##########################################################################
#
# TRIQS: a Toolbox for Research in Interacting Quantum Systems
#
# Copyright (C) 2011 by M. Aichhorn, L. Pourovskii, V. Vildosola
# Copyright (c) 2022-2023 Simons Foundation
#
# TRIQS is free software: you can redistribute it and/or modify it under the
# terms of the GNU General Public License as published by the Free Software
# Foundation, either version 3 of the License, or (at your option) any later
# version.
#
# TRIQS is distributed in the hope that it will be useful, but WITHOUT ANY
# WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
# FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
# details.
#
# You should have received a copy of the GNU General Public License along with
# TRIQS. If not, see .
#
# Authors: M. Aichhorn, S. Beck, A. Hampel, L. Pourovskii, V. Vildosola
##########################################################################
import sys
import numpy
from warnings import warn
from triqs.gf import *
import triqs.utility.mpi as mpi
from .symmetry import *
import scipy.constants as cst
import os.path
__all__ = ['transport_distribution', 'conductivity_and_seebeck', 'write_output_to_hdf',
'init_spectroscopy', 'transport_function']
# ----------------- helper functions -----------------------
def read_transport_input_from_hdf(sum_k):
r"""
Reads the data for transport calculations from the hdf5 archive.
Parameters
----------
sum_k : sum_k object
triqs SumkDFT object
Returns
-------
sum_k : sum_k object
triqs SumkDFT object
"""
assert sum_k.dft_code in (
'wien2k', 'elk', 'w90'), "read_transport_input_from_hdf() is only implemented for wien2k and elk inputs"
if sum_k.dft_code in ('wien2k', 'elk'):
thingstoread = ['band_window_optics', 'velocities_k']
else:
thingstoread = ['band_window_optics']
sum_k.read_input_from_hdf(subgrp=sum_k.transp_data, things_to_read=thingstoread)
if (sum_k.dft_code == "wien2k"):
thingstoread = ['band_window', 'lattice_angles', 'lattice_constants',
'lattice_type', 'n_symmetries', 'rot_symmetries']
elif (sum_k.dft_code == "elk"):
thingstoread = ['band_window', 'n_symmetries', 'rot_symmetries',
'cell_vol']
elif (sum_k.dft_code == 'w90'):
thingstoread = ['band_window', 'n_symmetries', 'rot_symmetries']
sum_k.read_input_from_hdf(subgrp=sum_k.misc_data, things_to_read=thingstoread)
if (sum_k.dft_code == "wien2k"):
sum_k.cell_vol = cellvolume(
sum_k.lattice_type, sum_k.lattice_constants, sum_k.lattice_angles)[1]
return sum_k
def write_output_to_hdf(sum_k, things_to_save, subgrp='user_data'):
r"""
Saves data from a list into the HDF file. Prints a warning if a requested data is not found in SumkDFT object.
Parameters
----------
hdf_file : hdf5 archive
hd5 file
things_to_save : list of strings
List of datasets to be saved into the hdf5 file.
subgrp : string, optional
Name of hdf5 file subgroup in which the data are to be stored.
"""
if not (mpi.is_master_node()):
return # do nothing on nodes
with HDFArchive(sum_k.hdf_file, 'a') as ar:
if not subgrp in ar:
ar.create_group(subgrp)
for it, val in things_to_save.items():
if it in ["gf_struct_sumk", "gf_struct_solver",
"solver_to_sumk", "sumk_to_solver", "solver_to_sumk_block"]:
warn("It is not recommended to save '{}' individually. Save 'block_structure' instead.".format(it))
ar[subgrp][it] = val
def cellvolume(lattice_type, lattice_constants, latticeangle):
r"""
Determines the conventional und primitive unit cell volumes.
Parameters
----------
lattice_type : string
Lattice type according to the Wien2k convention (P, F, B, R, H, CXY, CYZ, CXZ).
lattice_constants : list of double
Lattice constants (a, b, c).
lattice angles : list of double
Lattice angles (:math:`\alpha, \beta, \gamma`).
Returns
-------
vol_c : double
Conventional unit cell volume.
vol_p : double
Primitive unit cell volume.
"""
a = lattice_constants[0]
b = lattice_constants[1]
c = lattice_constants[2]
c_al = numpy.cos(latticeangle[0])
c_be = numpy.cos(latticeangle[1])
c_ga = numpy.cos(latticeangle[2])
vol_c = a * b * c * \
numpy.sqrt(1 + 2 * c_al * c_be * c_ga -
c_al ** 2 - c_be ** 2 - c_ga ** 2)
det = {"P": 1, "F": 4, "B": 2, "R": 3,
"H": 1, "CXY": 2, "CYZ": 2, "CXZ": 2}
vol_p = vol_c / det[lattice_type]
return vol_c, vol_p
def fermi_dis(w, beta, der=0):
r"""
Fermi distribution.
.. math::
f(x) = 1/(e^x+1).
Parameters
----------
w : double
frequency
beta : double
inverse temperature
der : integer
order of derivative
Returns
-------
f : double
"""
exponent = numpy.longdouble(w * beta)
fermi = 1.0 / (numpy.exp(exponent) + 1)
if der == 0:
return fermi
elif der == 1:
return - beta * fermi ** 2 * numpy.exp(exponent)
else:
raise ValueError('higher order of derivative than 1 not implemented')
def recompute_w90_input_on_different_mesh(sum_k, seedname, nk_optics, pathname='./', calc_velocity=False, calc_inverse_mass=False, oc_select='both', oc_basis='h'):
r"""
Recomputes dft_input objects on a finer mesh using WannierBerri and Wannier90 input.
Parameters
----------
sum_k : sum_k object
triqs SumkDFT object
seedname: string
Wannier90 seedname
nk_optics: single integer/float or three integers
if single integer given, mesh is [nk_optics, nk_optics, nk_optics]
elif single float given, mesh is ceiling of *sum_k.kpts * nk_optics
elif three integers given, mesh is nk_optics
pathname : string, optional, default='./'
location of Wannier90 data
calc_velocity : boolean, optional, default=False
whether the velocity (first derivative of H(k)) is computed
calc_inverse_mass : boolean, optional, default=False
whether the inverse effective mass (second derivative of H(k)) is computed
oc_select : string, optional, default='both'
select contributions for optical conductivity from ['intra', 'inter', 'both']
oc_basis : string, optional, default='h'
gauge choice options 'h' for Hamiltonian/band and 'w' for Wannier basis
Returns
-------
sum_k : sum_k object
triqs SumkDFT object
things_to_store : dictionary
dictionary of datasets to be temporarily overwritten
"""
mpi.report('Starting Wannier interpolation...')
BOHRTOANG = cst.physical_constants['Bohr radius'][0]/cst.angstrom
HARTREETOEV = cst.physical_constants['Hartree energy'][0]/cst.eV
n_inequiv_spin_blocks = sum_k.SP + 1 - sum_k.SO
# set-up k mesh depending on input shape
# read in transport input and some checks
read_transport_input_from_hdf(sum_k)
# first check for right formatting of sum_k.nk_optics
assert len(nk_optics) in [1, 3], '"nk_optics" must be given as three integers or one float'
if len(nk_optics) == 1:
assert numpy.array(list(nk_optics)).dtype in (
int, float), '"nk_optics" single value must be float or integer'
if len(nk_optics) == 3:
assert numpy.array(list(nk_optics)).dtype == int, '"nk_optics" mesh must be integers'
if len(nk_optics) == 1:
interpolate_factor = nk_optics[0]
nk_x, nk_y, nk_z = list(
map(lambda i: int(numpy.ceil(interpolate_factor * len(set(sum_k.kpts[:, i])))), range(3)))
else:
nk_x, nk_y, nk_z = nk_optics
# check for spin calculation (not supported)
assert sum_k.SP == 0, 'spin dependent transport calculations are not supported.'
n_orb = numpy.max([sum_k.n_orbitals[ik][0] for ik in range(sum_k.n_k)])
# temporarily recompute the following quantities on a different mesh
things_to_modify = {'bz_weights': None, 'hopping': None, 'kpt_weights': None, 'kpts': None,
'n_k': None, 'n_orbitals': None, 'proj_mat': None, 'band_window': None, 'band_window_optics': None}
things_to_store = dict.fromkeys(things_to_modify, None)
# initialize variables
n_kpts = nk_x * nk_y * nk_z
kpts = numpy.zeros((n_kpts, 3))
hopping = numpy.zeros((n_kpts, 1, n_orb, n_orb), dtype=complex)
proj_mat = numpy.zeros(numpy.shape(
hopping[:, 0, 0, 0]) + numpy.shape(sum_k.proj_mat[0, :]), dtype=complex)
cell_volume = kpts = None
if calc_velocity:
velocities_k = None
if calc_inverse_mass:
inverse_mass = None
if mpi.is_master_node():
# try wannierberri import
try:
import wannierberri as wb
except ImportError:
print('ImportError: WannierBerri needs to be installed to run test "Py_w90_optics_Sr2RuO4"')
try:
mpi.MPI.COMM_WORLD.Abort(1)
except:
sys.exit()
# initialize WannierBerri system
shift_gamma = numpy.array([0.0, 0.0, 0.0])
# wberri = wb.System_w90(pathname + seedname, berry=True, fft='numpy')
# WannierBerri uses python multiprocessing which might conflict with mpi.
# if there's a segfault, uncomment the following line
wberri = wb.System_w90(pathname + seedname, berry=True, fft='numpy', npar=16)
grid = wb.Grid(wberri, NKdiv=1, NKFFT=[nk_x, nk_y, nk_z])
dataK = wb.data_K.Data_K(wberri, dK=shift_gamma, grid=grid, fftlib='numpy')
assert dataK.HH_K.shape == hopping[:, 0, :, :].shape, 'wberri / wannier Hamiltonian has different number of orbitals than SumK object. Disentanglement is not supported as of now.'
# read in hoppings and proj_mat
if oc_basis == 'h':
hopping[:, 0, range(hopping.shape[2]), range(hopping.shape[3])] = dataK.E_K
elif oc_basis == 'w':
hopping[:, 0, :, :] = dataK.HH_K
fake_proj_mat = numpy.zeros(numpy.shape(dataK.UU_K), dtype=complex)
fake_proj_mat[:, range(numpy.shape(fake_proj_mat)[1]), range(
numpy.shape(fake_proj_mat)[2])] = 1. + 1j*0.0
for isp in range(n_inequiv_spin_blocks):
iorb = 0
for icrsh in range(sum_k.n_corr_shells):
dim = sum_k.corr_shells[icrsh]['dim']
if oc_basis == 'h':
proj_mat[:, isp, icrsh, 0:dim, :] = dataK.UU_K[:, iorb:iorb+dim, :]
elif oc_basis == 'w':
proj_mat[:, isp, icrsh, 0:dim, :] = fake_proj_mat[:, iorb:iorb+dim, :]
iorb += dim
if calc_velocity:
# velocity: [k x n_orb x n_orb x R]
def _commutator(A, B):
term1 = numpy.einsum('kmo, kona -> kmna', A, B)
term2 = numpy.einsum('kmoa, kon -> kmna', B, A)
return term1 - term2
# in the band basis
# vh_alpha = Hhbar_alpha + i [Hh, Ahbar_alpha]
if oc_basis == 'h':
# first term
Hhbar_alpha = dataK.Xbar('Ham', 1)
# second term
c_Hh_Ahbar_alpha = _commutator(hopping[:, 0, :, :], dataK.Xbar('AA'))
velocities_k = (Hhbar_alpha + 1j * c_Hh_Ahbar_alpha) / HARTREETOEV / BOHRTOANG
# split into diag and offdiag elements, corresponding to intra- and interband contributions
v_diag = numpy.zeros(numpy.shape(velocities_k), dtype=complex)
v_diag[:, range(numpy.shape(velocities_k)[1]),
range(numpy.shape(velocities_k)[2]), :] = velocities_k[:, range(numpy.shape(velocities_k)[1]),
range(numpy.shape(velocities_k)[2]), :]
v_offdiag = velocities_k.copy()
v_offdiag[:, range(numpy.shape(velocities_k)[1]), range(
numpy.shape(velocities_k)[2]), :] = 0. + 1j*0.0
if oc_select == 'intra':
velocities_k = v_diag
elif oc_select == 'inter':
velocities_k = v_offdiag
elif oc_select == 'both':
velocities_k = v_diag + v_offdiag
# in the orbital basis
# vw_alpha = Hw_alpha + i [Hw, Aw_alpha]
elif oc_basis == 'w':
# first term
Hw_alpha_R = dataK.Ham_R.copy()
# following three lines copied from wannierberri/data_K.py
shape_cR = numpy.shape(dataK.cRvec_wcc)
Hw_alpha_R = 1j * Hw_alpha_R.reshape((Hw_alpha_R.shape) + (1, )) * dataK.cRvec_wcc.reshape(
(shape_cR[0], shape_cR[1], dataK.system.nRvec) + (1, ) * len(Hw_alpha_R.shape[3:]) + (3, ))
Hw_alpha = dataK.fft_R_to_k(Hw_alpha_R, hermitean=False)[dataK.select_K]
# second term
Aw_alpha = dataK.fft_R_to_k(dataK.AA_R, hermitean=True)
c_Hw_Aw_alpha = _commutator(hopping[:, 0, :, :], Aw_alpha)
velocities_k = (Hw_alpha + 1j * c_Hw_Aw_alpha) / HARTREETOEV / BOHRTOANG
if calc_inverse_mass:
V_dot_D = numpy.einsum('kmnab, knoab -> kmoab', dataK.Xbar('Ham', 1)
[:, :, :, :, None], dataK.D_H[:, :, :, None, :])
V_dot_D_dagger = V_dot_D.conj().transpose(0, 2, 1, 3, 4)
V_curly = numpy.einsum('knnab -> knab', V_dot_D + V_dot_D_dagger)
del2E_H_diag = numpy.einsum('knnab->knab', dataK.Xbar('Ham', 2)).real
inverse_mass = del2E_H_diag + V_curly
# read in rest from dataK
cell_volume = dataK.cell_volume / BOHRTOANG ** 3
kpts = dataK.kpoints_all
# broadcast everything
sum_k.cell_vol = mpi.bcast(cell_volume)
kpts = mpi.bcast(kpts)
hopping = mpi.bcast(hopping)
proj_mat = mpi.bcast(proj_mat)
if calc_velocity:
velocities_k = mpi.bcast(velocities_k)
if calc_inverse_mass:
inverse_mass = mpi.bcast(inverse_mass)
# write interpolated sumk quantities into "things_to_modify"
things_to_modify['n_k'] = n_kpts
things_to_modify['n_orbitals'] = numpy.full((n_kpts, 1), n_orb)
for key in ['bz_weights', 'kpt_weights']:
things_to_modify[key] = numpy.full(n_kpts, 1/n_kpts)
n_inequiv_spin_blocks = sum_k.SP + 1 - sum_k.SO
for key in ['band_window', 'band_window_optics']:
things_to_modify[key] = [numpy.full((n_kpts, 2), sum_k.band_window[isp][0])
for isp in range(n_inequiv_spin_blocks)]
things_to_modify['kpts'] = kpts
things_to_modify['hopping'] = hopping
things_to_modify['proj_mat'] = proj_mat
if calc_velocity:
sum_k.velocities_k = None
things_to_modify['velocities_k'] = velocities_k
if calc_inverse_mass:
sum_k.inverse_mass = None
things_to_modify['inverse_mass'] = inverse_mass
# now save previous sum_k instances into "things_to_store" and overwrite
# TODO: decide whether this should be undone after the run
for key in things_to_modify:
things_to_store[key] = getattr(sum_k, key)
setattr(sum_k, key, things_to_modify[key])
# write velocities to file
if calc_velocity:
if mpi.is_master_node():
ar = HDFArchive(sum_k.hdf_file, 'a')
ar['dft_transp_input']['velocities_k'] = velocities_k
if calc_inverse_mass:
if mpi.is_master_node():
ar = HDFArchive(sum_k.hdf_file, 'a')
ar['dft_transp_input']['inverse_mass'] = inverse_mass
return sum_k, things_to_store
# ----------------- transport -----------------------
def init_spectroscopy(sum_k, code='wien2k', w90_params={}):
r"""
Reads all necessary quantities for transport calculations from transport subgroup of the hdf5 archive.
Performs checks on input. Uses interpolation if code=wannier90.
Parameters
----------
sum_k : sum_k object
triqs SumkDFT object
code : string
DFT code from which velocities are being read. Options: 'wien2k', 'wannier90'
w90_params : dictionary, optional
additional keywords necessary in case code == 'wannier90'
Returns
-------
sum_k : sum_k object
triqs SumkDFT object, interpolated
"""
mpi.report('Initializing optical conductivity...')
# up and down are equivalent if SP = 0
n_inequiv_spin_blocks = sum_k.SP + 1 - sum_k.SO
# ----------------- set-up input from DFT -----------------------
if code in ('wien2k', 'elk'):
# Check if wien converter was called and read transport subgroup form
# hdf file
if mpi.is_master_node():
ar = HDFArchive(sum_k.hdf_file, 'r')
if not (sum_k.transp_data in ar):
raise IOError(
"transport_distribution: No %s subgroup in hdf file found! Call convert_transp_input first." % sum_k.transp_data)
# check if outputs file was converted
if not ('n_symmetries' in ar['dft_misc_input']):
raise IOError(
"transport_distribution: n_symmetries missing. Check if case.outputs file is present and call convert_misc_input() or convert_dft_input().")
sum_k = read_transport_input_from_hdf(sum_k)
elif code in ('wannier90'):
required_entries = ['seedname', 'nk_optics']
assert all(entry in w90_params for entry in required_entries), 'Please provide additional keywords "seedname" and "nk_optics" for "code = "wannier90""'
# check if spin-unpolarized
assert n_inequiv_spin_blocks == 1, "Spin-polarized optical conductivity calculations not implemented with Wannier90"
# check some of the input
pathname = w90_params['pathname'] if 'pathname' in w90_params else './'
assert all(isinstance(name, str) for name in [
'seedname', 'pathname']), f'Check pathname {w90_params["pathname"]} and seedname {w90_params["seedname"]}'
for file_ending in ['.wout', '_hr.dat', '.chk', '.mmn', '.eig']:
filename = [pathname, w90_params['seedname'], file_ending]
assert os.path.isfile(
''.join(filename)), f'Filename {"".join(filename)} does not exist!'
calc_velocity = w90_params['calc_velocity'] if 'calc_velocity' in w90_params else True
calc_inverse_mass = w90_params['calc_inverse_mass'] if 'calc_inverse_mass' in w90_params else False
assert all(isinstance(name, bool) for name in [
calc_velocity, calc_inverse_mass]), f'Parameter {calc_velocity} or {calc_inverse_mass} not bool!'
# select contributions to be used
oc_select = w90_params['oc_select'] if 'oc_select' in w90_params else 'both'
assert oc_select in ['intra', 'inter',
'both'], '"oc_select" needs to be either ["intra", "inter", "both"]'
# gauge choice options 'h' for Hamiltonian and 'w' for Wannier
oc_basis = w90_params['oc_basis'] if 'oc_basis' in w90_params else 'h'
assert oc_basis in ['h', 'w'], '"oc_basis" needs to be either ["h", "w"]'
# finally, make sure oc_select is 'both' for oc_basis = 'w'
if oc_basis == 'w' and oc_select != 'both':
warn(f'"oc_select" must be "both" for "oc_basis" = "w"!')
oc_select = 'both'
# further checks for calc_inverse_mass
if calc_inverse_mass:
assert oc_basis == 'h', '"calc_inverse_mass" only implemented for "oc_basis" == "h"'
assert oc_select == 'both', '"oc_select" not implemented for "calc_inverse_mass"'
# print some information
mpi.report(f'{"Basis choice [h (Hamiltonian), w (Wannier)]:":<60s} {oc_basis}')
mpi.report(f'{"Contributions from [intra(-band), inter(-band), both]:":<60s} {oc_select}')
# recompute sum_k instances on denser grid
sum_k, _ = recompute_w90_input_on_different_mesh(sum_k, w90_params['seedname'], nk_optics=w90_params['nk_optics'], pathname=pathname,
calc_velocity=calc_velocity, calc_inverse_mass=calc_inverse_mass, oc_select=oc_select, oc_basis=oc_basis)
# k-dependent-projections.
# to be checked. But this should be obsolete atm, works for both cases
# k_dep_projection is nowhere used
# assert sum_k.k_dep_projection == 0, "transport_distribution: k dependent projection is not implemented!"
return sum_k
# Uses .data of only GfReFreq objects.
def transport_distribution(sum_k, beta, directions=['xx'], energy_window=None, Om_mesh=[0.0], with_Sigma=False, n_om=None, broadening=0.0, code='wien2k'):
r"""
Calculates the transport distribution
.. math::
\Gamma_{\alpha\beta}\left(\omega+\Omega/2, \omega-\Omega/2\right) = \frac{1}{V} \sum_k Tr\left(v_{k,\alpha}A_{k}(\omega+\Omega/2)v_{k,\beta}A_{k}\left(\omega-\Omega/2\right)\right)
in the direction :math:`\alpha\beta`. The velocities :math:`v_{k}` are read from the transport subgroup of the hdf5 archive.
Parameters
----------
sum_k : sum_k object
triqs SumkDFT object
beta : double
Inverse temperature :math:`\beta`.
directions : list of string, optional
:math:`\alpha\beta` e.g.: ['xx','yy','zz','xy','xz','yz'].
energy_window : list of double, optional
Specifies the upper and lower limit of the frequency integration for :math:`\Omega=0.0`. The window is automatically enlarged by the largest :math:`\Omega` value,
hence the integration is performed in the interval [energy_window[0]-max(Om_mesh), energy_window[1]+max(Om_mesh)].
Om_mesh : list of double, optional
:math:`\Omega` frequency mesh of the optical conductivity. For the conductivity and the Seebeck coefficient :math:`\Omega=0.0` has to be
part of the mesh. In the current version Om_mesh is repined to the mesh provided by the self-energy! The actual mesh is printed on the screen and given as output.
with_Sigma : boolean, optional
Determines whether the calculation is performed with or without self energy. If this parameter is set to False the self energy is set to zero (i.e. the DFT band
structure :math:`A(k,\omega)` is used). Note: For with_Sigma=False it is necessary to specify the parameters energy_window, n_om and broadening.
n_om : integer, optional
Number of equidistant frequency points in the interval [energy_window[0]-max(Om_mesh), energy_window[1]+max(Om_mesh)]. This parameters is only used if
with_Sigma = False.
broadening : double, optional
Lorentzian broadening. It is necessary to specify the boradening if with_Sigma = False, otherwise this parameter can be set to 0.0.
code : string
DFT code from which velocities are being read. Options: 'wien2k', 'wannier90'
Returns
-------
Gamma_w : dictionary of double matrices
transport distribution function in each direction, frequency given by Om_mesh_out and omega
omega : list of double
omega vector
Om_mesh_out : list of double
frequency mesh of the optical conductivity recomputed on the mesh provided by the self energy
"""
mpi.report('Computing transport distribution...')
n_inequiv_spin_blocks = sum_k.SP + 1 - sum_k.SO
# up and down are equivalent if SP = 0
# positive om_mesh
assert all(
Om >= 0.0 for Om in Om_mesh), "transport_distribution: Om_mesh should not contain negative values!"
# Check if energy_window is sufficiently large and correct
if (energy_window[0] >= energy_window[1] or energy_window[0] >= 0 or energy_window[1] <= 0):
assert 0, "transport_distribution: energy_window wrong!"
if (abs(fermi_dis(energy_window[0], beta) * fermi_dis(-energy_window[0], beta)) > 1e-5
or abs(fermi_dis(energy_window[1], beta) * fermi_dis(-energy_window[1], beta)) > 1e-5):
mpi.report(
"\n####################################################################")
mpi.report(
"transport_distribution: WARNING - energy window might be too narrow!")
mpi.report(
"####################################################################\n")
# ----------------- calculate A(k,w) -----------------------
# Define mesh for Green's function and in the specified energy window
if (with_Sigma == True):
omega = numpy.array([round(x.real, 12)
for x in sum_k.Sigma_imp[0].mesh])
mesh = None
mu = sum_k.chemical_potential
n_om = len(omega)
mpi.report("Using omega mesh provided by Sigma!")
if energy_window:
# Find according window in Sigma mesh
ioffset = numpy.sum(
omega < energy_window[0] - max(Om_mesh))
omega = omega[numpy.logical_and(
omega >= energy_window[0] - max(Om_mesh), omega <= energy_window[1] + max(Om_mesh))]
n_om = len(omega)
# Truncate Sigma to given omega window
# In the future there should be an option in gf to manipulate the mesh (e.g. truncate) directly.
# For now we stick with this:
for icrsh in range(sum_k.n_corr_shells):
Sigma_save = sum_k.Sigma_imp[icrsh].copy()
spn = sum_k.spin_block_names[sum_k.corr_shells[icrsh]['SO']]
def glist(): return [GfReFreq(target_shape=(block_dim, block_dim), window=(omega[
0], omega[-1]), n_points=n_om) for block, block_dim in sum_k.gf_struct_sumk[icrsh]]
sum_k.Sigma_imp[icrsh] = BlockGf(
name_list=spn, block_list=glist(), make_copies=False)
for i, g in sum_k.Sigma_imp[icrsh]:
for iL in g.indices[0]:
for iR in g.indices[0]:
for iom in range(n_om):
g.data[iom, int(iL), int(iR)] = Sigma_save[
i].data[ioffset + iom, int(iL), int(iR)]
else:
assert n_om is not None, "transport_distribution: Number of omega points (n_om) needed to calculate transport distribution!"
assert energy_window is not None, "transport_distribution: Energy window needed to calculate transport distribution!"
assert broadening != 0.0 and broadening is not None, "transport_distribution: Broadening necessary to calculate transport distribution!"
omega = numpy.linspace(
energy_window[0] - max(Om_mesh), energy_window[1] + max(Om_mesh), n_om)
mesh = MeshReFreq(energy_window[0] -
max(Om_mesh), energy_window[1] + max(Om_mesh), n_om)
mu = 0.0
dir_to_int = {'x': 0, 'y': 1, 'z': 2}
# Define mesh for optic conductivity
d_omega = round(numpy.abs(omega[0] - omega[1]), 12)
iOm_mesh = numpy.array([round((Om / d_omega), 0) for Om in Om_mesh])
temp_Om_mesh = iOm_mesh * d_omega
if mpi.is_master_node():
print("Chemical potential: ", mu)
print("Using n_om = %s points in the energy_window [%s,%s]" % (
n_om, omega[0], omega[-1]), end=' ')
print("where the omega vector is:")
print(omega)
print("Calculation requested for Omega mesh: ", numpy.array(Om_mesh))
print("Omega mesh automatically repined to: ", temp_Om_mesh)
Gamma_w = {direction: numpy.zeros((len(temp_Om_mesh), n_om),
dtype=numpy.float64) for direction in directions}
# Sum over all k-points
ikarray = numpy.array(list(range(sum_k.n_k)))
for ik in mpi.slice_array(ikarray):
# Calculate G_w for ik and initialize A_kw
G_w = sum_k.lattice_gf(ik, mu, broadening=broadening, mesh=mesh, with_Sigma=with_Sigma)
A_kw = [numpy.zeros((sum_k.n_orbitals[ik][isp], sum_k.n_orbitals[ik][isp], n_om), dtype=numpy.complex128)
for isp in range(n_inequiv_spin_blocks)]
for isp in range(n_inequiv_spin_blocks):
# copy data from G_w (swapaxes is used to have omega in the 3rd
# dimension)
A_kw[isp] = copy.deepcopy(G_w[sum_k.spin_block_names[sum_k.SO][
isp]].data.swapaxes(0, 1).swapaxes(1, 2))
# calculate A(k,w) for each frequency
for iw in range(n_om):
A_kw[isp][:, :, iw] = -1.0 / (2.0 * numpy.pi * 1j) * (
A_kw[isp][:, :, iw] - numpy.conjugate(numpy.transpose(A_kw[isp][:, :, iw])))
# Akw_write[ik] = A_kw[isp].copy() * sum_k.bz_weights[ik]
b_min = max(sum_k.band_window[isp][
ik, 0], sum_k.band_window_optics[isp][ik, 0])
b_max = min(sum_k.band_window[isp][
ik, 1], sum_k.band_window_optics[isp][ik, 1])
A_i = slice(
b_min - sum_k.band_window[isp][ik, 0], b_max - sum_k.band_window[isp][ik, 0] + 1)
v_i = slice(b_min - sum_k.band_window_optics[isp][
ik, 0], b_max - sum_k.band_window_optics[isp][ik, 0] + 1)
# loop over all symmetries
for R in sum_k.rot_symmetries:
# get transformed velocity under symmetry R
if code in ('wien2k'):
vel_R = copy.deepcopy(sum_k.velocities_k[isp][ik])
elif code in ('wannier90'):
vel_R = copy.deepcopy(sum_k.velocities_k[ik])
for nu1 in range(sum_k.band_window_optics[isp][ik, 1] - sum_k.band_window_optics[isp][ik, 0] + 1):
for nu2 in range(sum_k.band_window_optics[isp][ik, 1] - sum_k.band_window_optics[isp][ik, 0] + 1):
vel_R[nu1][nu2][:] = numpy.dot(
R, vel_R[nu1][nu2][:])
# calculate Gamma_w for each direction from the velocities
# vel_R and the spectral function A_kw
for direction in directions:
for iw in range(n_om):
for iq in range(len(temp_Om_mesh)):
if (iw + iOm_mesh[iq] >= n_om or omega[iw] < -temp_Om_mesh[iq] + energy_window[0] or omega[iw] > temp_Om_mesh[iq] + energy_window[1]):
continue
Gamma_w[direction][iq, iw] += (numpy.dot(numpy.dot(numpy.dot(vel_R[v_i, v_i, dir_to_int[direction[0]]], A_kw[isp][A_i, A_i, int(iw + iOm_mesh[iq])]),
vel_R[v_i, v_i, dir_to_int[direction[1]]]), A_kw[isp][A_i, A_i, iw]).trace().real * sum_k.bz_weights[ik])
for direction in directions:
Gamma_w[direction] = (mpi.all_reduce(Gamma_w[direction]) / sum_k.cell_vol / sum_k.n_symmetries)
return Gamma_w, omega, temp_Om_mesh
def transport_function(beta, directions, hopping, velocities, energy_window, n_om, rot_symmetries):
r"""
Calculates the transport function
.. math::
\Phi_\alpha\beta(\omega) = \sum_k v_{k,\alpha} v_{k,\beta} \delta(\omega-\varepsilon)
in the direction :math:`\alpha\beta`.
Parameters
----------
beta : double
Inverse temperature :math:`\beta`.
directions : list of string, optional
:math:`\alpha\beta` e.g.: ['xx','yy','zz','xy','xz','yz'].
hopping : double array
Hamiltonian in band basis :math:`\epsilon(k)`
veolcities : complex array
matrix elements derivative of Hamiltonian :math:`\frac{d\epsilon(k)}{dk}`
energy_window : list of double
Specifies the upper and lower limit of the frequency integration for :math:`\Omega=0.0`. The window is automatically enlarged by the largest :math:`\Omega` value,
hence the integration is performed in the interval [energy_window[0]-max(Om_mesh), energy_window[1]+max(Om_mesh)].
n_om : integer
Number of equidistant frequency points in the interval [energy_window[0]-max(Om_mesh), energy_window[1]+max(Om_mesh)]. This parameters is only used if
with_Sigma = False.
rot_symmetries : list of 3 x 3 matrices
rotational symmetries to restore the full FBZ
Returns
-------
transp_func : dictionary of double array
transport function in each direction, frequencies given by energy_window
"""
mpi.report('Computing transport function...')
# check that velocities are computed on the FBZ
assert numpy.shape(rot_symmetries)[
0] == 1, 'Using symmetries currently not implemented for transport function.'
dir_to_int = {'x': 0, 'y': 1, 'z': 2}
tol = 1/beta
orb_1, orb_2 = velocities.shape[1:3]
ws = numpy.linspace(energy_window[0], energy_window[1], n_om)
transp_func = {direction: numpy.zeros(shape=(ws.shape[0])) for direction in directions}
for ct, w in enumerate(ws):
idx = numpy.where(numpy.abs(hopping[:, 0, range(orb_1), range(orb_2)].real - w) <= tol)
fermi_wg = fermi_dis(hopping[:, 0, range(orb_1), range(orb_2)]
[idx].real - w, beta, 1)/fermi_dis(0., beta, 1)
for direction in directions:
dir_a, dir_b = [dir_to_int[x] for x in direction]
matrix_product = numpy.einsum(
'kmn, kno -> kmo', velocities[:, :, :, dir_a], velocities[:, :, :, dir_b])
transp_func[direction][ct] = numpy.sum(
fermi_wg * matrix_product[:, range(orb_1), range(orb_2)][idx]).real
return transp_func
def transport_coefficient(Gamma_w, omega, Om_mesh, spin_polarization, direction, iq, n, beta, method=None):
r"""
Calculates the transport coefficient A_n in a given direction for a given :math:`\Omega`. The required members (Gamma_w, directions, Om_mesh) have to be obtained first
by calling the function :meth:`transport_distribution `. For n>0 A is set to NaN if :math:`\Omega` is not 0.0.
Parameters
----------
Gamma_w : dictionary of double matrices
transport distribution function in each direction, frequency given by Om_mesh_out and omega
omega : list of double
omega vector
Om_mesh : list of double
frequency mesh of the optical conductivity recomputed on the mesh provided by the self energy
spin_polarization : integer
Boolean-type integer whether system is spin-polarized or not
direction : string
:math:`\alpha\beta` e.g.: 'xx','yy','zz','xy','xz','yz'.
iq : integer
Index of :math:`\Omega` point in the member Om_mesh.
n : integer
Number of the desired moment of the transport distribution.
beta : double
Inverse temperature :math:`\beta`.
method : string
Integration method: cubic spline and scipy.integrate.quad ('quad'), simpson rule ('simpson'), trapezoidal rule ('trapz'), rectangular integration (otherwise)
Note that the sampling points of the the self-energy are used!
Returns
-------
A : double
Transport coefficient.
"""
from scipy.interpolate import interp1d
from scipy.integrate import simpson, quad
if not (mpi.is_master_node()):
return None
if (Om_mesh[iq] == 0.0 or n == 0.0):
A = 0.0
# setup the integrand
if (Om_mesh[iq] == 0.0):
A_int = Gamma_w[direction][iq] * \
(fermi_dis(omega, beta) * fermi_dis(-omega, beta)) * (omega * beta)**n
elif (n == 0.0):
A_int = Gamma_w[direction][iq] * (fermi_dis(omega, beta) -
fermi_dis(omega + Om_mesh[iq], beta)) / (Om_mesh[iq] * beta)
# w-integration
if method == 'quad':
# quad on interpolated w-points with cubic spline
A_int_interp = interp1d(omega, A_int, kind='cubic')
A = quad(A_int_interp, min(omega), max(omega),
epsabs=1.0e-12, epsrel=1.0e-12, limit=500)
A = A[0]
elif method == 'simpson':
# simpson rule for w-grid
A = simpson(A_int, omega)
elif method == 'trapz':
# trapezoidal rule for w-grid
A = numpy.trapz(A_int, omega)
else:
# rectangular integration for w-grid (orignal implementation)
d_w = omega[1] - omega[0]
for iw in range(Gamma_w[direction].shape[1]):
A += A_int[iw] * d_w
A = A * numpy.pi * (2.0 - spin_polarization)
else:
A = numpy.nan
return A
def conductivity_and_seebeck(Gamma_w, omega, Om_mesh, SP, directions, beta, method=None):
r"""
Calculates the Seebeck coefficient and the optical conductivity by calling
:meth:`transport_coefficient `.
The required members (Gamma_w, directions, Om_mesh) have to be obtained first by calling the function
:meth:`transport_distribution `.
Parameters
----------
Gamma_w : dictionary of double matrices
transport distribution function in each direction, frequency given by Om_mesh_out and omega
omega : list of double
omega vector
Om_mesh : list of double
frequency mesh of the optical conductivity recomputed on the mesh provided by the self energy
spin_polarization : integer
Boolean-type integer whether system is spin-polarized or not
directions : list of string, optional
:math:`\alpha\beta` e.g.: ['xx','yy','zz','xy','xz','yz'].
beta : double
Inverse temperature :math:`\beta`.
method : string
Integration method: cubic spline and scipy.integrate.quad ('quad'), simpson rule ('simpson'), trapezoidal rule ('trapz'), rectangular integration (otherwise)
Note that the sampling points of the the self-energy are used!
Returns
-------
optic_cond : dictionary of double vectors
Optical conductivity in each direction and frequency given by Om_mesh.
seebeck : dictionary of double
Seebeck coefficient in each direction. If zero is not present in Om_mesh the Seebeck coefficient is set to NaN.
kappa : dictionary of double.
thermal conductivity in each direction. If zero is not present in Om_mesh the thermal conductivity is set to NaN
"""
mpi.report('Computing optical conductivity and kinetic coefficients...')
if not (mpi.is_master_node()):
return None, None, None
n_q = Gamma_w[directions[0]].shape[0]
# initialization
A0 = {direction: numpy.full((n_q,), numpy.nan) for direction in directions}
A1 = {direction: numpy.full((n_q,), numpy.nan) for direction in directions}
A2 = {direction: numpy.full((n_q,), numpy.nan) for direction in directions}
optic_cond = {direction: numpy.full((n_q,), numpy.nan) for direction in directions}
seebeck = {direction: numpy.nan for direction in directions}
kappa = {direction: numpy.nan for direction in directions}
for direction in directions:
for iq in range(n_q):
A0[direction][iq] = transport_coefficient(
Gamma_w, omega, Om_mesh, SP, direction, iq=iq, n=0, beta=beta, method=method)
A1[direction][iq] = transport_coefficient(
Gamma_w, omega, Om_mesh, SP, direction, iq=iq, n=1, beta=beta, method=method)
A2[direction][iq] = transport_coefficient(
Gamma_w, omega, Om_mesh, SP, direction, iq=iq, n=2, beta=beta, method=method)
print("A_0 in direction %s for Omega = %.2f %e a.u." %
(direction, Om_mesh[iq], A0[direction][iq]))
print("A_1 in direction %s for Omega = %.2f %e a.u." %
(direction, Om_mesh[iq], A1[direction][iq]))
print("A_2 in direction %s for Omega = %.2f %e a.u." %
(direction, Om_mesh[iq], A2[direction][iq]))
if ~numpy.isnan(A1[direction][iq]):
# Seebeck and kappa are overwritten if there is more than one Omega =
# 0 in Om_mesh
seebeck[direction] = - A1[direction][iq] / A0[direction][iq] * 86.17
kappa[direction] = A2[direction][iq] - \
A1[direction][iq]*A1[direction][iq]/A0[direction][iq]
kappa[direction] *= 293178.0
# factor for optical conductivity: hbar * velocity_Hartree_to_SI * volume_Hartree_to_SI * m_to_cm * 10^-4 final unit
convert_to_SI = cst.hbar * (cst.c * cst.fine_structure) ** 2 * \
(1/cst.physical_constants['Bohr radius'][0]) ** 3 * 1e-6
optic_cond[direction] = beta * convert_to_SI * A0[direction]
for iq in range(n_q):
print("Conductivity in direction %s for Omega = %.2f %f x 10^4 Ohm^-1 cm^-1" %
(direction, Om_mesh[iq], optic_cond[direction][iq]))
if not (numpy.isnan(A1[direction][iq])):
print("Seebeck in direction %s for Omega = 0.00 %f x 10^(-6) V/K" %
(direction, seebeck[direction]))
print("kappa in direction %s for Omega = 0.00 %f W/(m * K)" %
(direction, kappa[direction]))
return optic_cond, seebeck, kappa