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

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

.. _sphx_glr_api_gallery_condensed_topology_plot_haldane_edge_state_map.py:


The Haldane Model: a Chiral Edge State on an Arbitrary Boundary
========================================================================

:doc:`plot_haldane_phase_transition` exposes the Haldane model's chiral
edge states on a ribbon -- periodic in-plane, open along one
crystallographic direction. That is the easiest cut to compute, but it
leaves open whether the edge state depends on cutting *along* a lattice
direction the way graphene's zigzag zero modes do
(:doc:`plot_graphene_zigzag_flake`). It does not: a nonzero Chern number
guarantees a chiral state on *any* boundary of the sample, however it is
shaped. This example carves a circular disk out of the infinite lattice --
a boundary with no crystallographic meaning at all -- using
:func:`~physicskit.condensed.tight_binding.build_finite_cluster`'s ``keep``
predicate, and shows the in-gap states still hug that boundary in the
topological phase, and fail to in the trivial one.

.. GENERATED FROM PYTHON SOURCE LINES 18-26

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.condensed.models import haldane_lattice_hamiltonian
    from physicskit.condensed.tight_binding import Lattice, build_finite_cluster
    from physicskit.condensed.visualizers import plot_lattice_structure








.. GENERATED FROM PYTHON SOURCE LINES 27-33

Carving a disk out of the honeycomb lattice
----------------------------------------------
``build_finite_cluster``'s bounding box is a parallelogram of unit cells;
the ``keep`` predicate then discards every site farther than ``radius``
from the box's center, leaving a disk with an irregular, non-crystalline
edge.

.. GENERATED FROM PYTHON SOURCE LINES 33-43

.. code-block:: Python


    n_cells, radius = 14, 6.0
    lat = Lattice.honeycomb()
    center = np.array([n_cells / 2, n_cells / 2]) @ lat.lattice_vectors


    def in_disk(cell, orbital, position):
        return np.linalg.norm(position - center) <= radius









.. GENERATED FROM PYTHON SOURCE LINES 44-51

Topological vs. trivial: does an in-gap state hug this boundary?
----------------------------------------------------------------------
The Haldane model is a Chern insulator for :math:`|M| < 3\sqrt3\,t_2|\sin\phi|`
and trivial otherwise. In the topological phase, the handful of states
nearest mid-gap should sit at a density-weighted mean radius close to the
disk's edge; in the trivial phase, the states nearest mid-gap are just
ordinary near-degenerate bulk states with no boundary preference.

.. GENERATED FROM PYTHON SOURCE LINES 51-73

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(11, 5))

    for ax, (label, M) in zip(axes, [("topological (M=0)", 0.0), ("trivial (M=2.0)", 2.0)], strict=True):
        H_bulk = haldane_lattice_hamiltonian(t=1.0, t2=0.2, phi=np.pi / 2, M=M)
        H, positions, bonds = build_finite_cluster(H_bulk, n_cells=(n_cells, n_cells), keep=in_disk)
        eigenvalues, eigenvectors = np.linalg.eigh(H)

        mid = len(eigenvalues) // 2
        in_gap = slice(mid - 3, mid + 3)
        density = np.sum(np.abs(eigenvectors[:, in_gap]) ** 2, axis=1)

        r = np.linalg.norm(positions - center, axis=1)
        mean_r_sites = r.mean()
        mean_r_weighted = np.sum(r * density) / density.sum()
        print(f"{label}: mean site radius = {mean_r_sites:.2f}, density-weighted radius of near-gap states = {mean_r_weighted:.2f}  (disk radius = {radius:.1f})")

        plot_lattice_structure(positions, bonds, weights=density, ax=ax)
        ax.set_title(f"{label}\nnear-gap density, E~[{eigenvalues[in_gap][0]:.2f}, {eigenvalues[in_gap][-1]:.2f}]")

    fig.suptitle("Haldane model: a chiral edge state needs a Chern number, not a special edge")
    fig.tight_layout()



.. image-sg:: /api/gallery/condensed/topology/images/sphx_glr_plot_haldane_edge_state_map_001.png
   :alt: Haldane model: a chiral edge state needs a Chern number, not a special edge, topological (M=0) near-gap density, E~[-0.39, 0.39], trivial (M=2.0) near-gap density, E~[-1.15, 1.15]
   :srcset: /api/gallery/condensed/topology/images/sphx_glr_plot_haldane_edge_state_map_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    topological (M=0): mean site radius = 3.98, density-weighted radius of near-gap states = 5.55  (disk radius = 6.0)
    trivial (M=2.0): mean site radius = 3.98, density-weighted radius of near-gap states = 3.43  (disk radius = 6.0)





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

   **Total running time of the script:** (0 minutes 0.365 seconds)


.. _sphx_glr_download_api_gallery_condensed_topology_plot_haldane_edge_state_map.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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