.. _sphx_glr_auto_examples_plot_hubbardI_bethe.py:


Hubbard I solver Bethe lattice
==============================

Atomic limit expression of the self-energy
is described by

.. math:: \Sigma(\omega) =\frac{U n}{2} + \frac{ \frac{U^{2} n}{2} \left( 1 - \frac{n}{2}\right)}{\omega + \mu  - U \left(1 - \frac{n}{2}\right)}

This approximation is most accurate in the limit of strong coupling,
as there the atomic case is closer. Nevertheless it is possible to see
the formation of the Hubbard Bands and the redistribution of spectral
weight.




.. image:: /auto_examples/images/sphx_glr_plot_hubbardI_bethe_001.png
    :align: center


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

 Out::

    n_new 1.97901484639
    n_new 1.67550206029
    n_new 1.78121999181
    n_new 1.74447416817
    n_new 1.75728528387
    n_new 1.75279719806
    n_new 1.75437710222
    n_new 1.75382113619
    n_new 1.75401684774
    n_new 1.75394795984




|


.. code-block:: python


    # Created Mon Sep 28 15:25:30 2015
    # Author: Óscar Nájera

    from __future__ import division, absolute_import, print_function
    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.integrate import trapz
    import dmft.common as gf


    def hubbard_aprox(U, dmu, omega):

        mu = U / 2 + dmu
        n = 1.

        for _ in range(10):
            sigma = n * U / 2 + n / 2 * (1 - n / 2) * \
                U**2 / (omega + 0.05j + mu - (1 - n / 2) * U)
            gloc = gf.semi_circle_hiltrans(omega + mu - sigma)
            dos = -gloc.imag / np.pi
            n = 2 * trapz(dos * (omega < 0), omega)
            print('n_new', n)

        plt.plot(omega, -gloc.imag)

    omega = np.linspace(-8, 4, 600)
    hubbard_aprox(3, 2.051, omega)
    plt.show()

**Total running time of the script:** ( 0 minutes  0.065 seconds)



.. container:: sphx-glr-footer


  .. container:: sphx-glr-download

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



  .. container:: sphx-glr-download

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

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

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