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

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

.. _sphx_glr_api_gallery_condensed_topology_plot_chern_number.py:


The TKNN Invariant: Exactly Quantized Chern Numbers
==========================================================

Thouless, Kohmoto, Nightingale, and den Nijs showed that the quantized
Hall conductance is a topological invariant: the integral of the Berry
curvature of the occupied Bloch bands over the Brillouin zone, now called
the (first) Chern number :math:`C`. Because :math:`C` can only change when
a bulk gap closes, it is exact and disorder-independent -- the moment band
theory became *topological* band theory.
:func:`~physicskit.condensed.topology.compute_berry_curvature` and
:func:`~physicskit.condensed.topology.compute_chern_number` implement the
Fukui-Hatsugai-Suzuki lattice discretization of this integral, returning
exactly quantized integers for any gapped Bloch Hamiltonian.

.. GENERATED FROM PYTHON SOURCE LINES 16-23

.. code-block:: Python


    import numpy as np

    from physicskit.condensed.models import haldane_model
    from physicskit.condensed.topology import compute_berry_curvature, compute_chern_number
    from physicskit.condensed.visualizers import plot_berry_curvature








.. GENERATED FROM PYTHON SOURCE LINES 24-28

Any gapped two-band Bloch Hamiltonian -- here, Haldane's model
--------------------------------------------------------------------
The FHS algorithm needs nothing but a function ``H(k1, k2)``; it makes
no assumption about the model beyond a spectral gap.

.. GENERATED FROM PYTHON SOURCE LINES 28-31

.. code-block:: Python


    H = lambda k1, k2: haldane_model(k1, k2, t=1.0, t2=0.2, phi=np.pi / 2, M=0.0)








.. GENERATED FROM PYTHON SOURCE LINES 32-37

Berry curvature concentrated near the gapped Dirac points
------------------------------------------------------------------
The per-plaquette Berry curvature of the lower band is sharply peaked
near the honeycomb lattice's two (former) Dirac points, where the
time-reversal-breaking mass gap is smallest.

.. GENERATED FROM PYTHON SOURCE LINES 37-42

.. code-block:: Python


    F = compute_berry_curvature(H, grid_size=30, band_index=0)
    fig, ax = plot_berry_curvature(F)
    ax.set_title("Berry curvature of the lower Haldane band")




.. image-sg:: /api/gallery/condensed/topology/images/sphx_glr_plot_chern_number_001.png
   :alt: Berry curvature of the lower Haldane band
   :srcset: /api/gallery/condensed/topology/images/sphx_glr_plot_chern_number_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'Berry curvature of the lower Haldane band')



.. GENERATED FROM PYTHON SOURCE LINES 43-49

Integrating the curvature gives an exact integer
-----------------------------------------------------
Summing the curvature over the whole Brillouin zone and dividing by
:math:`2\pi` returns an *exact* integer for every band -- not
approximately quantized, but exactly, by construction of the FHS link
variables.

.. GENERATED FROM PYTHON SOURCE LINES 49-53

.. code-block:: Python


    C = compute_chern_number(H, grid_size=30)
    print("Chern numbers (per band):", C)
    print(f"raw curvature sum / 2*pi for the lower band: {F.sum() / (2 * np.pi):.10f}")




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

 .. code-block:: none

    Chern numbers (per band): [1, -1]
    raw curvature sum / 2*pi for the lower band: 1.0000000000





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

   **Total running time of the script:** (0 minutes 0.075 seconds)


.. _sphx_glr_download_api_gallery_condensed_topology_plot_chern_number.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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