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
def make_calc():
# ------------------------------------------------------------------
# -- Hubbard atom with two bath sites, Hamiltonian
p = ParameterCollection(
beta=1.0,
U=5.0,
nw=1,
nwf=20,
)
p.nwf_gf = 4 * p.nwf
p.mu = 0.5 * p.U
# ------------------------------------------------------------------
ca_up, cc_up = c('0', 0), c_dag('0', 0)
ca_do, cc_do = c('0', 1), c_dag('0', 1)
docc = cc_up * ca_up * cc_do * ca_do
nA = cc_up * ca_up + cc_do * ca_do
p.H = -p.mu * nA + p.U * docc
# ------------------------------------------------------------------
# -- Exact diagonalization
# Conversion from TRIQS to Pomerol notation for operator indices
# TRIQS: block_name, inner_index
# Pomerol: site_label, orbital_index, spin_name
index_converter = {
('0', 0): ('loc', 0, 'up'),
('0', 1): ('loc', 0, 'down'),
}
# -- Create Exact Diagonalization instance
ed = PomerolED(index_converter, verbose=True)
ed.diagonalize(p.H) # -- Diagonalize H
gf_struct = [['0', [0, 1]]]
# -- Single-particle Green's functions
p.G_iw = ed.G_iw(gf_struct, p.beta, n_iw=p.nwf_gf)['0']
# -- Particle-particle two-particle Matsubara frequency Green's function
opt = dict(beta=p.beta,
gf_struct=gf_struct,
blocks=set([("0", "0")]),
n_iw=p.nw,
n_inu=p.nwf)
p.G2_iw_ph = ed.G2_iw_inu_inup(channel='PH', **opt)[('0', '0')]
# ------------------------------------------------------------------
# -- Generalized susceptibility in magnetic PH channel
p.chi_m = Gf(mesh=p.G2_iw_ph.mesh, target_shape=[1, 1, 1, 1])
p.chi_m[0, 0, 0, 0] = p.G2_iw_ph[0, 0, 0, 0] - p.G2_iw_ph[0, 0, 1, 1]
p.chi0_m = chi0_from_gg2_PH(p.G_iw, p.chi_m)
p.label = r'Pomerol'
# ------------------------------------------------------------------
# -- Generalized susceptibility in PH channel
p.chi = chi_from_gg2_PH(p.G_iw, p.G2_iw_ph)
p.chi0 = chi0_from_gg2_PH(p.G_iw, p.G2_iw_ph)
p.gamma = inverse_PH(p.chi0) - inverse_PH(p.chi)
# ------------------------------------------------------------------
# -- Store to hdf5
filename = 'data_pomerol.h5'
with HDFArchive(filename, 'w') as res:
res['p'] = p