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quantumhall-matrixelements: Quantum Hall Landau-Level Matrix Elements

Docs DOI

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

  • Analytic Landau-level plane-wave form factors $F_{n',n}^\sigma(\mathbf{q})$.
  • Exchange kernels $X_{n_1 m_1 n_2 m_2}^\sigma(\mathbf{G})$.
  • Symmetric-gauge guiding-center and factorized density form factors.
  • Central one-body matrix elements, plane Haldane pseudopotentials, and LLL disk two-body reconstruction helpers.
  • 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 $n_&lt; = \min(n, n')$, $n_&gt; = \max(n, n')$, $L_n^\alpha$ 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, $V(q) = 2\pi e^2 / (\epsilon q)$. Units, $\kappa$ scaling, and the magnetic-field-sign convention are documented under Conventions.

Installation

From PyPI:

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,
)

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)      # built-in Coulomb

A custom interaction potential can be supplied as a callable of |q| in 1/ℓ_B units:

def V_screened(q, kappa=1.0, q_TF=0.5):
    return kappa * 2.0 * np.pi / (q + q_TF)

X_screened = get_exchange_kernels(
    Gs_dimless,
    thetas,
    nmax,
    potential=V_screened,
)

For more detailed examples, see the example scripts under examples/ and the tests under tests/. The documentation covers:

  • Plane-wave workflows — form factors, exchange kernels, Fock-matrix construction, memory guards, and backend choice.
  • Symmetric-gauge workflows — factorized density form factors, Haldane pseudopotentials, central one-body matrix elements, and LLL disk two-body reconstruction.

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

DOI

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).

Development

  • Run tests and coverage:

    pytest

  • Lint and type-check:

    ruff check .
mypy .

Authors and license

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