Why the zero-field $1/3$ state is weaker than the $2/3$ state

In twisted bilayer $\mathrm{MoTe_2}$ at $\theta \sim 3.5-3.9\,\mathrm{deg}$, fractional quantum anomalous Hall states are seen in transport at hole fillings $\nu = -2/3$ and $-3/5$ of the first Chern band (Cai et al., Park et al., Zeng et al., Nature 2023), while at $\nu = -1/3$ only optical signatures of a fragile state over a narrow displacement-field range have been reported (arXiv:2602.04561). Explain this asymmetry: mixing with the second moire band, Berry-curvature and quantum-metric inhomogeneity, or interaction-induced band renormalization favouring a charge-density wave at $1/3$. An answer is a calculation with independently fixed parameters reproducing the ground state and gap at each filling and its displacement-field range.

Band-mixing-inclusive exact diagonalization or DMRG with independently fixed interaction parameters predicting the $\nu = -1/3$ ground state and gap, checked against transport and local imaging of charge order at that filling.