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

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

.. _sphx_glr_api_gallery_rmt_paper_replications_localization_demo.py:


Anderson Localization in Power-Law Banded Matrices
==================================================

The power-law banded random matrix (PBRM) ensemble is a real symmetric
:math:`n \times n` matrix whose off-diagonal entry variance decays with
distance from the diagonal as

.. math::

    \mathrm{Var}(H_{ij}) = \left[1 + \left(\frac{|i-j|}{b}\right)^{2\alpha}
    \right]^{-1}, \qquad i \neq j,

with band-width parameter :math:`b` and decay exponent :math:`\alpha`
(default :math:`\alpha=1`, the multifractal critical point used here).
Small :math:`b` confines the effective coupling to nearby sites,
producing Anderson-localized eigenstates; large :math:`b` recovers an
ordinary (unbanded) GOE-like matrix with fully delocalized
eigenstates. The diagnostic is the inverse participation ratio of a
normalized eigenvector :math:`\psi`,

.. math::

    \mathrm{IPR}(\psi) = \sum_i |\psi_i|^4,

which is :math:`O(1)` (system-size independent) for a localized state
concentrated on a few sites, and :math:`O(1/n)` for a delocalized one
spread evenly over all :math:`n` sites. The delocalized reference value
is the exact mean IPR of a Haar-distributed unit vector at Dyson index
:math:`\beta`,

.. math::

    E[\mathrm{IPR}] = \frac{\beta/2 + 1}{n\beta/2 + 1},

which for :math:`\beta=1` (real, GOE-type vectors) reduces to
:math:`3/(n+2)`. This example reproduces the Anderson localization
transition in the power-law banded random matrix (PBRM) ensemble: the
inverse participation ratio (IPR) drops from O(1) (localized, small
band-width b) toward the delocalized Haar-vector theory value as b
grows.

Reference: A. D. Mirlin, Y. V. Fyodorov, F.-M. Dittes, J. Quezada,
T. H. Seligman, Phys. Rev. E 54 (1996) 3221.

Run:
    python examples/paper_replications/localization_demo.py

.. GENERATED FROM PYTHON SOURCE LINES 49-130



.. image-sg:: /api/gallery/rmt/paper_replications/images/sphx_glr_localization_demo_001.png
   :alt: PBRM localization transition -- N=300, Anderson localization transition, Eigenvector shape, Multifractal spectrum at the critical point (PBRM decay exponent=1)
   :srcset: /api/gallery/rmt/paper_replications/images/sphx_glr_localization_demo_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Saved localization_replication.png






|

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    import physicskit.rmt as rmt

    N = 300
    N_SAMPLES = 10
    SEED = 2026

    fig, axes = plt.subplots(1, 3, figsize=(16, 4.5))

    b_values = np.logspace(-1, 2, 15)
    ipr_means = []
    for b in b_values:
        ensemble = rmt.ensembles.PowerLawBandedEnsemble(n=N, b=b, alpha=2.0, seed=SEED)
        spectrum = ensemble.sample(n_samples=N_SAMPLES, return_eigenvectors=True)
        ipr_means.append(rmt.stats.inverse_participation_ratio(spectrum.eigenvectors).mean())

    theory = rmt.stats.ipr_theory(N, beta=1)
    axes[0].loglog(b_values, ipr_means, "o-", ms=4, color="steelblue", label="PBRM (empirical)")
    axes[0].axhline(theory, color="k", ls="--", lw=1.5, label="delocalized (Haar) theory")
    axes[0].set_xlabel("band-width parameter b")
    axes[0].set_ylabel("mean IPR")
    axes[0].set_title("Anderson localization transition")
    axes[0].legend(fontsize=8)

    # Eigenvector "shape" comparison: a localized state (small b) concentrates
    # on a handful of sites, a delocalized one (large b) spreads over all n.
    cases = [
        (0.5, "indianred", "localized (b=0.5)"),
        (50.0, "steelblue", "delocalized (b=50)"),
    ]
    for b, color, label in cases:
        ensemble = rmt.ensembles.PowerLawBandedEnsemble(n=N, b=b, alpha=2.0, seed=SEED)
        spectrum = ensemble.sample(n_samples=1, return_eigenvectors=True)
        mid = N // 2
        weights = np.abs(spectrum.eigenvectors[0][:, mid]) ** 2
        axes[1].plot(weights, color=color, label=label, alpha=0.8)

    axes[1].set_xlabel("site index")
    axes[1].set_ylabel(r"$|\psi_i|^2$ (mid-spectrum eigenvector)")
    axes[1].set_title("Eigenvector shape")
    axes[1].legend(fontsize=8)

    # --- Panel 3: multifractal singularity spectrum f(alpha) ---
    # At the ensemble's default decay exponent alpha=1 (unrelated to the
    # f(alpha) "alpha" below -- both symbols are standard in their own
    # literatures), PBRM sits at a genuine multifractal critical point for
    # ANY band-width b, with critical statistics that vary continuously
    # with b (see physicskit.rmt.ensembles.banded). rmt.stats.singularity_spectrum
    # estimates the Legendre transform f(alpha) of the mass exponent
    # tau(q) from finite-size scaling of the generalized IPR -- a genuinely
    # richer probe of the eigenvector statistics than the single mean-IPR
    # curve in panel 1, since it resolves how MULTIPLE moments q of the
    # wavefunction weights scale with system size, not just q=2.
    n_values_mf = [80, 120, 180, 260]
    q_values = np.array([-1.0, -0.5, 0.0, 0.5, 1.0, 1.5, 2.0, 3.0])

    for b_mf, color, mf_label in [
        (0.3, "indianred", "b=0.3 (near-localized critical)"),
        (3.0, "steelblue", "b=3.0 (near-delocalized critical)"),
    ]:

        def factory(n: int, seed: int | np.random.Generator | None = None, b_mf: float = b_mf) -> rmt.ensembles.PowerLawBandedEnsemble:
            return rmt.ensembles.PowerLawBandedEnsemble(n=n, b=b_mf, seed=seed)

        mf_alpha, f_alpha = rmt.stats.singularity_spectrum(factory, n_values_mf, q_values, n_samples=10, seed=SEED)
        axes[2].plot(mf_alpha, f_alpha, "o-", color=color, label=mf_label)

    axes[2].plot(1.0, 1.0, "k*", ms=12, label="fully delocalized point")
    axes[2].set_xlabel(r"$\alpha$ (singularity/Hölder exponent)")
    axes[2].set_ylabel(r"$f(\alpha)$")
    axes[2].set_title("Multifractal spectrum at the\ncritical point (PBRM decay exponent=1)")
    axes[2].legend(fontsize=7)

    fig.suptitle(f"PBRM localization transition -- N={N}")
    fig.tight_layout()
    out_path = "localization_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 2.122 seconds)


.. _sphx_glr_download_api_gallery_rmt_paper_replications_localization_demo.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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