quantumhall-matrixelements: Quantum Hall Landau-Level Matrix Elements

Landau-level plane-wave form factors and exchange kernels for quantum Hall systems in a small, reusable package (useful for Hartree-Fock and related calculations). It provides:

• Analytic Landau-level plane-wave form factors $F_{n',n}^\sigma(\mathbf{q})$.
• Exchange kernels $X_{n_1 m_1 n_2 m_2}^\sigma(\mathbf{G})$.
• Symmetry diagnostics for verifying kernel implementations.

Plane-Wave Landau-level Form Factors

For $\sigma = \mathrm{sgn}(qB_z)$, where $q$ is the charge of the carrier and $B_z$ is the magnetic field direction, The plane-wave matrix element $F^\sigma_{n',n}(\mathbf{q}) = \langle n' | e^{i \mathbf{q} \cdot \mathbf{R}_\sigma} | n \rangle$ can be written as

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

where $n_&lt; = \min(n, n')$, $n_&gt; = \max(n, n')$, and $L_n^\alpha$ are the generalized Laguerre polynomials, and $\ell_B$ is the magnetic length.

Exchange Kernels

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

where $V(q)$ is the interaction potential. For the Coulomb interaction, $V(q) = \frac{2\pi e^2}{\epsilon q}$.

Units and Interaction Strength

The package performs calculations in dimensionless units where lengths are scaled by $\ell_B$. The interaction strength is parameterized by a dimensionless prefactor $\kappa$.

• Coulomb interaction: The code assumes a potential of the form $V(q) = \kappa \frac{2\pi e^2}{q \ell_B}$ (in effective dimensionless form).
  • If you set kappa = 1.0, the resulting exchange kernels are in units of the Coulomb energy scale $E_C = e^2 / (\epsilon \ell_B)$.
  • To express results in units of the cyclotron energy $\hbar \omega_c$, set $\kappa = E_C / (\hbar \omega_c) = (e^2/\epsilon \ell_B) / (\hbar \omega_c)$.
• Custom potential: Provide a callable potential(q) that returns values in your desired energy units. The integration measure $d^2q/(2\pi)^2$ introduces a factor of $1/\ell_B^2$, so ensure your potential scaling is consistent.

Installation

From PyPI (once published):

pip install quantumhall-matrixelements

From a local checkout (development install):

pip install -e .[dev]

Basic usage

import numpy as np
from quantumhall_matrixelements import (
    get_form_factors,
    get_exchange_kernels,
)

# Simple G set: G0=(0,0), G+=(1,0), G-=(-1,0)
Gs_dimless = np.array([0.0, 1.0, 1.0])
thetas = np.array([0.0, 0.0, np.pi])
nmax = 2

F = get_form_factors(Gs_dimless, thetas, nmax)          # shape (nG, nmax, nmax)
X = get_exchange_kernels(Gs_dimless, thetas, nmax)      # default 'gausslegendre' backend

print("F shape:", F.shape)
print("X shape:", X.shape)

To use a user-provided interaction, pass a callable directly as potential:

def V_coulomb(q, kappa=1.0):
    # q is in 1/ℓ_B units; this returns V(q) in Coulomb units
    return kappa * 2.0 * np.pi / q

X_coulomb = get_exchange_kernels(
    Gs_dimless,
    thetas,
    nmax,
    method="gausslegendre",
    potential=lambda q: V_coulomb(q, kappa=1.0),
)

For more detailed examples, see the example scripts under examples/ and the tests under tests/.

Magnetic-field sign

The public APIs expose a sign_magneticfield keyword that represents $\sigma = \mathrm{sgn}(q B_z)$, the sign of the charge–field product. The default sign_magneticfield=-1 matches the package's internal convention (electrons in a positive $B_z$). Passing sign_magneticfield=+1 returns the appropriate complex-conjugated form factors or exchange kernels for the opposite field direction without requiring any manual phase adjustments:

F_plusB = get_form_factors(Gs_dimless, thetas, nmax, sign_magneticfield=+1)
X_plusB = get_exchange_kernels(Gs_dimless, thetas, nmax, method="hankel", sign_magneticfield=+1)

Citation

If you use the package quantumhall-matrixelements in academic work, you must cite:

Sparsh Mishra and Tobias Wolf, quantumhall-matrixelements: Quantum Hall Landau-Level Matrix Elements, version 0.1.0, 2025.
DOI: https://doi.org/10.5281/zenodo.17807688

A machine-readable CITATION.cff file is included in the repository and can be used with tools that support it (for example, GitHub’s “Cite this repository” button).

Backends and Reliability

The package provides two backends for computing exchange kernels:

1. gausslegendre (Default)

   • Method: Gauss-Legendre quadrature mapped from $[-1, 1]$ to $[0, \infty)$ via a rational mapping.
   • Pros: Fast and numerically stable for all Landau-level indices ($n$).
   • Cons: May require tuning nquad for extremely large momenta or indices ($n &gt; 100$).
   • Recommended for: General usage, especially for $n \ge 10$.

2. hankel

   • Method: Discrete Hankel transform.
   • Pros: High precision and stability.
   • Cons: Significantly slower than quadrature methods.
   • Recommended for: Reference calculations and verifying the Gauss–Legendre backend.

Notes

The following wavefunction is used to find all matrix elements:

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

Development

• Run tests and coverage:

  pytest

• Lint and type-check:

  ruff check .
  mypy .

Authors and license

• Authors: Dr. Tobias Wolf, Sparsh Mishra
• Copyright © 2025 Tobias Wolf
• License: MIT (see LICENSE).