.. _sphx_glr_auto_examples_IPT_plot_ipt.py:


=======================================================
IPT Solver Existence of Metalic and insulating solution
=======================================================

Within the iterative perturbative theory (IPT) the aim is to express the
self-energy of the impurity problem as

.. math:: \Sigma(\tau) \approx U^2 \mathcal{G}^0(\tau)^3

the contribution of the Hartree-term is not included here as it is cancelled




.. code-block:: python

    from __future__ import division, absolute_import, print_function

    import numpy as np
    import matplotlib.pylab as plt


    from dmft.ipt_imag import dmft_loop
    from dmft.common import greenF, tau_wn_setup, pade_continuation, fermi_dist
    from dmft.utils import optical_conductivity

    U = 2.7
    BETA = 100.
    tau, w_n = tau_wn_setup(dict(BETA=BETA, N_MATSUBARA=2**9))
    omega = np.linspace(-4, 4, 600)
    eps_k = np.linspace(-1, 1, 61)
    dos = np.exp(-2 * eps_k**2) / np.sqrt(np.pi / 2)
    de = eps_k[1] - eps_k[0]
    dosde = (dos * de).reshape(-1, 1)
    nf = fermi_dist(omega, BETA)







Insulator Calculations
----------------------



.. code-block:: python


    ig_iwn, is_iwn = dmft_loop(
        U, 0.5, -1.j / (w_n - 1 / w_n), w_n, tau, conv=1e-12)
    igw = pade_continuation(ig_iwn, w_n, omega, np.arange(100))
    isigma_w = pade_continuation(is_iwn, w_n, omega, np.arange(100))
    lat_gf = 1 / (np.add.outer(-eps_k, omega + 4e-2j) - isigma_w)
    lat_Aw = -lat_gf.imag / np.pi
    icond = optical_conductivity(lat_Aw, lat_Aw, nf, omega, dosde)








Metal Calculations
------------------



.. code-block:: python


    mg_iwn, s_iwn = dmft_loop(U, 0.5, greenF(w_n), w_n, tau, conv=1e-10)
    mgw = pade_continuation(mg_iwn, w_n, omega, np.arange(100))
    msigma_w = omega - 0.25 * mgw - 1 / mgw
    lat_gf = 1 / (np.add.outer(-eps_k, omega + 4e-2j) - msigma_w)
    lat_Aw = -lat_gf.imag / np.pi
    mcond = optical_conductivity(lat_Aw, lat_Aw, nf, omega, dosde)







Plots
-----



.. code-block:: python


    fig_giw, giw_ax = plt.subplots()
    fig_gw, gw_ax = plt.subplots()
    fig_sw, sw_ax = plt.subplots(1, 2, sharex=True)
    fig_cond, cond_ax = plt.subplots()

    giw_ax.plot(w_n, ig_iwn.imag, 's-', label='Insulator Solution')
    gw_ax.plot(omega, -igw.imag / np.pi, label='Insulator Solution')
    sw_ax[0].plot(omega, isigma_w.real, label='Insulator Solution')
    sw_ax[1].plot(omega, isigma_w.imag, label='Insulator Solution')
    cond_ax.plot(omega, icond, label='Insulator Solution')
    giw_ax.plot(w_n, mg_iwn.imag, 's-', label='Metal Solution')
    gw_ax.plot(omega, -mgw.imag / np.pi, label='Metal Solution')
    sw_ax[0].plot(omega, msigma_w.real, label='Metal Solution')
    sw_ax[1].plot(omega, msigma_w.imag, label='Metal Solution')
    cond_ax.plot(omega, mcond, label='Metal Solution')

    giw_ax.plot(w_n, -1 / w_n, label='$1/w$ tail')
    giw_ax.plot(w_n, -1 / w_n + U**2 / 4 / w_n**3,
                label='$w^{-1} + M_3 w^{-3}$ tail')

    cut = int(6.5 * BETA / np.pi)
    giw_ax.set_xlim([0, 6.5])
    giw_ax.set_ylim([mg_iwn.imag.min() * 1.1, 0])

    giw_ax.set_ylabel(r'$G(i\omega_n)$')
    giw_ax.set_xlabel(r'$i\omega_n$')
    giw_ax.set_title(r'$G(i\omega_n)$ at $\beta= {}$, $U= {}$'.format(BETA, U))
    giw_ax.legend(loc=0)

    gw_ax.set_ylabel(r'$A(\omega)$')
    gw_ax.set_xlabel(r'$\omega$')
    gw_ax.set_title(r'Spectral function at $\beta= {}$, $U= {}$'.format(BETA, U))
    gw_ax.legend(loc=0)

    sw_ax[0].set_ylabel(r'$\Re e \Sigma(\omega)$')
    sw_ax[0].set_ylim([-25, 25])
    sw_ax[1].set_ylabel(r'$\Im m \Sigma(\omega)$')
    sw_ax[0].set_xlabel(r'$\omega$')
    sw_ax[1].set_xlabel(r'$\omega$')
    sw_ax[0].set_title(r'Self-Energy at $\beta= {}$, $U= {}$'.format(BETA, U))
    sw_ax[0].legend(loc=0)
    sw_ax[1].legend(loc=0)

    cond_ax.legend(loc=0)
    cond_ax.set_xlabel(r'$\nu$')
    cond_ax.set_ylabel(r'$\sigma(\nu)$')
    cond_ax.set_ylim([0, .2])
    cond_ax.set_xlim([0, 4])



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


    *

      .. image:: /auto_examples/IPT/images/sphx_glr_plot_ipt_001.png
            :scale: 47

    *

      .. image:: /auto_examples/IPT/images/sphx_glr_plot_ipt_002.png
            :scale: 47

    *

      .. image:: /auto_examples/IPT/images/sphx_glr_plot_ipt_003.png
            :scale: 47

    *

      .. image:: /auto_examples/IPT/images/sphx_glr_plot_ipt_004.png
            :scale: 47




**Total running time of the script:** ( 0 minutes  0.593 seconds)



.. container:: sphx-glr-footer


  .. container:: sphx-glr-download

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



  .. container:: sphx-glr-download

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

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

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