.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/condensed/laughlin/plot_laughlin_correlation_hole.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_condensed_laughlin_plot_laughlin_correlation_hole.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_condensed_laughlin_plot_laughlin_correlation_hole.py:


The Fractional Quantum Hall Effect: Laughlin's Correlation Hole
======================================================================

Tsui, Stormer, and Gossard's 1982 discovery of a Hall plateau at filling
:math:`\nu=1/3` -- a fraction the single-particle Landau-level picture
behind the *integer* effect cannot explain, since a partially filled
Landau level is gapless without interactions -- was explained the
following year by Laughlin's many-body trial wavefunction,

.. math::

    \Psi_m(z_1,\dots,z_N) = \prod_{i<j}(z_i-z_j)^m\,
    e^{-\sum_i|z_i|^2/4\ell_B^2}, \qquad m=3 \text{ for } \nu=1/3.

:math:`|\Psi_m|^2` doubles as the Boltzmann weight of a classical 2D
plasma (Laughlin's "plasma analogy"), letting
:func:`~physicskit.condensed.laughlin.laughlin_metropolis_sweep` sample it
directly with ordinary Metropolis Monte Carlo -- no diagonalization of any
kind. The resulting pair correlation function
(:func:`~physicskit.condensed.laughlin.laughlin_pair_correlation`) shows
the wavefunction's defining feature: particles actively avoid each other
far more strongly than Pauli exclusion alone requires, a "correlation
hole" that vanishes at zero separation and whose quasihole excitations
carry the fractional charge :math:`e/m`.

.. GENERATED FROM PYTHON SOURCE LINES 29-35

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.condensed.laughlin import laughlin_metropolis_sweep, laughlin_pair_correlation, laughlin_radial_density
    from physicskit.statphys.core.monte_carlo import seed_numba_random








.. GENERATED FROM PYTHON SOURCE LINES 36-41

Sampling the Laughlin plasma
----------------------------------
N particles start in a random disk of the mean-field droplet radius,
then equilibrate under repeated Metropolis sweeps of the exact
:math:`|\Psi_m|^2` weight before any measurement is taken.

.. GENERATED FROM PYTHON SOURCE LINES 41-54

.. code-block:: Python


    rng = np.random.default_rng(0)  # initial positions
    seed_numba_random(0)  # Metropolis moves (drawn inside Numba-compiled code)
    N, m = 40, 3
    R0 = np.sqrt(2 * m * N)
    r0 = R0 * np.sqrt(rng.random(N))
    theta0 = rng.uniform(0, 2 * np.pi, N)
    z = (r0 * np.cos(theta0) + 1j * r0 * np.sin(theta0)).astype(np.complex128)

    n_equilibrate = 2000
    for _ in range(n_equilibrate):
        laughlin_metropolis_sweep(z, m=m, step=1.0)








.. GENERATED FROM PYTHON SOURCE LINES 55-59

Measuring the correlation hole and the droplet's flat-top density
-------------------------------------------------------------------------
A handful of sweeps between samples decorrelates successive snapshots
enough for an unbiased pair-correlation and density estimate.

.. GENERATED FROM PYTHON SOURCE LINES 59-87

.. code-block:: Python


    n_samples, n_decorrelate = 400, 5
    samples = []
    for _ in range(n_samples):
        for _ in range(n_decorrelate):
            laughlin_metropolis_sweep(z, m=m, step=1.0)
        samples.append(z.copy())
    samples = np.array(samples)

    r_g, g = laughlin_pair_correlation(samples, m=m, r_max=10.0, n_bins=50)
    r_n, density = laughlin_radial_density(samples, r_max=1.3 * R0, n_bins=40)
    bulk_density = 1.0 / (2 * np.pi * m)

    fig, axes = plt.subplots(1, 2, figsize=(11, 4.3))
    axes[0].plot(r_g, g, color="steelblue")
    axes[0].axhline(1.0, color="black", lw=0.5, ls="--")
    axes[0].set_xlabel(r"separation $r$  ($\ell_B$)")
    axes[0].set_ylabel("g(r)")
    axes[0].set_title(f"Correlation hole (m = {m})")

    axes[1].plot(r_n, density / bulk_density, color="firebrick")
    axes[1].axhline(1.0, color="black", lw=0.5, ls="--", label="bulk density")
    axes[1].set_xlabel(r"radius $r$  ($\ell_B$)")
    axes[1].set_ylabel(r"$\rho(r) / \rho_{\text{bulk}}$")
    axes[1].set_title("Incompressible droplet: flat interior, soft edge")
    axes[1].legend(fontsize=8)
    fig.tight_layout()




.. image-sg:: /api/gallery/condensed/laughlin/images/sphx_glr_plot_laughlin_correlation_hole_001.png
   :alt: Correlation hole (m = 3), Incompressible droplet: flat interior, soft edge
   :srcset: /api/gallery/condensed/laughlin/images/sphx_glr_plot_laughlin_correlation_hole_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 88-97

``g(r)`` vanishes at zero separation and climbs to the uncorrelated value
of 1 within a few magnetic lengths -- particles are kept apart far more
effectively than Pauli exclusion alone would manage, exactly the
correlation the :math:`(z_i-z_j)^m` prefactor was built to enforce. The
density profile is close to the exact bulk value
:math:`\rho=1/(2\pi m\,\ell_B^2)` through most of the droplet's interior,
falling off only within a few magnetic lengths of its edge -- the
real-space picture of the same incompressible liquid that makes the
quantized Hall plateau itself possible.

.. GENERATED FROM PYTHON SOURCE LINES 97-99

.. code-block:: Python


    plt.show()








.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.295 seconds)


.. _sphx_glr_download_api_gallery_condensed_laughlin_plot_laughlin_correlation_hole.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_laughlin_correlation_hole.ipynb <plot_laughlin_correlation_hole.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_laughlin_correlation_hole.py <plot_laughlin_correlation_hole.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_laughlin_correlation_hole.zip <plot_laughlin_correlation_hole.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_