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

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

.. _sphx_glr_api_gallery_condensed_topology_plot_graphene_dirac_cone.py:


Graphene: A Massless Dirac Cone on a Honeycomb Lattice
================================================================

Geim and Novoselov's isolation of a single atomic layer of graphite
produced the first genuinely 2D crystal -- and its low-energy
quasiparticles obey a massless relativistic (Dirac) equation rather than
the ordinary Schrodinger equation, turning a tabletop material into a
laboratory for relativistic quantum phenomena.
:func:`~physicskit.condensed.models.graphene_hamiltonian` reproduces this
linear dispersion near the honeycomb lattice's Brillouin zone corners.

.. GENERATED FROM PYTHON SOURCE LINES 13-19

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.condensed.models import graphene_hamiltonian








.. GENERATED FROM PYTHON SOURCE LINES 20-26

Sampling H(k) through the Dirac point
--------------------------------------------
The two inequivalent Brillouin-zone corners :math:`K=(2\pi/3, 4\pi/3)`
and :math:`K'` are where the honeycomb lattice's two sublattice bands
touch. Scanning a line of momenta through :math:`K` shows the gap close
exactly at the center and the bands disperse linearly on either side.

.. GENERATED FROM PYTHON SOURCE LINES 26-43

.. code-block:: Python


    K = np.array([2 * np.pi / 3, 4 * np.pi / 3])
    q = np.linspace(-0.3, 0.3, 201)
    bands = np.array([np.linalg.eigvalsh(graphene_hamiltonian(*(K + [dq, 0]))) for dq in q])

    fig, ax = plt.subplots(figsize=(6, 4))
    ax.plot(q, bands, color="C0")
    ax.axvline(0, color="gray", ls="--")
    ax.set_xlabel(r"$q$ (distance from Dirac point $K$)")
    ax.set_ylabel("Energy")
    ax.set_title(r"Massless Dirac cone: $E(q) \sim \pm v_F|q|$")
    fig.tight_layout()

    gap_at_K = bands[len(q) // 2, 1] - bands[len(q) // 2, 0]
    slope = (bands[-1, 1] - bands[len(q) // 2, 1]) / (q[-1] - q[len(q) // 2])
    print(f"gap exactly at K: {gap_at_K:.2e} (closes)")
    print(f"local slope dE/dq away from K: {slope:.4f} (linear, not quadratic)")



.. image-sg:: /api/gallery/condensed/topology/images/sphx_glr_plot_graphene_dirac_cone_001.png
   :alt: Massless Dirac cone: $E(q) \sim \pm v_F|q|$
   :srcset: /api/gallery/condensed/topology/images/sphx_glr_plot_graphene_dirac_cone_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    gap exactly at K: 8.01e-16 (closes)
    local slope dE/dq away from K: 0.9963 (linear, not quadratic)





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

   **Total running time of the script:** (0 minutes 0.033 seconds)


.. _sphx_glr_download_api_gallery_condensed_topology_plot_graphene_dirac_cone.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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