We construct and study quantum trimer models and resonating $\mathrm{SU}\left(3\right)$-singlet models on the kagome lattice, which generalize quantum dimer models and the resonating valence bond wave functions to a trimer and $\mathrm{SU}\left(3\right)$ setting. We demonstrate that these models carry a ${\mathbb{Z}}_3$ symmetry which originates in the structure of trimers and the $\mathrm{SU}\left(3\right)$ representation theory, and which becomes the only symmetry under renormalization. Based on this, we construct simple and exact parent Hamiltonians for the model which exhibit a topological ninefold degenerate ground space. A combination of analytical reasoning and numerical analysis reveals that the quantum order ultimately displayed by the model depends on the relative weight assigned to different types of trimers—it can display either ${\mathbb{Z}}_3$ topological order or form a symmetry-broken trimer crystal, and in addition possesses a point with an enhanced $\mathrm{U}\left(1\right)$ symmetry and critical behavior. Our results accordingly hold for the $\mathrm{SU}\left(3\right)$ model, where the two natural choices for trimer weights give rise to either a topological spin liquid or a system with symmetry-broken order, respectively. Our work thus demonstrates the suitability of resonating trimer and $\mathrm{SU}\left(3\right)$-singlet ansatzes to model $\mathrm{SU}\left(3\right)$ topological spin liquids on the kagome lattice.

• Received 19 June 2020
• Revised 13 August 2020
• Accepted 14 August 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.033382

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society