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

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

.. _sphx_glr_api_gallery_statphys_spin_glass_plot_sk_overlap_distribution.py:


The Sherrington-Kirkpatrick model and Parisi's replica overlap
=========================================================================

Where the Edwards-Anderson model puts random +/-J bonds on a lattice, the
Sherrington-Kirkpatrick (SK) model in 1975 went to the opposite extreme:
every spin couples to *every* other spin, with independent Gaussian bonds

.. math::

    H = -\sum_{i<j} J_{ij}\, s_i s_j, \qquad
    J_{ij} \sim \mathcal{N}\!\left(0, \frac{J^2}{N}\right).

The model's mean-field solution turned out to be far subtler than anyone
expected: the naive "replica trick" ansatz gave a negative entropy at low
temperature, a clear sign something was wrong, until Giorgio Parisi's
1979-1980 replica-symmetry-breaking solution revealed that the low-
temperature phase is not one frozen state but an infinite, hierarchically
organized family of them. The numerically observable fingerprint of this
structure is the distribution of the replica overlap

.. math::

    q = \frac{1}{N}\sum_i s_i^{(1)} s_i^{(2)}

between two independently thermalized replicas sharing the same disorder,
pooled across *many* independent disorder realizations. Above the
transition, :math:`P(q)` is a single narrow peak at :math:`q=0`
(replica-symmetric, self-averaging). Below it, finite-size simulations
already show a much broader, flatter, non-Gaussian :math:`P(q)` -- the
finite-:math:`N` shadow of the continuous distribution Parisi's solution
predicts exactly as :math:`N \to \infty`.

.. GENERATED FROM PYTHON SOURCE LINES 34-41

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.statphys.chapters.spin_glass import SherringtonKirkpatrick
    from physicskit.statphys.visualizers.spin_glass_render import plot_overlap_distribution








.. GENERATED FROM PYTHON SOURCE LINES 42-47

High temperature: a single peak at q = 0
------------------------------------------------------------
Well above the spin-glass transition (:math:`T_{SG} \approx J/k_B` for
the SK model), the two replicas decorrelate completely and the pooled
overlap distribution is a single narrow peak centered on zero.

.. GENERATED FROM PYTHON SOURCE LINES 47-50

.. code-block:: Python

    model_hot = SherringtonKirkpatrick(N=64, J=1.0, seed=0)
    samples_hot = model_hot.overlap_distribution(beta=0.3, n_disorder=25, n_equil=300, n_measure=60)








.. GENERATED FROM PYTHON SOURCE LINES 51-57

Low temperature: a broad, structured P(q)
------------------------------------------------------------
Below the transition, a fresh model instance (a new disorder realization
for its first sample) is thermalized at much lower temperature. The
resulting pooled distribution is visibly broader and no longer a single
peak -- finite-size Monte Carlo's glimpse of replica symmetry breaking.

.. GENERATED FROM PYTHON SOURCE LINES 57-70

.. code-block:: Python

    model_cold = SherringtonKirkpatrick(N=64, J=1.0, seed=1)
    samples_cold = model_cold.overlap_distribution(beta=2.0, n_disorder=25, n_equil=300, n_measure=60)

    fig, ax = plt.subplots(figsize=(6, 4))
    plot_overlap_distribution(samples_hot, ax=ax, label=r"$T=3.3\,J/k_B$ (paramagnetic)")
    plot_overlap_distribution(samples_cold, ax=ax, label=r"$T=0.5\,J/k_B$ (spin-glass)")
    ax.set_title("Sherrington-Kirkpatrick replica overlap distribution")
    plt.tight_layout()
    plt.show()

    print(f"High-T: mean|q|={abs(samples_hot).mean():.3f}, std(q)={samples_hot.std():.3f}")
    print(f"Low-T:  mean|q|={abs(samples_cold).mean():.3f}, std(q)={samples_cold.std():.3f}")




.. image-sg:: /api/gallery/statphys/spin_glass/images/sphx_glr_plot_sk_overlap_distribution_001.png
   :alt: Sherrington-Kirkpatrick replica overlap distribution
   :srcset: /api/gallery/statphys/spin_glass/images/sphx_glr_plot_sk_overlap_distribution_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    High-T: mean|q|=0.105, std(q)=0.132
    Low-T:  mean|q|=0.472, std(q)=0.512




.. GENERATED FROM PYTHON SOURCE LINES 71-79

Mean overlap across a full temperature sweep
------------------------------------------------------------
Rather than just the two endpoints above, sweeping :math:`\langle |q|
\rangle` continuously through :math:`T_{SG} \approx J/k_B` shows the
freezing transition directly: :math:`\langle |q| \rangle` stays near zero
in the paramagnetic phase and rises sharply as :math:`T` drops below
:math:`T_{SG}`, the SK analogue of the Edwards-Anderson order parameter
curve, but built from the same pooled-disorder machinery used above.

.. GENERATED FROM PYTHON SOURCE LINES 79-95

.. code-block:: Python

    T_values = np.linspace(0.3, 3.0, 8)
    mean_abs_q = []
    for T in T_values:
        model_T = SherringtonKirkpatrick(N=64, J=1.0, seed=2)
        samples_T = model_T.overlap_distribution(beta=1.0 / T, n_disorder=15, n_equil=200, n_measure=40)
        mean_abs_q.append(np.mean(np.abs(samples_T)))

    plt.figure(figsize=(6, 4))
    plt.plot(T_values, mean_abs_q, marker="o")
    plt.axvline(1.0, color="k", linestyle="--", linewidth=1, alpha=0.6, label=r"$T_{SG} \approx J/k_B$")
    plt.xlabel("Temperature")
    plt.ylabel(r"$\langle |q| \rangle$")
    plt.title("Sherrington-Kirkpatrick freezing transition")
    plt.legend()
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/statphys/spin_glass/images/sphx_glr_plot_sk_overlap_distribution_002.png
   :alt: Sherrington-Kirkpatrick freezing transition
   :srcset: /api/gallery/statphys/spin_glass/images/sphx_glr_plot_sk_overlap_distribution_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.395 seconds)


.. _sphx_glr_download_api_gallery_statphys_spin_glass_plot_sk_overlap_distribution.py:

.. only:: html

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

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

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

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.. only:: html

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

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