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

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

.. _sphx_glr_api_gallery_non_hermitian_plot_skin_effect.py:


The Non-Hermitian Skin Effect
===================================

Hatano and Nelson (1996) studied a chain whose hoppings to the right
and to the left differ, :math:`(H\psi)_n = t_R\psi_{n+1} + t_L\psi_{n-1}`
-- a non-reciprocal, non-Hermitian lattice. Twenty years later it became
the paradigm of a phenomenon with no Hermitian counterpart, the *skin
effect* (Yao and Wang; Kunst, Edvardsson, Budich and Bergholtz, 2018):

* on a ring, the spectrum :math:`E(k) = t_Re^{ik} + t_Le^{-ik}` is an
  ellipse, winding once around every energy inside it (spectral winding
  number :math:`W = \pm1`);
* on an open chain it collapses onto a real segment,
  :math:`|E| < 2\sqrt{t_Rt_L}`, and *every* eigenstate piles up at one
  end -- the bulk becomes a skin.

:meth:`~tbkit.kspace.KSpace.spectral_winding` and
:meth:`~tbkit.kspace.KSpace.finite_ham` compute both. The open-chain
bands live on a generalized Brillouin zone, see
``plot_generalized_brillouin_zone.py``.

.. GENERATED FROM PYTHON SOURCE LINES 23-36

.. code-block:: Python

    import numpy as np
    import matplotlib.pyplot as plt

    import tbkit.lattices as lattices
    from tbkit.bridges import finite_system
    from tbkit.kspace import KSpace
    from tbkit.plot import Plot

    t_right, t_left = 1., 0.5
    hn = KSpace(lattices.chain())
    hn.set_hopping([{'i': 0, 'j': 0, 'R': (1,), 't': t_right},
                            {'i': 0, 'j': 0, 'R': (-1,), 't': t_left}], hermitian=False)








.. GENERATED FROM PYTHON SOURCE LINES 37-44

The lattice
-----------
Ten sites of the Hatano-Nelson chain
(:func:`~tbkit.bridges.finite_system`): one site per cell and
nearest-neighbour bonds, non-reciprocal -- each carries :math:`t_R` one
way and :math:`t_L` the other. The widths drawn are those of
:math:`t_R`.

.. GENERATED FROM PYTHON SOURCE LINES 44-47

.. code-block:: Python


    fig_lat = Plot(finite_system(hn, 10)).lattice(plt_hop=True, ms=12, figsize=(7, 1.2))




.. image-sg:: /api/gallery/non_hermitian/images/sphx_glr_plot_skin_effect_001.png
   :alt: plot skin effect
   :srcset: /api/gallery/non_hermitian/images/sphx_glr_plot_skin_effect_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 48-50

Ring and open chain
-----------------------

.. GENERATED FROM PYTHON SOURCE LINES 50-82

.. code-block:: Python


    n = 60
    ring = np.linalg.eigvals(hn.finite_ham(n, periodic=True))
    en, vec = np.linalg.eig(hn.finite_ham(n))
    # compared as sets: E(k = pi) is real, and its angle (+pi or -pi) depends on
    # the sign of a roundoff imaginary part
    expected = [t_right*np.exp(1j*k) + t_left*np.exp(-1j*k) for k in 2*np.pi*np.arange(n)/n]
    assert np.allclose(np.sort_complex(np.round(ring, 8)), np.sort_complex(np.round(expected, 8)))
    assert np.max(np.abs(en.imag)) < 1e-8 and np.max(np.abs(en.real)) < 2*np.sqrt(t_right*t_left)
    winding = hn.spectral_winding(0.)
    print('Spectral winding about E = 0: {:.3f}; open-chain spectrum real, within +-{:.3f}.'
              .format(winding, 2*np.sqrt(t_right*t_left)))
    assert np.isclose(winding, 1.)

    # every eigenstate on the left end
    weight = np.abs(vec) ** 2 / np.sum(np.abs(vec) ** 2, axis=0)
    left = weight[:n // 4].sum(axis=0)
    print('Weight on the leftmost quarter: at least {:.0%} for every eigenstate.'.format(left.min()))
    assert left.min() > 0.9

    fig, axes = plt.subplots(1, 2, figsize=(10, 4))
    axes[0].plot(ring.real, ring.imag, 'ob', ms=3, label='ring')
    axes[0].plot(en.real, en.imag, 'or', ms=3, label='open chain')
    axes[0].set_xlabel(r'Re $E$')
    axes[0].set_ylabel(r'Im $E$')
    axes[0].legend()
    axes[0].set_aspect('equal')
    axes[1].semilogy(np.arange(n), weight[:, np.argsort(en.real)[::10]], lw=1)
    axes[1].set_xlabel('site')
    axes[1].set_ylabel(r'$|\psi_n|^2$')
    axes[1].set_title('open chain: every state on the left end')
    fig.set_layout_engine('tight')



.. image-sg:: /api/gallery/non_hermitian/images/sphx_glr_plot_skin_effect_002.png
   :alt: open chain: every state on the left end
   :srcset: /api/gallery/non_hermitian/images/sphx_glr_plot_skin_effect_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Spectral winding about E = 0: 1.000; open-chain spectrum real, within +-1.414.
    Weight on the leftmost quarter: at least 100% for every eigenstate.





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

   **Total running time of the script:** (0 minutes 0.088 seconds)


.. _sphx_glr_download_api_gallery_non_hermitian_plot_skin_effect.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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