Energy Corrections

The energy reported during training (total_energy) is complex-valued because the magnetic kinetic energy term is complex. The electronic variational energy \(E_v\) is its real part — the imaginary component is a finite-sampling artifact whose expectation value vanishes. \(E_v\) includes only the kinetic and electron-electron Coulomb terms evaluated on the sphere. It does not include the background charge contribution or finite-size corrections needed to compare with literature values.

To obtain the corrected energy \(E_c\) per electron reported in our paper, apply the following post-processing steps:

Background Charge

We add the neutralizing background contribution on the Haldane sphere (Eq. 7 of our paper; see also [Jain, Composite Fermions, Cambridge University Press, 2007]):

\[E_\text{bg} = -\kappa\hbar\omega_c \frac{N^2}{2\sqrt{Q}}\]

where \(\kappa = (e^2/\epsilon\ell)/(\hbar\omega_c)\) is the Landau level mixing parameter (corresponds to interaction_strength in the code), \(Q = \text{flux}/2\), and \(N\) is the total electron count.

Density Correction

We subtract the zero-point energy \(N\omega_c/2\) and apply a density correction factor to reduce finite-size dependence (Eq. 8 of our paper; Morf, Phys. Rev. B 33, 2221 (1986)):

\[E_c = \sqrt{\frac{2Q\nu}{N}}\left(E_v + E_\text{bg} - \frac{N\omega_c}{2}\right)\]

where \(\nu\) is the filling factor. The flux-filling relationship on the sphere is \(2Q = N/\nu - \mathcal{S}\), where \(\mathcal{S}\) is the topological shift (e.g., \(\mathcal{S} = 3\) for the Laughlin \(\nu = 1/3\) state). All energies are in units of \(\hbar\omega_c\kappa\).

Quasiparticle and Quasihole Excitations

On the Haldane sphere, quasiparticle (qp) and quasihole (qh) excitations are ground states at shifted flux values. For the \(\nu = 1/3\) state:

\[2Q^{\text{qp/qh}} = 3(N-1) \mp 1\]

Their background charge contribution differs from the ground state:

\[E_\text{bg}^{\text{qp/qh}} = -\kappa\hbar\omega_c \frac{N^2 - q^2}{2\sqrt{Q^{\text{qp/qh}}}}\]

where \(|q| = 1/3\) is the fractional excess charge. We apply the density correction to each term separately to get the corrected transport gap:

\[E_c^{\text{gap}} = \sqrt{\frac{2Q^{\text{qp}}\nu}{N}}\left(E_v^{\text{qp}} + E_\text{bg}^{\text{qp}} - \frac{N\omega_c}{2}\right) + \sqrt{\frac{2Q^{\text{qh}}\nu}{N}}\left(E_v^{\text{qh}} + E_\text{bg}^{\text{qh}} - \frac{N\omega_c}{2}\right) - 2E_c\]