The task sits within a stream of work on neural network wave functions for strongly correlated electrons. Real-valued architectures with self-attention for moiré systems without a field have already been built, as have complex approaches for the fractional quantum Hall effect. However, the question of whether a complex internal representation offers a practical advantage in variational search within the magnetic moiré continuum remained open. This work fills that gap by introducing complex attention and magnetic boundary conditions into a single scheme.
A neural network wave function, Complex Psiformer, was constructed. Each orbital is formed as the product of a fixed magnetic section carrying the boundary phase law and a trainable periodic coefficient. The coefficient is computed by a network with complex hidden features and HMCVA attention blocks, where attention weights are real and messages are complex. Antisymmetry is ensured by Slater determinants. Optimization was performed with the variational Monte Carlo method using minimum-step stochastic reconfiguration (MinSR). The complex and real versions of Psiformer were compared, as were neural network Hartree-Fock bases, on systems of 12 electrons in supercells of 18 and 25 moiré cells.
After the same number of optimization steps, Complex Psiformer reached lower energy than Real Psiformer in both supercells. In the 25-cell system, the average gain over five runs was 1.458 meV per electron with smaller scatter. For a separately trained two-electron system, the complex state came 16.891 meV per electron closer to configuration interaction than the real one. In the 18-cell system at flux n_phi=2, the complex state showed a sharper honeycomb charge pattern: 95.8% of the difference in the structure factor comes from the average density contribution. When the flux is increased to n_phi=4, the modulation of the average density weakens, but connected fluctuations persist. Gauge-invariant current maps revealed qualitative differences in local circulation.
The work stands out from neighboring studies in that it systematically tests, for the first time, the advantage of a complex internal representation in the magnetic moiré continuum under an equal optimization budget. Neighboring studies either limited themselves to zero field or used real hidden features with complex readout. Here complexity permeates the entire hidden representation and the attention mechanism, yielding a measurable energy gain. This suggests that the structure of quantum mechanics can guide more efficient variational search.
The topic could develop toward isolating the contributions of individual components: radial activation, the attention rule, and residual stabilization. Comparisons with a matched hyperparameter tuning budget will be needed. Studying alternative magnetic sections and repeated optimizations to assess trends with flux is also promising. Efficient derivative computation and transferable training could extend tests to larger systems.
The results will influence the modeling of quantum phases in moiré semiconductors and the design of neural network ansatze for problems involving a magnetic field.
The next step is component-wise comparisons with matched tuning budgets to understand the origin of the energy gain. Reference densities and currents are also needed to verify the physical consequences of the improvement.
The work connects to the open problem of accurately describing the competition between charge order and Hofstadter states in moiré lattices as magnetic flux varies. It also touches on the question of the role of complex representations in variational neural network states for strongly correlated fermions.
In the 25-cell system, the average gain of Complex Psiformer over Real Psiformer is 1.458 meV per electron, roughly 14 times smaller than the room-temperature thermal energy of 25 meV. Such small energy differences are nevertheless reproducibly stable across five independent runs.
The idea of a neural network that learns to 'see' a quantum phase echoes the image of self-learning quantum computers in Greg Egan's novel Schild's Ladder, where states of matter are computed as variational approximations.