.. _sphx_glr_auto_examples_IPT_plot_crossover.py:


===========================
IPT Transition at crossover
===========================

Study the end of the crossover region



.. code-block:: python


    from __future__ import division, absolute_import, print_function

    from dmft.ipt_imag import dmft_loop
    from dmft.common import greenF, tau_wn_setup, fit_gf
    from dmft.twosite import matsubara_Z

    import numpy as np
    import matplotlib.pylab as plt


    def hysteresis(beta, u_range):
        log_g_sig = []
        tau, w_n = tau_wn_setup(dict(BETA=beta, N_MATSUBARA=beta))
        g_iwn = greenF(w_n)
        for u_int in u_range:
            g_iwn, sigma = dmft_loop(u_int, 0.5, g_iwn, w_n, tau)
            log_g_sig.append((g_iwn, sigma))
        return log_g_sig


    U = np.linspace(2.2, 2.5, 100)
    results = []
    betarange = [16, 17.85, 19.2, 20., 21.3]
    for beta in betarange:
        results.append(hysteresis(beta, U))








Quasiparticle weight
--------------------



.. code-block:: python


    figz, axz = plt.subplots()
    for beta, result in zip(betarange, results):
        u_zet = [matsubara_Z(sigma.imag, beta) for _, sigma in result]
        axz.plot(U, u_zet, '+-', label='$\\beta={}$'.format(beta))
    axz.set_title('Quasiparticle weigth')
    axz.legend(loc=0)
    axz.set_ylabel('Z')
    axz.set_xlabel('U/D')





.. image:: /auto_examples/IPT/images/sphx_glr_plot_crossover_001.png
    :align: center




Spectral density at Fermi level
-------------------------------



.. code-block:: python


    figf, axf = plt.subplots()
    for beta, result in zip(betarange, results):
        tau, w_n = tau_wn_setup(dict(BETA=beta, N_MATSUBARA=3))
        u_fl = [-fit_gf(w_n, g_iwn.imag)(0.)for g_iwn, _ in result]
        axf.plot(U, u_fl, 'x:', label='$\\beta={}$'.format(beta))

    axf.set_ylabel('Dos(0)')
    axz.set_title('Density of states at Fermi level')
    axf.set_xlabel('U/D')





.. image:: /auto_examples/IPT/images/sphx_glr_plot_crossover_002.png
    :align: center




Double occupation
-----------------

Is proportional to the Potential energy which is defined in
:ref:`potential_energy` the tail of this function behaves as

.. math:: \Sigma G(i\omega_n\rightarrow \infty)= \frac{U^2}{8(i\omega_n)^2}

Then to find out the double occupation one uses the relation

.. math:: \langle n_\uparrow n_\downarrow \rangle = \frac{2\langle V \rangle}{U}+\frac{1}{4}





.. code-block:: python



    figd, axd = plt.subplots()
    for beta, result in zip(betarange, results):
        tau, w_n = tau_wn_setup(dict(BETA=beta, N_MATSUBARA=beta))
        V = np.asarray([2 * (0.5 * s * g + u**2 / 8. / w_n**2).real.sum() / beta
                        for (g, s), u in zip(result, U)]) - 0.25 * beta * U**2 / 8.

        D = 2. / U * V + 0.25
        axd.plot(U, D, '-', label='$\\beta={}$'.format(beta))

    axd.set_title('Double occupation')
    axd.legend(loc=0)
    axd.set_ylabel(r'$\langle n_\uparrow n_\downarrow \rangle$')
    axd.set_xlabel('U/D')



.. image:: /auto_examples/IPT/images/sphx_glr_plot_crossover_003.png
    :align: center




**Total running time of the script:** ( 0 minutes  1.452 seconds)



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



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