quantumhall-matrixelements 0.1.0

Landau-level plane-wave form factors and exchange kernels for quantum Hall systems.

quantumhall-matrixelements: Quantum Hall Landau-Level Matrix Elements

Landau-level plane-wave form factors and exchange kernels for quantum Hall systems.

This library factors out the continuum matrix-element kernels used in Hartree–Fock and related calculations into a small, reusable package. It provides:

• Analytic Landau-level plane-wave form factors $F_{n',n}(\mathbf{q})$.
• Exchange kernels $X_{n_1 m_1 n_2 m_2}(\mathbf{G})$ computed via:
  • Gauss-Legendre quadrature with rational mapping (gausslegendre backend, default).
  • Generalized Gauss–Laguerre quadrature (gausslag backend).
  • Hankel-transform based integration (hankel backend).
• Symmetry diagnostics for verifying kernel implementations on a given G-grid.

Backends and Reliability

The package provides three backends for computing exchange kernels, each with different performance and stability characteristics:

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 ($n > 100$).
   • Recommended for: General usage, especially for large $n$ ($n \ge 10$).

2. gausslag:

   • Method: Generalized Gauss-Laguerre quadrature.
   • Pros: Very fast for small $n$.
   • Cons: Numerically unstable for large $n$ ($n \ge 12$) due to high-order Laguerre polynomial roots.
   • Recommended for: Small systems ($n < 10$) where speed is critical.

3. hankel:

   • Method: Discrete Hankel transform.
   • Pros: High precision and stability.
   • Cons: Significantly slower than quadrature methods.
   • Recommended for: Reference calculations and verifying other backends.

Mathematical Definitions

Plane-Wave Form Factors

The form factors are the matrix elements of the plane-wave operator $e^{i \mathbf{q} \cdot \mathbf{R}}$ in the Landau level basis $|n\rangle$:

$$ F_{n',n}(\mathbf{q}) = \langle n' | e^{i \mathbf{q} \cdot \mathbf{R}} | n \rangle $$

Analytically, these are given by:

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

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

Exchange Kernels

The exchange kernels $X_{n_1 m_1 n_2 m_2}(\mathbf{G})$ are defined as the Fourier transform of the interaction potential weighted by the form factors:

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

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 $V(q) = \kappa \frac{2\pi \ell_B}{q \ell_B}$ (in effective dimensionless form).

  • If you set kappa = 1.0, the resulting exchange kernels will be in units of the Coulomb energy scale $E_C = e^2 / (\epsilon \ell_B)$.
  • If you want the results in units of the cyclotron energy $\hbar \omega_c$, you should set $\kappa = E_C / (\hbar \omega_c) = (e^2/\epsilon \ell_B) / (\hbar \omega_c)$.

• General Potential: For a general $V(q)$, the function V_of_q should return 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)

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

Citation

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

Tobias Wolf, quantumhall-matrixelements: Quantum Hall Landau-Level Matrix Elements, version 0.1.0, 2025.
DOI: https://doi.org/10.5281/zenodo.17646027

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

• Author: Tobias Wolf
• Copyright © 2025 Tobias Wolf
• License: MIT (see LICENSE).