Determine the thermodynamic phase boundary under magnetic flux

Determine the thermodynamic phase boundary separating competing charge-ordered and Hofstadter states in the interacting magnetic moiré continuum model, beyond the finite-supercell comparisons of Real and Complex Psiformer states.

Background

The paper studies independently optimized finite-cell neural-wavefunction states across several integer flux sectors and observes nonmonotonic changes in density correlations and weakening honeycomb mean-density modulation at higher flux. However, the calculations use finite supercells, a single optimization endpoint per flux sector, and model-dependent magnetic-section assignments. Consequently, the reported changes cannot by themselves locate a thermodynamic transition or phase boundary between charge order and Hofstadter-like behavior.