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

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

.. _sphx_glr_api_gallery_condensed_tight_binding_plot_bloch_bands.py:


Bloch's Theorem: From an Infinite Chain to H(k)
====================================================

Felix Bloch's theorem turns the Schrodinger equation for an electron in an
infinite periodic crystal into a finite eigenvalue problem :math:`H(k)
u_{nk} = E_n(k) u_{nk}` at each crystal momentum :math:`k`, periodic on the
Brillouin zone. :class:`~physicskit.condensed.tight_binding.Hamiltonian`
builds :math:`H(k)` by Bloch-summing real-space hoppings, and
:meth:`~physicskit.condensed.tight_binding.Hamiltonian.bands` diagonalizes
it -- reproducing the monatomic chain's exact dispersion
:math:`E(k) = -2t\cos k` with no continuum limit or approximation involved.

.. GENERATED FROM PYTHON SOURCE LINES 14-21

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.condensed.tight_binding import Hamiltonian, Lattice, build_finite_cluster
    from physicskit.condensed.visualizers import plot_lattice_structure








.. GENERATED FROM PYTHON SOURCE LINES 22-26

A monatomic chain with a single real-space hopping
------------------------------------------------------
One orbital per unit cell, nearest-neighbor hopping ``t=1`` -- the
simplest possible crystal.

.. GENERATED FROM PYTHON SOURCE LINES 26-31

.. code-block:: Python


    lat = Lattice.chain(a=1.0)
    H = Hamiltonian(lat, onsite=[0.0])
    H.add_hopping(0, 0, (1,), -1.0)








.. GENERATED FROM PYTHON SOURCE LINES 32-39

Bloch-summing and diagonalizing at every k in the Brillouin zone
---------------------------------------------------------------------
:meth:`Hamiltonian.bloch` assembles :math:`H(k) = -t(e^{ik} + e^{-ik}) =
-2t\cos k` from the single real-space hopping added above;
:meth:`Hamiltonian.bands` diagonalizes it. The result matches the
textbook dispersion to machine precision -- there is no approximation
between the real-space model and the Bloch-summed band.

.. GENERATED FROM PYTHON SOURCE LINES 39-43

.. code-block:: Python


    k_grid = np.linspace(0, 2 * np.pi, 200)
    bands = np.array([H.bands([k]) for k in k_grid])








.. GENERATED FROM PYTHON SOURCE LINES 44-51

The real-space structure Bloch's theorem is summing over
---------------------------------------------------------------------
:math:`H(k)` above is nothing but this same finite chain's single
repeated hopping, re-summed with a phase :math:`e^{ikR}` per unit cell
translation :math:`R`. :func:`build_finite_cluster` truncates that
translational symmetry to a finite, open segment so the real-space
structure Bloch's theorem abstracts away can be drawn directly.

.. GENERATED FROM PYTHON SOURCE LINES 51-72

.. code-block:: Python


    H_finite, positions, bonds = build_finite_cluster(H, n_cells=10)

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(11, 4))

    ax1.plot(k_grid, bands[:, 0], lw=2.5, label="H.bands(k)")
    ax1.plot(k_grid, -2 * np.cos(k_grid), "k--", lw=1.5, label=r"$E(k)=-2t\cos k$")
    ax1.set_xlabel("k")
    ax1.set_ylabel("Energy")
    ax1.set_title("Bloch band (reciprocal space)")
    ax1.legend()

    plot_lattice_structure(positions, bonds, ax=ax2)
    ax2.set_ylim(-0.5, 0.5)
    ax2.set_title(f"The same chain, open and finite ({H_finite.shape[0]} sites)")

    fig.suptitle("Tight-binding chain: Bloch sum reproduces the exact dispersion")
    fig.tight_layout()

    max_error = np.max(np.abs(bands[:, 0] - (-2 * np.cos(k_grid))))
    print(f"max deviation from exact dispersion: {max_error:.2e}")



.. image-sg:: /api/gallery/condensed/tight_binding/images/sphx_glr_plot_bloch_bands_001.png
   :alt: Tight-binding chain: Bloch sum reproduces the exact dispersion, Bloch band (reciprocal space), The same chain, open and finite (10 sites)
   :srcset: /api/gallery/condensed/tight_binding/images/sphx_glr_plot_bloch_bands_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    max deviation from exact dispersion: 0.00e+00





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

   **Total running time of the script:** (0 minutes 0.060 seconds)


.. _sphx_glr_download_api_gallery_condensed_tight_binding_plot_bloch_bands.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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