def bubble_PI_wk(g_wk):
r""" Compute the particle-hole bubble from the single particle lattice Green's function
.. math::
\Pi_{abcd}(i\omega_n, k) = -
\mathcal{F}_{\tau, \mathbf{r} \rightarrow i\omega_n, \mathbf{k}}
\left\{ G_{d\bar{a}}(\tau, \mathbf{r}) G_{b\bar{c}}(-\tau, -\mathbf{r}) \right\}
Parameters
----------
g_wk : TRIQS Green's function (rank 2) on Matsubara and Brillouinzone meshes
Single particle lattice Green's function.
Returns
-------
PI_wk : TRIQS Green's function (rank 4) on Matsubara and Brillouinzone meshes
Particle hole bubble.
"""
nw = len(g_wk.mesh.components[0]) // 2
g_wr = fourier_wk_to_wr(g_wk)
g_tr = fourier_wr_to_tr(g_wr)
del g_wr
PI_tr = chi0_tr_from_grt_PH(g_tr)
del g_tr
PI_wr = chi_wr_from_chi_tr(PI_tr, nw=nw)
del PI_tr
PI_wk = chi_wk_from_chi_wr(PI_wr)
del PI_wr
return PI_wk
def bubble_setup(beta, mu, tb_lattice, nk, nw, sigma_w=None):
print tprf_banner(), "\n"
print 'beta =', beta
print 'mu =', mu
print 'sigma =', (not (sigma == None))
norb = tb_lattice.NOrbitalsInUnitCell
print 'nk =', nk
print 'nw =', nw
print 'norb =', norb
print
ntau = 4 * nw
ntot = np.prod(nk) * norb**4 + np.prod(nk) * (nw + ntau) * norb**2
nbytes = ntot * np.complex128().nbytes
ngb = nbytes / 1024.**3
print 'Approx. Memory Utilization: %2.2f GB\n' % ngb
periodization_matrix = np.diag(np.array(list(nk), dtype=np.int32))
#print 'periodization_matrix =\n', periodization_matrix
bz = BrillouinZone(tb_lattice.bl)
bzmesh = MeshBrillouinZone(bz, periodization_matrix)
print '--> ek'
e_k = ek_tb_dispersion_on_bzmesh(tb_lattice, bzmesh, bz)
if sigma is None:
print '--> g0k'
wmesh = MeshImFreq(beta=beta, S='Fermion', n_max=nw)
g_wk = lattice_dyson_g0_wk(mu=mu, e_k=e_k, mesh=wmesh)
else:
print '--> gk'
sigma_w = strip_sigma(nw, beta, sigma)
g_wk = lattice_dyson_g_wk(mu=mu, e_k=e_k, sigma_w=sigma_w)
print '--> gr_from_gk (k->r)'
g_wr = fourier_wk_to_wr(g_wk)
del g_wk
print '--> grt_from_grw (w->tau)'
g_tr = fourier_wr_to_tr(g_wr)
del g_wr
if sigma is None:
return g_tr
else:
return g_tr, sigma_w
def imtime_bubble_chi0_wk(g_wk, nw=1):
wmesh, kmesh = g_wk.mesh.components
norb = g_wk.target_shape[0]
beta = wmesh.beta
nw_g = len(wmesh)
nk = len(kmesh)
ntau = 4 * nw_g
ntot = np.prod(nk) * norb**4 + np.prod(nk) * (nw_g + ntau) * norb**2
nbytes = ntot * np.complex128().nbytes
ngb = nbytes / 1024.**3
if mpi.is_master_node():
print tprf_banner(), "\n"
print 'beta =', beta
print 'nk =', nk
print 'nw =', nw_g
print 'norb =', norb
print
print 'Approx. Memory Utilization: %2.2f GB\n' % ngb
mpi.report('--> fourier_wk_to_wr')
g_wr = fourier_wk_to_wr(g_wk)
del g_wk
mpi.report('--> fourier_wr_to_tr')
g_tr = fourier_wr_to_tr(g_wr)
del g_wr
if nw == 1:
mpi.report('--> chi0_w0r_from_grt_PH (bubble in tau & r)')
chi0_wr = chi0_w0r_from_grt_PH(g_tr)
del g_tr
else:
mpi.report('--> chi0_tr_from_grt_PH (bubble in tau & r)')
chi0_tr = chi0_tr_from_grt_PH(g_tr)
del g_tr
mpi.report('--> chi_wr_from_chi_tr')
chi0_wr = chi_wr_from_chi_tr(chi0_tr, nw=nw)
del chi_tr
mpi.report('--> chi_wk_from_chi_wr (r->k)')
chi0_wk = chi_wk_from_chi_wr(chi0_wr)
del chi0_wr
return chi0_wk
def test_square_lattice_chi00():
# ------------------------------------------------------------------
# -- Discretizations
n_k = (2, 2, 1)
nw_g = 500
nnu = 400
nw = 1
# ------------------------------------------------------------------
# -- tight binding parameters
beta = 20.0
mu = 0.0
t = 1.0
h_loc = np.array([
[-0.3, -0.5],
[-0.5, .4],
])
T = -t * np.array([
[1., 0.23],
[0.23, 0.5],
])
# ------------------------------------------------------------------
# -- tight binding
print '--> tight binding model'
t_r = TBLattice(
units=[(1, 0, 0), (0, 1, 0)],
hopping={
# nearest neighbour hopping -t
(0, 0): h_loc,
(0, +1): T,
(0, -1): T,
(+1, 0): T,
(-1, 0): T,
},
orbital_positions=[(0, 0, 0)] * 2,
orbital_names=['up_0', 'do_0'],
)
e_k = t_r.on_mesh_brillouin_zone(n_k)
kmesh = e_k.mesh
wmesh = MeshImFreq(beta=beta, S='Fermion', n_max=nw_g)
print '--> g0_wk'
g0_wk = lattice_dyson_g0_wk(mu=mu, e_k=e_k, mesh=wmesh)
print '--> g0_wr'
g0_wr = fourier_wk_to_wr(g0_wk)
print '--> g0_tr'
g0_tr = fourier_wr_to_tr(g0_wr)
# ------------------------------------------------------------------
# -- anaytic chi00
print '--> chi00_wk analytic'
chi00_wk_analytic = lindhard_chi00_wk(e_k=e_k, nw=nw, beta=beta, mu=mu)
print '--> chi00_wr analytic'
chi00_wr_analytic = chi_wr_from_chi_wk(chi00_wk_analytic)
# ------------------------------------------------------------------
# -- imtime chi00
print '--> chi0_tr_from_grt_PH'
chi00_tr = chi0_tr_from_grt_PH(g0_tr)
print '--> chi_wr_from_chi_tr'
chi00_wr = chi_wr_from_chi_tr(chi00_tr, nw=1)
print '--> chi_w0r_from_chi_tr'
chi00_wr_ref = chi_w0r_from_chi_tr(chi00_tr)
print '--> chi0_w0r_from_grt_PH'
chi00_wr_opt = chi0_w0r_from_grt_PH(g0_tr)
print 'dchi00_wr =', np.max(
np.abs(chi00_wr_analytic.data - chi00_wr.data))
print 'dchi00_wr_ref =', np.max(
np.abs(chi00_wr_analytic.data - chi00_wr_ref.data))
print 'dchi00_wr_opt =', np.max(
np.abs(chi00_wr_analytic.data - chi00_wr_opt.data))
np.testing.assert_array_almost_equal(chi00_wr_analytic.data,
chi00_wr.data,
decimal=8)
np.testing.assert_array_almost_equal(chi00_wr_analytic.data,
chi00_wr_ref.data,
decimal=4)
np.testing.assert_array_almost_equal(chi00_wr_analytic.data,
chi00_wr_opt.data,
decimal=4)
print '--> chi_wk_from_chi_wr'
chi00_wk_imtime = chi_wk_from_chi_wr(chi00_wr)
# ------------------------------------------------------------------
# -- imtime chi00 helper function
chi00_wk_imtime_2 = imtime_bubble_chi0_wk(g0_wk, nw=1)
# ------------------------------------------------------------------
# -- imfreq chi00
print '--> chi00_wnr'
chi00_wnr = chi0r_from_gr_PH(nw=1, nnu=nnu, gr=g0_wr)
print '--> chi00_wnk'
chi00_wnk = chi0q_from_chi0r(chi00_wnr)
# -- Test per k and w calculator for chi0_wnk
print '--> chi00_wnk_ref'
from triqs_tprf.lattice import chi0q_from_g_wk_PH
chi00_wnk_ref = chi0q_from_g_wk_PH(nw=1, nnu=nnu, g_wk=g0_wk)
diff = np.max(np.abs(chi00_wnk.data - chi00_wnk_ref.data))
print 'chi00_wnk diff =', diff
np.testing.assert_array_almost_equal(chi00_wnk.data, chi00_wnk_ref.data)
print '--> chi00_wk_imfreq'
chi00_wk_imfreq = chi0q_sum_nu(chi00_wnk)
print '--> chi00_wk_imfreq_tail_corr'
chi00_wk_imfreq_tail_corr = chi0q_sum_nu_tail_corr_PH(chi00_wnk)
# ------------------------------------------------------------------
# -- Compare results
def cf_chi_w0(chi1, chi2, decimal=9):
chi1, chi2 = chi1[Idx(0), :].data, chi2[Idx(0), :].data
diff = np.linalg.norm(chi1 - chi2)
print '|dchi| =', diff
np.testing.assert_array_almost_equal(chi1, chi2, decimal=decimal)
print '--> Cf analytic with imtime'
cf_chi_w0(chi00_wk_analytic, chi00_wk_imtime, decimal=7)
print '--> Cf analytic with imtime 2'
cf_chi_w0(chi00_wk_analytic, chi00_wk_imtime_2, decimal=4)
print '--> Cf analytic with imfreq'
cf_chi_w0(chi00_wk_analytic, chi00_wk_imfreq, decimal=2)
print '--> Cf analytic with imfreq (tail corr)'
cf_chi_w0(chi00_wk_analytic, chi00_wk_imfreq_tail_corr, decimal=5)
def imtime_bubble_chi0_wk(g_wk, nw=1):
ncores = multiprocessing.cpu_count()
wmesh, kmesh = g_wk.mesh.components
norb = g_wk.target_shape[0]
beta = wmesh.beta
nw_g = len(wmesh)
nk = len(kmesh)
ntau = 2 * nw_g
# -- Memory Approximation
ng_tr = ntau * np.prod(nk) * norb**2 # storing G(tau, r)
ng_wr = nw_g * np.prod(nk) * norb**2 # storing G(w, r)
ng_t = ntau * norb**2 # storing G(tau)
nchi_tr = ntau * np.prod(nk) * norb**4 # storing \chi(tau, r)
nchi_wr = nw * np.prod(nk) * norb**4 # storing \chi(w, r)
nchi_t = ntau * norb**4 # storing \chi(tau)
nchi_w = nw * norb**4 # storing \chi(w)
nchi_r = np.prod(nk) * norb**4 # storing \chi(r)
if nw == 1:
ntot_case_1 = ng_tr + ng_wr
ntot_case_2 = ng_tr + nchi_wr + ncores * (nchi_t + 2 * ng_t)
ntot_case_3 = 4 * nchi_wr
ntot = max(ntot_case_1, ntot_case_2, ntot_case_3)
else:
ntot_case_1 = ng_tr + nchi_tr + ncores * (nchi_t + 2 * ng_t)
ntot_case_2 = nchi_tr + nchi_wr + ncores * (nchi_w + nchi_t)
ntot = max(ntot_case_1, ntot_case_2)
nbytes = ntot * np.complex128().nbytes
ngb = nbytes / 1024.**3
if mpi.is_master_node():
print tprf_banner(), "\n"
print 'beta =', beta
print 'nk =', nk
print 'nw =', nw_g
print 'norb =', norb
print
print 'Approx. Memory Utilization: %2.2f GB\n' % ngb
mpi.report('--> fourier_wk_to_wr')
g_wr = fourier_wk_to_wr(g_wk)
del g_wk
mpi.report('--> fourier_wr_to_tr')
g_tr = fourier_wr_to_tr(g_wr)
del g_wr
if nw == 1:
mpi.report('--> chi0_w0r_from_grt_PH (bubble in tau & r)')
chi0_wr = chi0_w0r_from_grt_PH(g_tr)
del g_tr
else:
mpi.report('--> chi0_tr_from_grt_PH (bubble in tau & r)')
chi0_tr = chi0_tr_from_grt_PH(g_tr)
del g_tr
mpi.report('--> chi_wr_from_chi_tr')
chi0_wr = chi_wr_from_chi_tr(chi0_tr, nw=nw)
del chi0_tr
mpi.report('--> chi_wk_from_chi_wr (r->k)')
chi0_wk = chi_wk_from_chi_wr(chi0_wr)
del chi0_wr
return chi0_wk
def gw_sigma_wk(Wr_wk, g_wk, fft_flag=False):
r""" GW self energy :math:`\Sigma(i\omega_n, \mathbf{k})` calculator
Fourier transforms the screened interaction and the single-particle
Green's function to imagiary time and real space.
.. math::
G_{ab}(\tau, \mathbf{r}) = \mathcal{F}^{-1}
\left\{ G_{ab}(i\omega_n, \mathbf{k}) \right\}
.. math::
W^{(r)}_{abcd}(\tau, \mathbf{r}) = \mathcal{F}^{-1}
\left\{ W^{(r)}_{abcd}(i\omega_n, \mathbf{k}) \right\}
computes the GW self-energy as the product
.. math::
\Sigma_{ab}(\tau, \mathbf{r}) =
\sum_{cd} W^{(r)}_{abcd}(\tau, \mathbf{r}) G_{cd}(\tau, \mathbf{r})
and transforms back to frequency and momentum
.. math::
\Sigma_{ab}(i\omega_n, \mathbf{k}) =
\mathcal{F} \left\{ \Sigma_{ab}(\tau, \mathbf{r}) \right\}
Parameters
----------
V_k : TRIQS Green's function (rank 4) on a Brillouinzone mesh
static bare interaction :math:`V_{abcd}(\mathbf{k})`
Wr_wk : TRIQS Green's function (rank 4) on Matsubara and Brillouinzone meshes
retarded screened interaction :math:`W^{(r)}_{abcd}(i\omega_n, \mathbf{k})`
g_wk : TRIQS Green's function (rank 2) on Matsubara and Brillouinzone meshes
single particle Green's function :math:`G_{ab}(i\omega_n, \mathbf{k})`
Returns
-------
sigma_wk : TRIQS Green's function (rank 2) on Matsubara and Brillouinzone meshes
GW self-energy :math:`\Sigma_{ab}(i\omega_n, \mathbf{k})`
"""
if fft_flag:
nw = len(g_wk.mesh.components[0]) // 2
ntau = nw * 6 + 1
mpi.report('g wk -> wr')
g_wr = fourier_wk_to_wr(g_wk)
mpi.report('g wr -> tr')
g_tr = fourier_wr_to_tr(g_wr, nt=ntau)
del g_wr
mpi.report('W wk -> wr')
Wr_wr = chi_wr_from_chi_wk(Wr_wk)
mpi.report('W wr -> tr')
Wr_tr = chi_tr_from_chi_wr(Wr_wr, ntau=ntau)
del Wr_wr
mpi.report('sigma tr')
sigma_tr = cpp_gw_sigma_tr(Wr_tr, g_tr)
del Wr_tr
del g_tr
mpi.report('sigma tr -> wr')
sigma_wr = fourier_tr_to_wr(sigma_tr, nw=nw)
del sigma_tr
mpi.report('sigma wr -> wk')
sigma_wk = fourier_wr_to_wk(sigma_wr)
del sigma_wr
else:
sigma_wk = cpp_gw_sigma_wk_serial_fft(Wr_wk, g_wk)
return sigma_wk
