Topological Criticality with Charge-\(3e\) Fluctuations in the \(\mathrm{SU}(3)\) Triangular-Lattice Hofstadter--Hubbard Model

A transition between an integer quantum Hall insulator and a chiral spin liquid can become critical without closing the single-electron gap, raising the question of which composite excitations govern the change in topological order. We investigate this question in the \(\mathrm{SU}(3)\) triangular-lattice Hofstadter--Hubbard model at one particle per site and magnetic flux \(π/3\) per triangle. Using infinite density matrix renormalization group calculations, we identify a weak-coupling integer quantum Hall phase and an interaction-driven \(\mathrm{SU}(3)_1\) chiral spin liquid. Near the transition, both the neutral and charge-\(3e\) correlation lengths increase substantially, whereas the single-electron correlation length remains short, providing evidence for a Chern--Simons--Higgs critical theory, where a gauge-invariant monopole carries the quantum numbers of a fermionic \(\mathrm{SU}(3)\)-singlet trion. We discuss the implications for doping the system and recent experiments on doping \(ν=-2/3\) fractional quantum anomalous Hall in tMoTe\(_2\).