.. _sphx_glr_auto_examples_plot_fourier.py:


Fourier Transforms
==================

Transforming the non-interacting Green's Function for the Bethe
lattice.  Using the analytical expression for the tails. It is
important to note that to get good enough resolution in the time
domain at least :math:`\beta` Matsubara frequencies are needed.

If the Greens Functions is very asymmetric one better use
:math:`2\beta` Matsubara frequencies as for the Fourier transform
improves with a large interval of zeros.



.. code-block:: python


    # Author: Óscar Nájera

    from __future__ import division, absolute_import, print_function
    import dmft.common as gf
    import matplotlib.pyplot as plt
    plt.matplotlib.rcParams.update({'figure.figsize': (8, 8), 'axes.labelsize': 22,
                                    'axes.titlesize': 22, 'figure.autolayout': True})

    tau, w_n = gf.tau_wn_setup(dict(BETA=64, N_MATSUBARA=128))
    fig, ax = plt.subplots(2, 1)
    c_v = ['b', 'g', 'r', 'y']

    for mu, c in zip([0, 0.3, 0.6, 1.], c_v):
        giw = gf.greenF(w_n, mu=mu)
        gtau = gf.gw_invfouriertrans(giw, tau, w_n, [1., -mu, 0.25])

        ax[0].plot(w_n, giw.real, c+'o:', label=r'$\Re G$, $\mu={}$'.format(mu))
        ax[0].plot(w_n, giw.imag, c+'s:')
        ax[1].plot(tau, gtau, c, lw=2, label=r'$\mu={}$'.format(mu))

    ax[0].set_xlim([0, 6])
    ax[0].set_xlabel(r'$i\omega_n$')
    ax[0].set_ylabel(r'$G(i\omega_n)$')
    ax[1].set_xlabel(r'$\tau$')
    ax[1].set_ylabel(r'$G(\tau)$')
    ax[1].set_xlim([0, 64])
    ax[0].legend(loc=0)
    ax[1].legend(loc=0)




.. image:: /auto_examples/images/sphx_glr_plot_fourier_001.png
    :align: center




Review on resolution
--------------------

Here comes to evidence the need of having a long contribution of
zeros from the tail subtracted Matsubara Green function. There is
a reduced noise in the function transformed with its third moment
coming from the self-consistency(Black line)



.. code-block:: python



    fig, ax = plt.subplots(2, 1, sharex=True, sharey=True)
    ax[1].set_xlabel(r'$\tau$')
    ax[1].set_ylabel(r'$G(\tau)$')
    c_v = ['b', 'g', 'r', 'y']

    for mu, c in zip([0, 0.3, 0.6, 1.], c_v):
        tau, w_n = gf.tau_wn_setup(dict(BETA=64, N_MATSUBARA=64))
        giw = gf.greenF(w_n, mu=mu)
        gtau = gf.gw_invfouriertrans(giw, tau, w_n, [1., -mu, 0.])
        ax[0].semilogy(tau, -gtau, c, lw=2, label=r'$\mu={}$'.format(mu))
        gtau = gf.gw_invfouriertrans(giw, tau, w_n, [1., -mu, 0.25])
        ax[0].semilogy(tau, -gtau, 'k', lw=1)

        tau, w_n = gf.tau_wn_setup(dict(BETA=64, N_MATSUBARA=128))
        giw = gf.greenF(w_n, mu=mu)
        gtau = gf.gw_invfouriertrans(giw, tau, w_n, [1., -mu, 0.])
        ax[1].semilogy(tau, -gtau, c, lw=2, label=r'$\mu={}$'.format(mu))
        gtau = gf.gw_invfouriertrans(giw, tau, w_n, [1., -mu, 0.25])
        ax[1].semilogy(tau, -gtau, 'k', lw=1)

    ax[0].set_ylabel(r'$\ln (-G(\tau))$')
    ax[0].set_title(r'$\beta$ frequencies')
    ax[1].set_ylabel(r'$\ln (-G(\tau))$')
    ax[1].set_title(r'$2\beta$ frequencies')
    ax[1].set_xlabel(r'$\tau$')
    ax[1].set_xlim([0, 64])
    ax[0].legend(loc=0)
    ax[1].legend(loc=0)




.. image:: /auto_examples/images/sphx_glr_plot_fourier_002.png
    :align: center




Including the self-energy of the atomic limit so to have a third moment
-----------------------------------------------------------------------

Here the there are the 3 improvements. Large tail subtracted
Matsubara Green function, tail from the Self-Energy and tail of the
self-consistency



.. code-block:: python


    fig, ax = plt.subplots(2, 1)
    sigma = 2.5**2/4/(1j*w_n)
    giiw = gf.greenF(w_n, mu=0, sigma=sigma)
    gitau = gf.gw_invfouriertrans(giiw, tau, w_n, [1., 0, 2.5**2/4+0.25])
    ax[0].plot(w_n, giiw.imag)
    ax[1].plot(tau, gitau)

    ax[0].set_xlim([0, 6])
    ax[0].set_title('Green function in the atomic limit')
    ax[0].set_xlabel(r'$i\omega_n$')
    ax[0].set_ylabel(r'$\Im m G(i\omega_n)$')
    ax[1].set_xlabel(r'$\tau$')
    ax[1].set_ylabel(r'$G(\tau)$')
    ax[1].set_xlim([0, 64])




.. image:: /auto_examples/images/sphx_glr_plot_fourier_003.png
    :align: center





.. code-block:: python


    fig, ax = plt.subplots(2, 1, sharex=True, sharey=True)
    tau, w_n = gf.tau_wn_setup(dict(BETA=64, N_MATSUBARA=64))
    sigma = 2.5**2/4/(1j*w_n)
    giiw = gf.greenF(w_n, mu=0, sigma=sigma)
    gitau = gf.gw_invfouriertrans(giiw, tau, w_n, [1., 0, 2.5**2/4+0.25])
    ax[0].semilogy(tau, -gitau, 'k')
    gitau = gf.gw_invfouriertrans(giiw, tau, w_n, [1., 0, 2.5**2/4])
    ax[0].semilogy(tau, -gitau)

    tau, w_n = gf.tau_wn_setup(dict(BETA=64, N_MATSUBARA=128))
    sigma = 2.5**2/4/(1j*w_n)
    giiw = gf.greenF(w_n, mu=0, sigma=sigma)
    gitau = gf.gw_invfouriertrans(giiw, tau, w_n, [1., 0, 2.5**2/4+0.25])
    ax[1].semilogy(tau, -gitau, 'k', lw=2)
    gitau = gf.gw_invfouriertrans(giiw, tau, w_n, [1., 0, 2.5**2/4])
    ax[1].semilogy(tau, -gitau)
    gitau = gf.gw_invfouriertrans(giiw, tau, w_n, [1., 0, 2.5**2/4+0.25])
    ax[1].semilogy(tau, gf.np.abs(gitau), 'r')

    ax[0].set_title('Green function in the atomic limit')
    ax[0].set_ylabel(r'$\ln (-G(\tau))$')
    ax[0].set_title(r'$\beta$ frequencies')
    ax[1].set_ylabel(r'$\ln (-G(\tau))$')
    ax[1].set_title(r'$2\beta$ frequencies')
    ax[1].set_xlabel(r'$\tau$')
    ax[1].set_xlim([0, 64])



.. image:: /auto_examples/images/sphx_glr_plot_fourier_004.png
    :align: center




**Total running time of the script:** ( 0 minutes  1.064 seconds)



.. container:: sphx-glr-footer


  .. container:: sphx-glr-download

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



  .. container:: sphx-glr-download

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

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

    `Generated by Sphinx-Gallery <https://sphinx-gallery.readthedocs.io>`_