.. _sphx_glr_auto_examples_IPT_plot_free_transition.py:


Transition on free energy
=========================

Using the formalism presented in [1]_ I calculate the stability of the
IPT solution within the coexistence region.

.. [1] Moeller, G., V. Dobrosavljevi\'c, & Ruckenstein, A. E. (1999). RKKY
  interactions and the Mott transition. Physical Review B, 59(10),
  6846–6854. http://dx.doi.org/10.1103/physrevb.59.6846




.. code-block:: python

    # Created Fri Apr 21 15:11:05 2017
    # Author: Óscar Nájera

    from __future__ import division, absolute_import, print_function
    import numpy as np
    from scipy.integrate import trapz
    import matplotlib.pylab as plt

    from dmft.ipt_imag import dmft_loop, single_band_ipt_solver
    from dmft.common import greenF, tau_wn_setup


    def mix(gmet, gins, l):
        return (1 - l) * gins + l * gmet


    def one_loop(giw, t, u_int, w_n, tau):
        iw_n = 1j * w_n
        g_0_iwn = 1. / (iw_n - t**2 * giw)
        g_iwn, sigma_iwn = single_band_ipt_solver(u_int, g_0_iwn, w_n, tau)
        return g_iwn


    def free_energy_change(beta, u_int, mix_grid):

        tau, w_n = tau_wn_setup(dict(BETA=beta, N_MATSUBARA=2**11))

        ig_iwn, is_iwn = dmft_loop(
            u_int, 0.5, -1.j / (w_n - 1 / w_n), w_n, tau, conv=1e-10)

        mg_iwn, ms_iwn = dmft_loop(u_int, 0.5, greenF(w_n), w_n, tau, conv=1e-10)

        solution_diff = mg_iwn - ig_iwn

        integrand = []
        for l in mix_grid:
            g_in = mix(mg_iwn, ig_iwn, l)
            g_grad = one_loop(g_in, 0.5, U, w_n, tau) - g_in
            integrand.append(np.dot(g_grad, solution_diff).real / beta)

        return np.array([0] + [trapz(integrand[:i], mix_grid[:i])
                               for i in range(2, len(mix_grid) + 1)])


    mix_grid = np.linspace(0, 1, 201)







High in the coexistence region



.. code-block:: python

    U = 2.6
    for temp in [0.003, 0.009, 0.015, 0.021, 0.027, 0.033, 0.037]:
        beta = 1 / temp
        delta_f = free_energy_change(beta, U, mix_grid)
        plt.plot(mix_grid, delta_f, label='T={:.3}'.format(1 / beta))
    plt.xlabel('$l$ Admixture of metallic solution')
    plt.ylabel(r'$\Delta F(l)$: Change in free energy')
    plt.title('$U/D={}$'.format(U))
    plt.legend(loc=0)
    plt.show()




.. image:: /auto_examples/IPT/images/sphx_glr_plot_free_transition_001.png
    :align: center




Lower in the coexistence region



.. code-block:: python


    U = 3
    for beta in [52., 57.38, 64., 100.]:
        delta_f = free_energy_change(beta, U, mix_grid)
        plt.plot(mix_grid, delta_f, label='T={:.3}'.format(1 / beta))
    plt.xlabel('$l$ Admixture of metallic solution')
    plt.ylabel(r'$\Delta F(l)$: Change in free energy')
    plt.title('$U/D={}$'.format(U))
    plt.legend(loc=0)



.. image:: /auto_examples/IPT/images/sphx_glr_plot_free_transition_002.png
    :align: center




**Total running time of the script:** ( 0 minutes  9.189 seconds)



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



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