.. _sphx_glr_auto_examples_plot_hubbardIII.py:


=========================
Hubbard III approximation
=========================

In this approach the aim is to find the band dispersion of the
insulating system but including the effects of the bath. For this one
approximates the local Green's function by

.. math: G(\omega) = \frac{1}{\mathcal{G}_0^{-1}-\frac{U^2}{4\mathcal{G}_0^{-1}}}

And the using the self consistency equation of the Bethe lattice

.. math: \mathcal{G}_0^{-1}=\omega - t^2G

Here this equation can be solved analytically but for current purposes
 here it will be solved by fixed point iteration.



.. code-block:: python


    # Created Fri Apr  1 14:44:59 2016
    # Author: Óscar Nájera

    from __future__ import absolute_import, division, print_function

    import numpy as np
    import matplotlib.pyplot as plt

    import dmft.common as gf
    from dmft.plot import plot_band_dispersion


    w = np.linspace(-3, 3, 800)
    g0_1 = w + 1e-6j
    U = 2.

    for i in range(2000):
        g0_1 = w - .25 / (g0_1 - U**2 / 4. / g0_1)







The Self-Energy
---------------



.. code-block:: python


    plt.figure()
    plt.plot(w, (U**2 / 4 / g0_1).real, label=r"Real")
    plt.plot(w, (U**2 / 4 / g0_1).imag, label=r"Imag")

    plt.ylabel(r'$\Sigma(\omega)$')
    plt.xlabel(r'$\omega$')
    plt.title(r'$\Sigma(\omega)$ at $U= {}$'.format(U))
    plt.legend(loc=0)
    plt.ylim([-1.5, 1])





.. image:: /auto_examples/images/sphx_glr_plot_hubbardIII_001.png
    :align: center




The Green Function
------------------



.. code-block:: python


    plt.figure()
    plt.plot(w, (1 / (w - U**2 / 4 / g0_1)).real, label=r"Real")
    plt.plot(w, (1 / (w - U**2 / 4 / g0_1)).imag, label=r"Imag")
    plt.plot(w, (gf.semi_circle_hiltrans(w - U**2 / 4 / g0_1)).imag, label=r"Imag")

    plt.ylabel(r'$G(\omega)$')
    plt.xlabel(r'$\omega$')
    plt.title(r'$G(\omega)$ at $U= {}$'.format(U))
    plt.legend(loc=0)





.. image:: /auto_examples/images/sphx_glr_plot_hubbardIII_002.png
    :align: center




The Band Dispersion
-------------------



.. code-block:: python


    eps_k = np.linspace(-1, 1, 61)
    lat_gf = 1 / (np.add.outer(-eps_k, w + 8e-2j) - U**2 / 4 / g0_1)
    Aw = -lat_gf.imag / np.pi

    plot_band_dispersion(w, Aw, 'Hubbard III band dispersion', eps_k)



.. rst-class:: sphx-glr-horizontal


    *

      .. image:: /auto_examples/images/sphx_glr_plot_hubbardIII_003.png
            :scale: 47

    *

      .. image:: /auto_examples/images/sphx_glr_plot_hubbardIII_004.png
            :scale: 47




**Total running time of the script:** ( 0 minutes  0.586 seconds)



.. container:: sphx-glr-footer


  .. container:: sphx-glr-download

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



  .. container:: sphx-glr-download

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

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

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