Conventions

Magnetic-field sign

The package uses

\[ \sigma = \mathrm{sgn}(q B_z) \]

as its public field-orientation convention. The default is sign_magneticfield = -1, which corresponds to the package’s electron in positive B_z convention.

Passing sign_magneticfield=+1 returns the opposite-field quantity with the appropriate phase convention already applied. No manual conjugation or angle flip is required.

Momentum and magnetic-length units

The basic inputs are polar wavevector arrays q_magnitudes and q_angles. They must have the same shape; the public APIs do not broadcast them.

All formulas depend on the dimensionless combination

\[ |q| \ell_B. \]

This means:

• If your inputs are already dimensionless |q| l_B, pass them directly and leave lB=1.

• If your inputs are physical wavevectors in inverse-length units, pass those values together with the desired lB.

The same rule applies to the symmetric-gauge form-factor helpers.

Landau-gauge form factor

The implemented plane-wave matrix element is

\[ F_{n',n}^{\sigma}(\mathbf{q}) = i^{|n-n'|} e^{i \sigma (n' - n)\theta_{\mathbf{q}}} \sqrt{\frac{n_{<}!}{n_{>}!}} \left( \frac{| \mathbf{q} | \ell_B}{\sqrt{2}} \right)^{|n-n'|} L_{n_<}^{|n-n'|}\!\left( \frac{| \mathbf{q} |^2 \ell_B^2}{2} \right) e^{-| \mathbf{q} |^2 \ell_B^2 / 4}, \]

where n_< = min(n, n') and n_> = max(n, n').

The underlying Landau-gauge wavefunction convention is

\[ \Psi_{nX}^{\sigma}(x, y) = \frac{e^{i \sigma X y \ell_B^{-2}}}{\sqrt{L_y}} i^n \phi_n(x - X), \qquad X = \sigma k_y \ell_B^2. \]

Exchange kernel

The exchange kernel is

\[ X_{n_1 m_1 n_2 m_2}^{\sigma}(\mathbf{G}) = \int \frac{d^2 q}{(2\pi)^2} V(q) F_{m_1,n_1}^{\sigma}(\mathbf{q}) F_{n_2,m_2}^{\sigma}(-\mathbf{q}) e^{i \sigma (\mathbf{q} \times \mathbf{G})_z \ell_B^2}. \]

For most calculations, method="laguerre" is the right backend. The full materialized exchange tensor scales like O(nmax^4) per G, so large jobs usually belong on the compressed or Fock-constructor APIs.

The explicit tensor order is (G, n1, m1, n2, m2). In this Landau-gauge notation, m1 and m2 are still Landau-level labels, not symmetric-gauge guiding-center orbitals.

Symmetric gauge

In the symmetric-gauge basis |n, m>, the density operator factorizes as

\[ \langle n', m' | e^{i \mathbf{q} \cdot \mathbf{r}} | n, m \rangle = F_{n',n}^{\sigma}(\mathbf{q}) G_{m',m}^{-\sigma}(\mathbf{q}). \]

The guiding-center sector therefore has the opposite chirality to the cyclotron sector. The helper get_guiding_center_form_factors(...) handles that sign flip internally.

Coulomb scaling

For the built-in Coulomb interaction, the package assumes

\[ V(q) = \kappa \frac{2 \pi}{q}, \]

with q expressed in 1 / l_B units. With kappa=1, the resulting exchange quantities are in the Coulomb energy scale e^2 / (\epsilon l_B). If you pass a callable potential(q), its return value sets the overall energy units.