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

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

.. _sphx_glr_api_gallery_magnetic_field_plot_hofstadter_butterfly.py:


Hofstadter's Butterfly
===========================

Sweeping the flux per plaquette continuously (rather than fixing it, as
the Aharonov-Bohm ring example does) reveals a self-similar, fractal
spectrum: at rational flux p/q, each Landau-level-like band splits into q
sub-bands, and the whole pattern is periodic in the flux with period one
flux quantum. This script sweeps
:meth:`~tbkit.system.System.set_magnetic_field` over a finite square-lattice
flake and plots the resulting "butterfly".

.. GENERATED FROM PYTHON SOURCE LINES 13-36

.. code-block:: Python

    import numpy as np
    import matplotlib.pyplot as plt

    from tbkit.lattice import Lattice
    from tbkit.system import System
    from tbkit.plot import Plot


    N1, N2 = 16, 16  # flake size


    def flake(alpha):
        lat = Lattice(unit_cell=[{'tag': 'a', 'r0': (0., 0.)}], prim_vec=[(1., 0.), (0., 1.)])
        lat.get_lattice(n1=N1, n2=N2)
        sys = System(lat)
        sys.set_hopping([{'n': 1, 't': 1.}])
        if alpha != 0.:
            sys.set_magnetic_field(alpha=alpha)
        sys.get_ham()
        sys.get_eig()
        return sys









.. GENERATED FROM PYTHON SOURCE LINES 37-42

The flake
------------
A square-lattice flake. The flux swept below threads each of its square
plaquettes; the Peierls substitution multiplies the bonds drawn here by
a phase, leaving their magnitude (and so this picture) unchanged.

.. GENERATED FROM PYTHON SOURCE LINES 42-45

.. code-block:: Python


    fig0 = Plot(flake(alpha=0.)).lattice(plt_hop=True, ms=7, figsize=(5, 5))




.. image-sg:: /api/gallery/magnetic_field/images/sphx_glr_plot_hofstadter_butterfly_001.png
   :alt: plot hofstadter butterfly
   :srcset: /api/gallery/magnetic_field/images/sphx_glr_plot_hofstadter_butterfly_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 46-59

Sanity checks that hold at every flux
--------------------------------------------
These hold regardless of the (fractal, hard to predict in detail)
spectrum shape itself:

1. :math:`|E|\le 4|t|` always (Gershgorin: at most 4 neighbors of hopping
   magnitude :math:`|t|=1`, and the Peierls substitution changes their
   phase, never their magnitude).
2. The spectrum is symmetric under :math:`E \to -E` (the square lattice
   is bipartite, and that chiral symmetry survives the Peierls
   substitution since it only ever multiplies hoppings by a phase).
3. The spectrum is periodic in :math:`\alpha` with period 1 (one flux
   quantum per plaquette is gauge-equivalent to no flux at all).

.. GENERATED FROM PYTHON SOURCE LINES 59-75

.. code-block:: Python


    alphas = np.linspace(0., 1., 161)
    spectra = [flake(alpha).en.real for alpha in alphas]

    max_abs_e = max(np.max(np.abs(en)) for en in spectra)
    print('max|E| over the whole sweep: {:.4f} (Gershgorin bound: 4.0000)'.format(max_abs_e))
    assert max_abs_e <= 4. + 1e-8

    sym_ok = all(np.allclose(np.sort(en), -np.sort(en)[::-1], atol=1e-6) for en in spectra)
    print('Particle-hole symmetry E -> -E holds at every flux: {}'.format(sym_ok))
    assert sym_ok

    periodic_ok = np.allclose(np.sort(spectra[0]), np.sort(spectra[-1]), atol=1e-6)
    print('Spectrum at alpha=0 matches alpha=1 (period-1 flux quantum): {}'.format(periodic_ok))
    assert periodic_ok





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

 .. code-block:: none

    max|E| over the whole sweep: 3.9319 (Gershgorin bound: 4.0000)
    Particle-hole symmetry E -> -E holds at every flux: True
    Spectrum at alpha=0 matches alpha=1 (period-1 flux quantum): True




.. GENERATED FROM PYTHON SOURCE LINES 76-78

The butterfly itself
---------------------------

.. GENERATED FROM PYTHON SOURCE LINES 78-85

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(6, 7))
    for alpha, en in zip(alphas, spectra):
        ax.plot(alpha*np.ones_like(en), en, '.', ms=0.5, color='b')
    ax.set_xlabel(r'flux per plaquette $\alpha$ (in units of $\Phi_0$)')
    ax.set_ylabel('$E$')
    ax.set_title("Hofstadter's butterfly ({}x{} flake)".format(N1, N2))



.. image-sg:: /api/gallery/magnetic_field/images/sphx_glr_plot_hofstadter_butterfly_002.png
   :alt: Hofstadter's butterfly (16x16 flake)
   :srcset: /api/gallery/magnetic_field/images/sphx_glr_plot_hofstadter_butterfly_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, "Hofstadter's butterfly (16x16 flake)")




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

   **Total running time of the script:** (0 minutes 0.937 seconds)


.. _sphx_glr_download_api_gallery_magnetic_field_plot_hofstadter_butterfly.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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