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

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

.. _sphx_glr_api_gallery_condensed_topology_plot_ssh_edge_states.py:


The SSH Model: Zak Phase and Protected Edge States
========================================================

Su, Schrieffer, and Heeger showed that a dimerized chain -- alternating
intracell hopping :math:`v` and intercell hopping :math:`w` -- is
topologically nontrivial whenever :math:`v < w`: the bulk Zak phase is
quantized to :math:`\pi`, and cutting the chain open exposes a pair of
protected, exponentially localized zero-energy states, one at each end.
This is the earliest example of the bulk-boundary correspondence.
:func:`~physicskit.condensed.models.ssh_hamiltonian` gives the closed-form
Bloch Hamiltonian, :func:`~physicskit.condensed.topology.zak_phase`
computes the bulk invariant from it, and
:func:`~physicskit.condensed.models.ssh_lattice_hamiltonian` together with
:func:`~physicskit.condensed.tight_binding.build_finite_cluster` exposes the
boundary modes on a finite, open chain -- and lets us plot the chain's
actual real-space structure rather than an abstract site-index axis.

.. GENERATED FROM PYTHON SOURCE LINES 19-28

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.condensed.models import ssh_hamiltonian, ssh_lattice_hamiltonian
    from physicskit.condensed.tight_binding import build_finite_cluster
    from physicskit.condensed.topology import zak_phase
    from physicskit.condensed.visualizers import plot_lattice_structure








.. GENERATED FROM PYTHON SOURCE LINES 29-34

The bulk invariant: Zak phase in the two dimerization regimes
-------------------------------------------------------------------
The Zak phase is quantized by chiral symmetry to either :math:`0`
(trivial, :math:`v>w`) or :math:`\pi` (topological, :math:`v<w`) -- with
nothing in between, since the bulk gap only closes at :math:`v=w`.

.. GENERATED FROM PYTHON SOURCE LINES 34-43

.. code-block:: Python


    topological = lambda k: ssh_hamiltonian(k, v=0.5, w=1.0)
    trivial = lambda k: ssh_hamiltonian(k, v=1.0, w=0.5)

    zak_topological = zak_phase(topological)
    zak_trivial = zak_phase(trivial)
    print(f"Zak phase, v<w (topological): {zak_topological:.4f}  (expect +-pi)")
    print(f"Zak phase, v>w (trivial):     {zak_trivial:.4f}  (expect 0)")





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

 .. code-block:: none

    Zak phase, v<w (topological): -3.1416  (expect +-pi)
    Zak phase, v>w (trivial):     -0.0000  (expect 0)




.. GENERATED FROM PYTHON SOURCE LINES 44-61

Bulk-boundary correspondence: the finite chain's real-space structure
------------------------------------------------------------------------
The Zak phase is a purely bulk (periodic-boundary) quantity. Cutting the
topological chain (:math:`v<w`) into a finite, open wire produces a pair
of near-degenerate mid-gap states pinned to zero energy -- exactly the
correspondence the Zak phase predicts. Because the two are numerically
degenerate, :func:`numpy.linalg.eigh` returns an arbitrary basis of that
two-dimensional subspace rather than the individual left- and
right-localized modes; summing both states' densities recovers the
physical picture, exponentially localized at *both* ends and vanishing on
every other (B-sublattice) site.

Drawing the chain's real structure -- rather than density against a bare
site index -- also makes the mechanism visible directly: bond linewidth
below is drawn proportional to hopping strength, so the weak :math:`v`
bond exposed at each open end is exactly what leaves those end sites
undercoordinated and hosting the protected zero mode.

.. GENERATED FROM PYTHON SOURCE LINES 61-86

.. code-block:: Python


    v, w, n_cells = 0.5, 1.0, 15
    H, positions, bonds = build_finite_cluster(ssh_lattice_hamiltonian(v=v, w=w), n_cells=n_cells)

    spectrum, states = np.linalg.eigh(H)
    edge_idx = np.argsort(np.abs(spectrum))[:2]
    density = np.abs(states[:, edge_idx[0]]) ** 2 + np.abs(states[:, edge_idx[1]]) ** 2

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(11, 4.5))

    ax1.bar(["trivial (v>w)", "topological (v<w)"], [abs(zak_trivial), abs(zak_topological)], color=["C0", "C1"])
    ax1.axhline(np.pi, color="gray", ls="--", label=r"$\pi$")
    ax1.set_ylabel("|Zak phase|")
    ax1.set_title("Bulk invariant")
    ax1.legend()

    plot_lattice_structure(positions, bonds, weights=density, ax=ax2)
    ax2.set_title(f"Zero-mode density, |E| = {abs(spectrum[edge_idx[0]]):.2e}")
    ax2.set_ylim(-0.5, 0.5)

    fig.suptitle("SSH model: Zak phase pi <=> protected zero-energy edge states")
    fig.tight_layout()

    left_weight, right_weight = density[: len(density) // 2].sum(), density[len(density) // 2 :].sum()
    print(f"zero-mode weight -- left half: {left_weight:.3f}, right half: {right_weight:.3f}")



.. image-sg:: /api/gallery/condensed/topology/images/sphx_glr_plot_ssh_edge_states_001.png
   :alt: SSH model: Zak phase pi <=> protected zero-energy edge states, Bulk invariant, Zero-mode density, |E| = 2.29e-05
   :srcset: /api/gallery/condensed/topology/images/sphx_glr_plot_ssh_edge_states_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    zero-mode weight -- left half: 1.000, right half: 1.000





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

   **Total running time of the script:** (0 minutes 0.068 seconds)


.. _sphx_glr_download_api_gallery_condensed_topology_plot_ssh_edge_states.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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