.. _sphx_glr_api_gallery_statmech_ising:

Ising model
------------

Exact results for the nearest-neighbor Ising (lattice-gas) model: Ising's
one-dimensional chain, Kramers-Wannier duality, and Onsager's
two-dimensional solution.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Ernst Ising (1925) solved the chain of spins s_i=\pm1 with nearest-neighbor coupling J and found no spontaneous magnetization at any temperature above zero. Ising1D solves it with the 2x2 transfer matrix, whose eigenvalues \lambda_\pm give 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 T=0), a correlation length that grows as e^{2J/k_BT} but stays finite, and a smooth heat-capacity maximum instead of a singularity.">

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  .. image:: /api/gallery/statmech/ising/images/thumb/sphx_glr_plot_01_ising_chain_transfer_matrix_thumb.png
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  :doc:`/api/gallery/statmech/ising/plot_01_ising_chain_transfer_matrix`

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      <div class="sphx-glr-thumbnail-title">Ising's one-dimensional chain: no phase transition</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Kramers and Wannier (1941) showed that the square-lattice Ising model&#x27;s high-temperature series at coupling K=J/k_BT equals its low-temperature series at the dual coupling K^*, with">

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  .. image:: /api/gallery/statmech/ising/images/thumb/sphx_glr_plot_02_kramers_wannier_duality_thumb.png
    :alt:

  :doc:`/api/gallery/statmech/ising/plot_02_kramers_wannier_duality`

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      <div class="sphx-glr-thumbnail-title">Kramers-Wannier duality and the exact Ising critical temperature</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Onsager (1944) computed the free energy of the square-lattice Ising model exactly: the first exact solution of a model with a genuine phase transition. Ising2DOnsager evaluates his free energy, internal energy, and heat capacity, together with Yang&#x27;s (1952) spontaneous magnetization m=[1-\sinh^{-4}(2J/k_BT)]^{1/8}. The heat capacity diverges logarithmically at T_c, and the magnetization vanishes with exponent \beta=1/8. Neither the mean-field (Bragg-Williams) theory, shown for comparison, nor Ising&#x27;s own one-dimensional chain reproduces these features.">

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  .. image:: /api/gallery/statmech/ising/images/thumb/sphx_glr_plot_03_onsager_exact_solution_thumb.png
    :alt:

  :doc:`/api/gallery/statmech/ising/plot_03_onsager_exact_solution`

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      <div class="sphx-glr-thumbnail-title">Onsager's exact solution of the two-dimensional Ising model</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/statmech/ising/plot_01_ising_chain_transfer_matrix
   /api/gallery/statmech/ising/plot_02_kramers_wannier_duality
   /api/gallery/statmech/ising/plot_03_onsager_exact_solution