.. 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_phase_transition.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_phase_transition.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_condensed_topology_plot_haldane_phase_transition.py:


The Haldane Model: A Chern Insulator with Zero Net Flux
==============================================================

Duncan Haldane showed that the integer quantum Hall effect does not
require a net magnetic field at all. Threading complex second-neighbor
hopping :math:`t_2 e^{i\phi}` through a honeycomb lattice, in a pattern
with zero *net* flux per unit cell but nonzero local (loop) curvature,
produces a Chern insulator with :math:`C=\pm1` -- the first quantum
anomalous Hall model, and the direct template for the
:math:`\mathbb{Z}_2` topological insulators discovered two decades later.
:func:`~physicskit.condensed.models.haldane_model` gives the Bloch
Hamiltonian used to track the bulk Chern number across the phase
transition, and :func:`~physicskit.condensed.models.haldane_lattice_hamiltonian`
builds the same model in real space to expose the chiral edge states that
bulk-boundary correspondence predicts.

.. 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, haldane_model
    from physicskit.condensed.tight_binding import build_ribbon
    from physicskit.condensed.topology import compute_chern_number








.. GENERATED FROM PYTHON SOURCE LINES 27-34

Crossing the topological phase transition
------------------------------------------------
Scanning the sublattice (Semenoff) mass :math:`M` across the critical
value :math:`M_c = 3\sqrt{3}\,t_2|\sin\phi|` drives the lower band's
Chern number from :math:`1` to :math:`0` exactly where the bulk gap
closes -- a topological invariant can only change value at a gap-closing
transition.

.. GENERATED FROM PYTHON SOURCE LINES 34-40

.. code-block:: Python


    t2, phi = 0.2, np.pi / 2
    M_c = 3 * np.sqrt(3) * t2 * abs(np.sin(phi))
    M_values = np.linspace(0, 2 * M_c, 15)
    chern = [compute_chern_number(lambda k1, k2: haldane_model(k1, k2, t2=t2, phi=phi, M=M), grid_size=24)[0] for M in M_values]








.. GENERATED FROM PYTHON SOURCE LINES 41-48

Bulk-boundary correspondence: chiral edge states in a ribbon
--------------------------------------------------------------------
A nonzero Chern number in the bulk (``M=0``) forces gapless states to
cross the bulk gap on any boundary. Cutting the honeycomb lattice into a
finite-width, open ribbon with :func:`build_ribbon` and plotting its
spectrum vs. the remaining periodic momentum shows exactly this: bands
that thread the bulk gap rather than staying confined to the bulk bands.

.. GENERATED FROM PYTHON SOURCE LINES 48-74

.. code-block:: Python


    H_bulk = haldane_lattice_hamiltonian(t=1.0, t2=t2, phi=phi, M=0.0)
    H_ribbon = build_ribbon(H_bulk, open_direction=1, n_cells=20)
    k_parallel = np.linspace(0, 2 * np.pi, 200)
    ribbon_bands = np.array([np.linalg.eigvalsh(H_ribbon([k])) for k in k_parallel])

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(11, 4))

    ax1.plot(M_values / M_c, chern, "o-")
    ax1.axvline(1.0, color="gray", ls="--", label=r"$M_c=3\sqrt{3}\,t_2|\sin\phi|$")
    ax1.set_xlabel(r"$M / M_c$")
    ax1.set_ylabel("Chern number (lower band)")
    ax1.set_title("Topological phase transition")
    ax1.legend()

    ax2.plot(k_parallel, ribbon_bands, color="C0", lw=0.8)
    ax2.set_xlabel(r"$k_\parallel$")
    ax2.set_ylabel("Energy")
    ax2.set_title("Ribbon spectrum: chiral edge states cross the bulk gap")

    fig.suptitle("Haldane model: Chern insulator from zero net flux")
    fig.tight_layout()

    min_over_k = np.min(np.abs(ribbon_bands), axis=1)
    print(f"bulk gap (away from edge crossings): ~{np.median(min_over_k[min_over_k > 0.5]):.3f}")
    print(f"minimum |E| reached by an edge-crossing state: {min_over_k.min():.4f}")



.. image-sg:: /api/gallery/condensed/topology/images/sphx_glr_plot_haldane_phase_transition_001.png
   :alt: Haldane model: Chern insulator from zero net flux, Topological phase transition, Ribbon spectrum: chiral edge states cross the bulk gap
   :srcset: /api/gallery/condensed/topology/images/sphx_glr_plot_haldane_phase_transition_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    bulk gap (away from edge crossings): ~1.028
    minimum |E| reached by an edge-crossing state: 0.0148





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

   **Total running time of the script:** (0 minutes 0.291 seconds)


.. _sphx_glr_download_api_gallery_condensed_topology_plot_haldane_phase_transition.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_phase_transition.ipynb <plot_haldane_phase_transition.ipynb>`

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

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

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

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


.. only:: html

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

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