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

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

.. _sphx_glr_api_gallery_topology_plot_zak_phase.py:


The Zak Phase: Where the SSH Chain's Electrons Sit
=======================================================

J. Zak showed in 1989 that the Berry phase acquired by a Bloch state
carried once across the Brillouin zone,

.. math::

    \gamma = \oint \mathbf{A}\cdot d\mathbf{k}\, ,\qquad
    \mathbf{A} = i\langle u_k|\partial_k u_k\rangle\, ,

is a property of the whole band -- the *Zak phase* -- and that
:math:`\gamma/2\pi` is the centre of the band's Wannier functions in
units of the lattice constant: the electronic polarization. With
inversion symmetry it is quantized to 0 or :math:`\pi`.

In the SSH chain (sites at :math:`x = 0` and :math:`x = 1/2`, intracell
hopping :math:`v`, intercell :math:`w`), the occupied band's Wannier
function sits on the stronger bond: at the intracell bond centre
:math:`x = 1/4` when :math:`v > w`, at the intercell one :math:`x = 3/4`
when :math:`w > v`. The two phases differ by :math:`\pi` -- half an
electron per cell moved by half a cell -- and the jump is what leaves
the end charges and zero modes of an open topological chain.
:meth:`~tbkit.kspace.KSpace.berry_phase` and
:meth:`~tbkit.kspace.KSpace.wannier_centers` compute both, from the
gauge-invariant discrete Wilson loop.

.. GENERATED FROM PYTHON SOURCE LINES 29-47

.. code-block:: Python

    import numpy as np
    import matplotlib.pyplot as plt

    from tbkit.lattice import Lattice
    from tbkit.bridges import finite_system
    from tbkit.kspace import KSpace
    from tbkit.plot import Plot


    def ssh(v, w):
        lat = Lattice(unit_cell=[{'tag': 'a', 'r0': (0., 0.)}, {'tag': 'b', 'r0': (0.5, 0.)}],
                             prim_vec=[(1., 0.)])
        chain = KSpace(lat)
        chain.set_hopping([{'i': 0, 'j': 1, 'R': (0,), 't': v},
                                    {'i': 1, 'j': 0, 'R': (1,), 't': w}])
        return chain









.. GENERATED FROM PYTHON SOURCE LINES 48-54

The lattice
-----------
Eight cells of the SSH chain (:func:`~tbkit.bridges.finite_system`),
orbitals a and b in two colours, bond widths proportional to the
hoppings: :math:`v` inside each cell, :math:`w` between cells, for
:math:`v > w` (top) and :math:`v < w` (bottom).

.. GENERATED FROM PYTHON SOURCE LINES 54-59

.. code-block:: Python


    fig_lat, axes_lat = plt.subplots(2, 1, figsize=(7, 2.2))
    Plot(finite_system(ssh(1., 0.4), 8)).lattice(plt_hop=True, ms=7, c=4., ax=axes_lat[0])
    Plot(finite_system(ssh(0.4, 1.), 8)).lattice(plt_hop=True, ms=7, c=4., ax=axes_lat[1])




.. image-sg:: /api/gallery/topology/images/sphx_glr_plot_zak_phase_001.png
   :alt: plot zak phase
   :srcset: /api/gallery/topology/images/sphx_glr_plot_zak_phase_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <Figure size 700x220 with 2 Axes>



.. GENERATED FROM PYTHON SOURCE LINES 60-62

The Wannier centre sits on the strong bond
-------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 62-68

.. code-block:: Python


    for v, w, centre in [(1., 0.4, 0.25), (0.4, 1., 0.75)]:
        x = ssh(v, w).wannier_centers(0, nk=200)[0]
        print('v = {}, w = {}: Wannier centre x = {:.6f}'.format(v, w, x))
        assert np.isclose(x, centre, atol=1e-8)





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

 .. code-block:: none

    v = 1.0, w = 0.4: Wannier centre x = 0.250000
    v = 0.4, w = 1.0: Wannier centre x = 0.750000




.. GENERATED FROM PYTHON SOURCE LINES 69-73

Across the transition: a jump of pi
-------------------------------------
The Zak phase is pinned while the gap stays open, and jumps by
:math:`\pi` where it closes, at :math:`v = w`.

.. GENERATED FROM PYTHON SOURCE LINES 73-90

.. code-block:: Python


    ratios = np.linspace(0.2, 2., 37)
    zak = np.array([ssh(r, 1.).berry_phase(0, nk=200) for r in ratios])
    centres = (zak / (2*np.pi)) % 1.
    topo = ratios < 1.
    assert np.allclose(centres[topo], 0.75, atol=1e-8)
    assert np.allclose(centres[~topo & ~np.isclose(ratios, 1.)], 0.25, atol=1e-8)
    print('Wannier centre 3/4 for v < w, 1/4 for v > w: a jump of half a cell (Zak phase pi).')

    fig, ax = plt.subplots(figsize=(6, 3.6))
    ax.plot(ratios, centres, 'o-b')
    ax.axvline(1., color='k', ls='--', lw=1)
    ax.set_xlabel('$v/w$')
    ax.set_ylabel(r'Wannier centre $\gamma/2\pi$')
    ax.set_title('Zak phase of the SSH chain')
    fig.set_layout_engine('tight')




.. image-sg:: /api/gallery/topology/images/sphx_glr_plot_zak_phase_002.png
   :alt: Zak phase of the SSH chain
   :srcset: /api/gallery/topology/images/sphx_glr_plot_zak_phase_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Wannier centre 3/4 for v < w, 1/4 for v > w: a jump of half a cell (Zak phase pi).




.. GENERATED FROM PYTHON SOURCE LINES 92-96

The polarization is a property of the band, not of the gauge
-------------------------------------------------------------------
Placing both orbitals on the cell origin (``positions=False``) shifts
each phase, but not the quantized difference between the two phases.

.. GENERATED FROM PYTHON SOURCE LINES 96-101

.. code-block:: Python


    diff = (ssh(0.4, 1.).berry_phase(0, positions=False)
                - ssh(1., 0.4).berry_phase(0, positions=False)) % (2*np.pi)
    assert np.isclose(diff, np.pi)
    print('Zak-phase difference in the periodic gauge: {:.6f} = pi.'.format(diff))




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

 .. code-block:: none

    Zak-phase difference in the periodic gauge: 3.141593 = pi.





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

   **Total running time of the script:** (0 minutes 0.077 seconds)


.. _sphx_glr_download_api_gallery_topology_plot_zak_phase.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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