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

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

.. _sphx_glr_api_gallery_condensed_topology_plot_iqhe_hall_conductance.py:


The Integer Quantum Hall Effect: Quantized Hall Conductance from Chern Numbers
====================================================================================

Von Klitzing, Dorda, and Pepper's 1980 discovery -- a two-dimensional
electron gas in a strong magnetic field develops a Hall conductance
quantized to extraordinary precision, :math:`\sigma_{xy} = \nu\,e^2/h` with
:math:`\nu` an exact integer -- is reproduced here on its lattice (Bloch)
incarnation, the Harper-Hofstadter model
(:func:`~physicskit.condensed.models.harper_hofstadter_hamiltonian`): a
square lattice threaded by a uniform flux :math:`p/q` per plaquette. At
flux :math:`1/3` the spectrum splits into three magnetic sub-bands; feeding
each one, in turn, into
:func:`~physicskit.condensed.topology.compute_chern_number` (the same TKNN
machinery used for the Haldane model) gives the exact integer Chern number
of that band, and summing the Chern numbers of every filled band below a
gap gives that gap's quantized Hall conductance directly -- the numerical
content of the TKNN formula that explains *why* von Klitzing's measurement
came out an integer at all.

.. GENERATED FROM PYTHON SOURCE LINES 23-29

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.condensed.models import harper_hofstadter_hamiltonian
    from physicskit.condensed.topology import compute_chern_number








.. GENERATED FROM PYTHON SOURCE LINES 30-34

Three magnetic sub-bands at flux 1/3
------------------------------------------
Odd ``q`` avoids the exact band touchings that occur for some even flux
denominators, where individual-band Chern numbers become ill-defined.

.. GENERATED FROM PYTHON SOURCE LINES 34-59

.. code-block:: Python


    p, q = 1, 3
    k2_grid = np.linspace(0, 2 * np.pi, 200)
    bands = np.array([np.linalg.eigvalsh(harper_hofstadter_hamiltonian(0.0, k2, p, q)) for k2 in k2_grid])

    chern_numbers = compute_chern_number(lambda k1, k2: harper_hofstadter_hamiltonian(k1, k2, p, q), grid_size=30)
    sigma_xy = np.cumsum(chern_numbers)
    print(f"flux = {p}/{q}: band Chern numbers = {chern_numbers} (sum = {sum(chern_numbers)})")
    print(f"quantized Hall conductance below each gap (units of e^2/h): {list(sigma_xy[:-1])}")

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(11, 4.3))
    for n in range(q):
        ax1.plot(k2_grid, bands[:, n], color="steelblue")
    ax1.set_xlabel(r"$k_2$")
    ax1.set_ylabel("E")
    ax1.set_title(f"Hofstadter sub-bands at flux {p}/{q} (k1 = 0)")

    ax2.bar(range(1, q + 1), sigma_xy, color="firebrick")
    ax2.axhline(0, color="black", lw=0.5)
    ax2.set_xticks(range(1, q + 1))
    ax2.set_xlabel("bands filled")
    ax2.set_ylabel(r"$\sigma_{xy}$  ($e^2/h$)")
    ax2.set_title("Quantized Hall conductance below each gap")
    fig.tight_layout()




.. image-sg:: /api/gallery/condensed/topology/images/sphx_glr_plot_iqhe_hall_conductance_001.png
   :alt: Hofstadter sub-bands at flux 1/3 (k1 = 0), Quantized Hall conductance below each gap
   :srcset: /api/gallery/condensed/topology/images/sphx_glr_plot_iqhe_hall_conductance_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    flux = 1/3: band Chern numbers = [-1, 2, -1] (sum = 0)
    quantized Hall conductance below each gap (units of e^2/h): [np.int64(-1), np.int64(1)]




.. GENERATED FROM PYTHON SOURCE LINES 60-65

Every gap's conductance is an exact integer, and the three bands' Chern
numbers sum to exactly zero -- the full three-band Hilbert space, taken
together, is topologically trivial, as it must be for any complete set of
bands of a lattice Hamiltonian; the nontrivial physics is entirely in how
that zero splits across the individual gaps.

.. GENERATED FROM PYTHON SOURCE LINES 65-67

.. code-block:: Python


    plt.show()








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

   **Total running time of the script:** (0 minutes 0.063 seconds)


.. _sphx_glr_download_api_gallery_condensed_topology_plot_iqhe_hall_conductance.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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