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.

References

The finite-cell comparison does not determine a thermodynamic phase boundary.

— Learning Quantum Matter through Attention in Complex Space  (2609.11792 - Jing et al., 10 Sep 2026) in Section 5.4, subsection “Flux-dependent density correlations”

The weaker minima cannot be so clearly defined. Each falls near the next commensurate fractions drawn in Fig.~\ref{fig:crystal}, such as $1/4$ and $2/5$, and could reflect a further crystal commensuration, a general integer Hofstadter gap, or a mixture the present data cannot separate.

— Microwave spectroscopic evidence of a moiré-commensurate electron crystal in monolayer graphene  (2610.21875 - Solodovnyk et al., 15 Oct 2026) in Discussion section, paragraph beginning “The weaker minima cannot be so clearly defined.”