.. _sphx_glr_dimer_lattice_plot_hubbard_I.py:


===================
Hubbard I for dimer
===================

The atomic self-energy is extracted and plotted into the lattice
Green's function to see the behavior of the insulating state.



.. code-block:: python

    # Created Mon Mar 14 13:56:37 2016
    # Author: Óscar Nájera

    from __future__ import division, absolute_import, print_function

    import matplotlib.pyplot as plt
    import numpy as np
    from dmft.common import gw_invfouriertrans
    from dmft.plot import plot_band_dispersion
    import dmft.dimer as dimer
    import slaveparticles.quantum.operators as op








Approximating Hubbard I
=======================

Here I use the molecule self-energy



.. code-block:: python


    def molecule_sigma(omega, U, mu, tp, beta):
        """Return molecule self-energy in the given frequency axis"""

        h_at, oper = dimer.hamiltonian(U, mu, tp)
        oper_pair = [[oper[0], oper[0]], [oper[0], oper[1]]]

        eig_e, eig_v = op.diagonalize(h_at.todense())
        gfsU = np.array([op.gf_lehmann(eig_e, eig_v, c.T, beta, omega, d)
                         for c, d in oper_pair])

        invg = dimer.mat_inv(gfsU[0], gfsU[1])
        plt.plot(omega.real, -gfsU[0].imag, label='Interacting')
        plt.plot(omega.real, -dimer.mat_inv(omega, -tp)[0].imag, label='Free')
        plt.xlabel(r'$\omega$')
        plt.ylabel(r'$A(\omega)$')
        plt.title(r'Isolated dimer $U={}$, $t_\perp={}$, $\beta={}$'.format(U, tp, beta))
        plt.legend(loc=0)

        return [omega - invg[0], -tp - invg[1]]








The Real axis Self-energy
-------------------------



.. code-block:: python


    def plot_self_energy(w, sd_w, so_w, U, mu, tp, beta):
        f, ax = plt.subplots(2, sharex=True)
        ax[0].plot(w, sd_w.real, label='Real')
        ax[0].plot(w, sd_w.imag, label='Imag')
        ax[1].plot(w, so_w.real, label='Real')
        ax[1].plot(w, so_w.imag, label='Imag')
        ax[0].legend(loc=0)
        ax[1].set_xlabel(r'$\omega$')
        ax[0].set_ylabel(r'$\Sigma_{AA}(\omega)$')
        ax[1].set_ylabel(r'$\Sigma_{AB}(\omega)$')
        ax[0].set_title(
            r'Isolated dimer $U={}$, $t_\perp={}$, $\beta={}$'.format(U, tp, beta))

    w = np.linspace(-3, 3, 800)
    U, mu, tp, beta = 2.15, 0., 0.3, 100.
    sd_w, so_w = molecule_sigma(w + 5e-5j, U, mu, tp, beta)
    plot_self_energy(w, sd_w, so_w, U, mu, tp, beta)




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


    *

      .. image:: /dimer_lattice/images/sphx_glr_plot_hubbard_I_001.png
            :scale: 47

    *

      .. image:: /dimer_lattice/images/sphx_glr_plot_hubbard_I_002.png
            :scale: 47




Hubbard I Band dispersion
-------------------------



.. code-block:: python


    eps_k = np.linspace(-1, 1, 61)
    lat_gf = dimer.mat_inv(np.add.outer(-eps_k, w + 5e-2j - sd_w), -tp - so_w)
    Aw = -lat_gf[0].imag / np.pi

    plot_band_dispersion(
        w, Aw, r'Hubbard I dimer $U={}$, $t_\perp={}$, $\beta={}$'.format(U, tp, beta), eps_k)




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


    *

      .. image:: /dimer_lattice/images/sphx_glr_plot_hubbard_I_003.png
            :scale: 47

    *

      .. image:: /dimer_lattice/images/sphx_glr_plot_hubbard_I_004.png
            :scale: 47




The Real axis Self-energy
-------------------------



.. code-block:: python


    w = np.linspace(-3, 3, 800)
    U, mu, tp, beta = 2.15, 0., 0.3, 5.
    sd_w, so_w = molecule_sigma(w + 5e-5j, U, mu, tp, beta)
    plot_self_energy(w, sd_w, so_w, U, mu, tp, beta)




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


    *

      .. image:: /dimer_lattice/images/sphx_glr_plot_hubbard_I_005.png
            :scale: 47

    *

      .. image:: /dimer_lattice/images/sphx_glr_plot_hubbard_I_006.png
            :scale: 47




Hubbard I Band dispersion
-------------------------



.. code-block:: python


    eps_k = np.linspace(-1, 1, 61)
    lat_gf = dimer.mat_inv(np.add.outer(-eps_k, w + 5e-2j - sd_w), -tp - so_w)
    Aw = -lat_gf[0].imag / np.pi
    plot_band_dispersion(
        w, Aw, r'Hubbard I dimer $U={}$, $t_\perp={}$, $\beta={}$'.format(U, tp, beta), eps_k)



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


    *

      .. image:: /dimer_lattice/images/sphx_glr_plot_hubbard_I_007.png
            :scale: 47

    *

      .. image:: /dimer_lattice/images/sphx_glr_plot_hubbard_I_008.png
            :scale: 47




**Total running time of the script:** ( 0 minutes  1.208 seconds)



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