.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/condensed/ginzburg_landau/plot_ginzburg_landau_healing.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_ginzburg_landau_plot_ginzburg_landau_healing.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_condensed_ginzburg_landau_plot_ginzburg_landau_healing.py:


Ginzburg-Landau Theory: Healing Length and Type I vs Type II
===========================================================================

Ginzburg and Landau described superconductivity with a free-energy
functional of a complex order parameter alone, no microscopic pairing
mechanism required. Minimizing it gives an equilibrium condensate density
:math:`|\psi_0|^2=-a/b` (:func:`~physicskit.condensed.ginzburg_landau.gl_equilibrium_order_parameter`)
below the transition, and two length scales -- the coherence length
:math:`\xi` (:func:`~physicskit.condensed.ginzburg_landau.gl_coherence_length`)
and the penetration depth :math:`\lambda`
(:func:`~physicskit.condensed.ginzburg_landau.gl_penetration_depth`) --
whose ratio, the Ginzburg-Landau parameter :math:`\kappa`
(:func:`~physicskit.condensed.ginzburg_landau.ginzburg_landau_parameter`),
alone decides Type I vs Type II behavior.

.. GENERATED FROM PYTHON SOURCE LINES 17-30

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.condensed.ginzburg_landau import (
        ginzburg_landau_parameter,
        gl_coherence_length,
        gl_equilibrium_order_parameter,
        gl_free_energy_density,
        gl_order_parameter_profile,
        gl_penetration_depth,
    )








.. GENERATED FROM PYTHON SOURCE LINES 31-35

The free energy's double-well shape below the transition
---------------------------------------------------------------------
For a < 0, f(psi) has degenerate minima at psi = +-psi0 rather than the
single minimum psi=0 of the normal state (a > 0).

.. GENERATED FROM PYTHON SOURCE LINES 35-42

.. code-block:: Python


    a, b = -1.0, 1.0
    psi0 = gl_equilibrium_order_parameter(a, b)
    psi_range = np.linspace(-1.5 * psi0, 1.5 * psi0, 200)
    free_energy = [gl_free_energy_density(psi, a, b) for psi in psi_range]
    print(f"equilibrium order parameter psi0 = {psi0:.4f}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    equilibrium order parameter psi0 = 1.0000




.. GENERATED FROM PYTHON SOURCE LINES 43-49

The order parameter heals from a boundary over the coherence length
---------------------------------------------------------------------
Pinned to zero at a boundary (e.g. a normal-superconducting interface),
the order parameter recovers its bulk value over a few coherence
lengths -- the exact analytic solution
:func:`~physicskit.condensed.ginzburg_landau.gl_order_parameter_profile`.

.. GENERATED FROM PYTHON SOURCE LINES 49-54

.. code-block:: Python


    xi = gl_coherence_length(a=a)
    x = np.linspace(0, 8 * xi, 300)
    profile = gl_order_parameter_profile(x, xi)








.. GENERATED FROM PYTHON SOURCE LINES 55-60

Type I vs Type II: the Ginzburg-Landau parameter kappa
---------------------------------------------------------------------
A denser condensate screens magnetic fields over a shorter penetration
depth. Sweeping psi0 sweeps kappa=lambda/xi across the 1/sqrt(2)
Type I / Type II boundary.

.. GENERATED FROM PYTHON SOURCE LINES 60-87

.. code-block:: Python


    psi0_values = np.linspace(0.2, 3.0, 40)
    kappas = [ginzburg_landau_parameter(xi, gl_penetration_depth(psi0=p)) for p in psi0_values]
    kappa_c = 1.0 / np.sqrt(2)

    fig, axes = plt.subplots(1, 3, figsize=(14, 4))

    axes[0].plot(psi_range / psi0, free_energy, lw=2.5)
    axes[0].axvline(1.0, color="gray", ls="--", lw=1)
    axes[0].axvline(-1.0, color="gray", ls="--", lw=1)
    axes[0].set_xlabel(r"$\psi/\psi_0$")
    axes[0].set_ylabel("free energy density f")
    axes[0].set_title("Double-well free energy (a<0)")

    axes[1].plot(x / xi, profile, lw=2.5)
    axes[1].set_xlabel(r"$x/\xi$")
    axes[1].set_ylabel(r"$\psi(x)/\psi_0$")
    axes[1].set_title("Order parameter healing at a boundary")

    axes[2].plot(psi0_values, kappas, lw=2.5)
    axes[2].axhline(kappa_c, color="C1", ls="--", label=r"$\kappa=1/\sqrt{2}$")
    axes[2].set_xlabel(r"$\psi_0$ (condensate density)")
    axes[2].set_ylabel(r"$\kappa=\lambda/\xi$")
    axes[2].set_title("Type I (below) vs Type II (above)")
    axes[2].legend()

    fig.tight_layout()



.. image-sg:: /api/gallery/condensed/ginzburg_landau/images/sphx_glr_plot_ginzburg_landau_healing_001.png
   :alt: Double-well free energy (a<0), Order parameter healing at a boundary, Type I (below) vs Type II (above)
   :srcset: /api/gallery/condensed/ginzburg_landau/images/sphx_glr_plot_ginzburg_landau_healing_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.090 seconds)


.. _sphx_glr_download_api_gallery_condensed_ginzburg_landau_plot_ginzburg_landau_healing.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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