def analytic_hubbard_atom(beta, U, nw, nwf, nwf_gf):
d = ParameterCollection()
d.beta, d.U, d.nw, d.nwf, d.nwf_gf = beta, U, nw, nwf, nwf_gf
g_iw = single_particle_greens_function(beta=beta, U=U, nw=nwf_gf)
d.G_iw = g_iw
# make block gf of the single gf
G_iw_block = BlockGf(name_list=['up', 'dn'], block_list=[g_iw, g_iw])
g_mat = block_iw_AB_to_matrix_valued(G_iw_block)
d.chi_m = chi_ph_magnetic(beta=beta, U=U, nw=nw, nwf=nwf)
d.chi0_m = chi0_from_gg2_PH(g_mat, d.chi_m)
# -- Numeric vertex from BSE
d.gamma_m_num = inverse_PH(d.chi0_m) - inverse_PH(d.chi_m)
# -- Analytic vertex
d.gamma_m = gamma_ph_magnetic(beta=beta, U=U, nw=nw, nwf=nwf)
# -- Analytic magnetization expecation value
# -- and static susceptibility
d.Z = 2. + 2 * np.exp(-beta * 0.5 * U)
d.m2 = 0.25 * (2 / d.Z)
d.chi_m_static = 2. * beta * d.m2
d.label = r'Analytic'
return d
def analytic_hubbard_atom(beta, U, nw, nwf, nwf_gf):
r""" Compute dynamical response functions for the Hubbard atom at half filling.
This function returns an object that contains the single-particle
Greens function :math:`G(\omega)`, the magnetic two-particle generalized susceptibility
:math:`\chi_m(\omega, \nu, \nu')`, and the corresponding bare bubble
:math:`\chi^{(0)}_m(\omega, \nu, \nu')`, and the magnetic vertex function
:math:`\Gamma_m(\omega, \nu, \nu')`.
This is implemented using analytical formulas from
Thunstrom et al. [PRB 98, 235107 (2018)]
please cite the paper if you use this function!
In particular this is one exact solution to the Bethe-Salpeter
equation, that is the infinite matrix inverse problem:
.. math::
\Gamma_m = [\chi^{(0)}_m]^{-1} - \chi_m^{-1}
Parameters
----------
beta : float
Inverse temperature.
U : float
Hubbard U interaction parameter.
nw : int
Number of bosonic Matsubara frequencies
in the computed two-particle response functions.
nwf : int
Number of fermionic Matsubara frequencies
in the computed two-particle response functions.
nwf_gf : int
Number of fermionic Matsubara frequencies
in the computed single-particle Greens function.
Returns
-------
p : ParameterCollection
Object containing all the response functions and some other
observables, `p.G_iw`, `p.chi_m`, `p.chi0_m`,
`p.gamma_m`, `p.Z`, `p.m2`, `p.chi_m_static`.
"""
d = ParameterCollection()
d.beta, d.U, d.nw, d.nwf, d.nwf_gf = beta, U, nw, nwf, nwf_gf
g_iw = single_particle_greens_function(beta=beta, U=U, nw=nwf_gf)
d.G_iw = g_iw
# make block gf of the single gf
G_iw_block = BlockGf(name_list=['up', 'dn'], block_list=[g_iw, g_iw])
g_mat = block_iw_AB_to_matrix_valued(G_iw_block)
d.chi_m = chi_ph_magnetic(beta=beta, U=U, nw=nw, nwf=nwf)
d.chi0_m = chi0_from_gg2_PH(g_mat, d.chi_m)
# -- Numeric vertex from BSE
d.gamma_m_num = inverse_PH(d.chi0_m) - inverse_PH(d.chi_m)
# -- Analytic vertex
d.gamma_m = gamma_ph_magnetic(beta=beta, U=U, nw=nw, nwf=nwf)
# -- Analytic magnetization expecation value
# -- and static susceptibility
d.Z = 2. + 2 * np.exp(-beta * 0.5 * U)
d.m2 = 0.25 * (2 / d.Z)
d.chi_m_static = 2. * beta * d.m2
d.label = r'Analytic'
return d