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

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

.. _sphx_glr_api_gallery_rmt_paper_replications_bdg_classes_demo.py:


Altland-Zirnbauer Classes: BdG Eigenvalue Symmetry
==================================================

Extending Dyson's threefold way (GOE/GUE/GSE) with particle-hole
(charge-conjugation) symmetry -- appropriate to Bogoliubov-de Gennes
mean-field Hamiltonians of disordered/chaotic superconductors -- yields
four more symmetry classes: D, C, CI, DIII. Each is a
:math:`2N \times 2N` Hermitian matrix built from a block structure such
as (class D, :math:`\beta=2`)

.. math::

    H = \begin{pmatrix} A & B \\ B^\dagger & -A^T \end{pmatrix},
    \qquad A^\dagger = A,\ \ B^T = -B,

with :math:`A` complex Hermitian and :math:`B` complex antisymmetric
(class C instead takes :math:`B` complex symmetric; CI and DIII use an
analogous block built from real symmetric, resp. purely-imaginary
skew-symmetric, :math:`X_1, X_2` blocks -- see
:mod:`physicskit.rmt.ensembles.bdg` for the full construction). This
block structure has a built-in charge-conjugation symmetry
:math:`\mathcal{C} H \mathcal{C}^{-1} = -H`, which forces the spectrum
to be symmetric about :math:`E=0`: eigenvalues come in :math:`\pm
\lambda` pairs (not degenerate copies -- generically distinct).

This example visualizes the Altland-Zirnbauer superconductor classes D,
C, CI, DIII: eigenvalue density, symmetric about E=0 by construction (the
defining particle-hole symmetry).

Note: the literature also reports a *finer* near-zero-energy signature
distinguishing classes C and D (a density dip vs. peak exactly at E=0,
tied to the presence/absence of a Majorana zero mode -- Beenakker and
collaborators). That effect lives at a much smaller energy scale than
this plot resolves and is not something this demo claims to show;
what's demonstrated here is the coarser, unambiguous structural
property (Hermiticity and the exact +-E pairing) validated in
``tests/test_bdg.py``.

References:
A. Altland, M. R. Zirnbauer, Phys. Rev. B 55 (1997) 1142.

Run:
    python examples/paper_replications/bdg_classes_demo.py

.. GENERATED FROM PYTHON SOURCE LINES 46-81



.. image-sg:: /api/gallery/rmt/paper_replications/images/sphx_glr_bdg_classes_demo_001.png
   :alt: Altland-Zirnbauer superconductor classes -- eigenvalue density (structural validation only: Hermiticity + exact +-E pairing, not a fitted curve), Class D (beta=2), Class C (beta=2), Class CI (beta=1), Class DIII (beta=4)
   :srcset: /api/gallery/rmt/paper_replications/images/sphx_glr_bdg_classes_demo_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Saved bdg_classes_replication.png






|

.. code-block:: Python


    import matplotlib.pyplot as plt

    import physicskit.rmt as rmt

    N = 150
    N_SAMPLES = 60
    SEED = 2026

    fig, axes = plt.subplots(1, 4, figsize=(15, 3.8), sharex=True)

    for ax, (label, cls) in zip(
        axes,
        [("D", rmt.ensembles.BdGClassD), ("C", rmt.ensembles.BdGClassC), ("CI", rmt.ensembles.BdGClassCI), ("DIII", rmt.ensembles.BdGClassDIII)],
        strict=True,
    ):
        ensemble = cls(n=N, seed=SEED)
        spectrum = ensemble.sample(n_samples=N_SAMPLES)
        scale = ensemble.natural_scale()
        eigs = spectrum.eigenvalues.ravel() / scale

        ax.hist(eigs, bins=100, density=True, alpha=0.5, color="steelblue")
        ax.axvline(0, color="k", lw=0.8, ls="--")
        ax.set_title(f"Class {label} (beta={cls.beta})")
        ax.set_xlabel("E (rescaled)")
        ax.set_xlim(-1.5, 1.5)

    axes[0].set_ylabel("density")
    fig.suptitle(
        "Altland-Zirnbauer superconductor classes -- eigenvalue density\n(structural validation only: Hermiticity + exact +-E pairing, not a fitted curve)"
    )
    fig.tight_layout()
    out_path = "bdg_classes_replication.png"
    fig.savefig(out_path, dpi=150)
    print(f"Saved {out_path}")


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

   **Total running time of the script:** (0 minutes 1.470 seconds)


.. _sphx_glr_download_api_gallery_rmt_paper_replications_bdg_classes_demo.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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