.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/statmech/ising/plot_01_ising_chain_transfer_matrix.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_statmech_ising_plot_01_ising_chain_transfer_matrix.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_statmech_ising_plot_01_ising_chain_transfer_matrix.py:


Ising's one-dimensional chain: no phase transition
=====================================================

Ernst Ising (1925) solved the chain of spins :math:`s_i=\pm1` with
nearest-neighbor coupling `J` and found no spontaneous magnetization at
any temperature above zero. :class:`~chemistrykit.statmech.Ising1D`
solves it with the 2x2 transfer matrix, whose eigenvalues
:math:`\lambda_\pm` give :math:`Z_N=\lambda_+^N+\lambda_-^N` exactly. The
plots show a checked partition function, a magnetization that is a smooth
function of field at every temperature (it only becomes a step at
:math:`T=0`), a correlation length that grows as :math:`e^{2J/k_BT}` but
stays finite, and a smooth heat-capacity maximum instead of a singularity.

.. GENERATED FROM PYTHON SOURCE LINES 17-27

.. code-block:: Python

    import itertools

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.constants import K_B
    from chemistrykit.statmech import Ising1D

    J = 100.0 * K_B  # coupling, J/k_B = 100 K








.. GENERATED FROM PYTHON SOURCE LINES 28-29

The transfer-matrix result equals brute-force enumeration of all 2^N states:

.. GENERATED FROM PYTHON SOURCE LINES 29-38

.. code-block:: Python


    N, T_check, h_check = 10, 150.0, 20.0 * K_B
    Z_exact = 0.0
    for spins in itertools.product((-1, 1), repeat=N):
        s = np.array(spins)
        E = -J * np.sum(s * np.roll(s, 1)) - h_check * np.sum(s)
        Z_exact += np.exp(-E / (K_B * T_check))
    print(f"brute force Z_10 = {Z_exact:.6e}, transfer matrix Z_10 = {Ising1D(J, h_check).partition_function(T_check, N):.6e}")





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

 .. code-block:: none

    brute force Z_10 = 1.123856e+04, transfer matrix Z_10 = 1.123856e+04




.. GENERATED FROM PYTHON SOURCE LINES 39-65

.. code-block:: Python

    fig, axes = plt.subplots(1, 3, figsize=(14, 4.2))
    h = np.linspace(-50.0, 50.0, 400) * K_B
    for T, color in [(25.0, "crimson"), (50.0, "darkorange"), (100.0, "seagreen"), (300.0, "steelblue")]:
        m = np.array([Ising1D(J, hi).magnetization(T) for hi in h])
        axes[0].plot(h / K_B, m, color=color, label=f"T = {T:.0f} K")
    axes[0].set_xlabel(r"field $h/k_B$ (K)")
    axes[0].set_ylabel("magnetization per spin")
    axes[0].set_title("m(h) is smooth for every T > 0")
    axes[0].legend()

    T = np.linspace(20.0, 400.0, 300)
    chain = Ising1D(J)
    axes[1].semilogy(T, chain.correlation_length(T), color="steelblue", label=r"exact $\xi = -1/\ln\tanh K$")
    axes[1].semilogy(T, 0.5 * np.exp(2 * J / (K_B * T)), color="gray", linestyle="--", label=r"$\frac{1}{2}e^{2J/k_BT}$")
    axes[1].set_xlabel("T (K)")
    axes[1].set_ylabel("correlation length (sites)")
    axes[1].set_title("Finite correlation length")
    axes[1].legend()

    axes[2].plot(T, chain.heat_capacity_per_spin(T) / K_B, color="crimson")
    axes[2].set_xlabel("T (K)")
    axes[2].set_ylabel(r"$c / k_B$ per spin")
    axes[2].set_title("Smooth heat capacity: no transition")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/statmech/ising/images/sphx_glr_plot_01_ising_chain_transfer_matrix_001.png
   :alt: m(h) is smooth for every T > 0, Finite correlation length, Smooth heat capacity: no transition
   :srcset: /api/gallery/statmech/ising/images/sphx_glr_plot_01_ising_chain_transfer_matrix_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.141 seconds)


.. _sphx_glr_download_api_gallery_statmech_ising_plot_01_ising_chain_transfer_matrix.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_ising_chain_transfer_matrix.ipynb <plot_01_ising_chain_transfer_matrix.ipynb>`

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

      :download:`Download Python source code: plot_01_ising_chain_transfer_matrix.py <plot_01_ising_chain_transfer_matrix.py>`

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

      :download:`Download zipped: plot_01_ising_chain_transfer_matrix.zip <plot_01_ising_chain_transfer_matrix.zip>`


.. only:: html

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

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