but have spontaneously broken parity symmetry for p odd but not even.

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