#!/usr/bin/env python3
# -*- coding: utf-8 -*-
################################################################################
#
# solid_dmft - A versatile python wrapper to perform DFT+DMFT calculations
# utilizing the TRIQS software library
#
# Copyright (C) 2018-2020, ETH Zurich
# Copyright (C) 2021, The Simons Foundation
# authors: A. Hampel, M. Merkel, and S. Beck
#
# solid_dmft 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.
#
# solid_dmft 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
# solid_dmft (in the file COPYING.txt in this directory). If not, see
# <http://www.gnu.org/licenses/>.
#
################################################################################
"""
Contains all the functions related to setting the chemical potential in the
next iteration.
"""
import numpy as np
import triqs.utility.mpi as mpi
from h5 import HDFArchive
from triqs.gf import BlockGf, GfImFreq, GfImTime, Fourier, MeshImFreq
try:
if mpi.is_master_node():
from solid_dmft.postprocessing import maxent_gf_latt
imported_maxent = True
except ImportError:
imported_maxent = False
[docs]
def _mix_chemical_potential(general_params, density_tot, density_required,
previous_mu, predicted_mu):
"""
Mixes the previous chemical potential and the predicted potential with linear
mixing:
new_mu = factor * predicted_mu + (1-factor) * previous_mu, with
factor = mu_mix_per_occupation_offset * |density_tot - density_required| + mu_mix_const
under the constrain of 0 <= factor <= 1.
Parameters
----------
general_params : dict
general parameters as a dict
density_tot : float
total occupation of the correlated system
density_required : float
required density for the impurity problem
previous_mu : float
the chemical potential from the previous iteration
predicted_mu : float
the chemical potential predicted by methods like the SumkDFT dichotomy
Returns
-------
new_mu : float
the chemical potential that results from the mixing
"""
mu_mixing = general_params['mu_mix_per_occupation_offset'] * abs(density_tot - density_required)
mu_mixing += general_params['mu_mix_const']
mu_mixing = max(min(mu_mixing, 1), 0)
new_mu = mu_mixing * predicted_mu + (1-mu_mixing) * previous_mu
mpi.report('Mixing dichotomy mu with previous iteration by factor {:.3f}'.format(mu_mixing))
mpi.report('New chemical potential: {:.3f}'.format(new_mu))
return new_mu
[docs]
def _initialize_lattice_gf(sum_k, general_params):
"""
Creates lattice Green's function (GF) that is averaged over orbitals,
blocks and spins. Returns lattice GF as input for an analytical
continuation as well as G_lattice(tau=beta/2) (proxy for the spectral
weight) and G_lattice(beta) (proxy for the total occupation).
Parameters
----------
sum_k : SumkDFT object
Sumk object to generate the lattice GF from.
general_params : dict
general parameters as dict.
Returns
-------
gf_lattice_iw : triqs.gf.BlockGf
trace of the lattice GF over all blocks, orbitals and spins in
Matsubara frequency.
g_betahalf : complex
the Fourier transform of gf_lattice_iw evaluated at tau=beta/2.
occupation : complex
the total density from gf_lattice_iw
"""
# Initializes lattice GF to zero for each process
mesh = sum_k.Sigma_imp[0].mesh
trace_gf_latt = GfImFreq(mesh=mesh, data=np.zeros((len(mesh), 1, 1), dtype=complex))
occupation = 0
# Takes trace over orbitals and spins
ikarray = np.arange(sum_k.n_k)
for ik in mpi.slice_array(ikarray):
gf_latt = sum_k.lattice_gf(ik)*sum_k.bz_weights[ik]
trace_gf_latt.data[:] += np.trace(sum(g.data for _, g in gf_latt), axis1=1, axis2=2).reshape(-1, 1, 1)
occupation += gf_latt.total_density()
trace_gf_latt << mpi.all_reduce(trace_gf_latt)
occupation = mpi.all_reduce(occupation)
# Lattice GF as BlockGf, required for compatibility with MaxEnt functions
gf_lattice_iw = BlockGf(name_list=['total'], block_list=[trace_gf_latt])
# Fourier transforms the lattice GF
gf_tau = GfImTime(beta=general_params['beta'], n_points=general_params['n_tau'], indices=[0])
gf_tau << Fourier(gf_lattice_iw['total'])
tau_mesh = np.array([float(m) for m in gf_tau.mesh])
middle_index = np.argmin(np.abs(tau_mesh-general_params['beta']/2))
samp = 10
# Extracts G_latt(tau) at beta/2
g_betahalf = np.mean(gf_tau.data[middle_index-samp:middle_index+samp, 0, 0])
mpi.report('Lattice Gf: occupation = {:.5f}'.format(occupation))
mpi.report(' G(beta/2) = {:.5f}'.format(g_betahalf))
return gf_lattice_iw, g_betahalf, occupation
[docs]
def _determine_band_edge(mesh, spectral_function, spectral_func_threshold, valence_band, edge_threshold=.2):
"""
Finds the band edge of a spectral function. This is done in two steps:
starting from the Fermi energy, looks for the first peak
(>spectral_func_threshold and local maximum on discrete grid). Then moves
back towards Fermi energy until the spectral function is smaller than the
fraction edge_threshold of the peak value.
Parameters
----------
mesh : numpy.ndarray of float
the real frequencies grid.
spectral_function : numpy.ndarray of float
the values of the spectral function on the grid.
spectral_func_threshold : float
Threshold for spectral function to cross before looking for peaks.
valence_band : bool
Determines if looking for valence band (i.e. the upper band edge) or
the conduction band (i.e. the lower band edge).
edge_threshold : float
Fraction of the peak value that defines the band edge value.
Returns
-------
float
The frequency value of the band edge.
"""
# Determines direction to move away from Fermi energy to find band edge
direction = 1 if valence_band else -1
# Starts closest to the Fermi energy
starting_index = np.argmin(np.abs(mesh))
print('Starting at index {} with A(omega={:.3f})={:.3f}'.format(starting_index, mesh[starting_index], spectral_function[starting_index]))
assert spectral_function[starting_index] < spectral_func_threshold
# Finds peak
peak_index = None
for i in range(starting_index+direction, mesh.shape[0] if valence_band else -1, direction):
# If A(omega) low, go further
if spectral_function[i] < spectral_func_threshold:
continue
# If spectral function still increasing, go further
if spectral_function[i-direction] < spectral_function[i]:
continue
peak_index = i-direction
break
assert peak_index is not None, 'Band peak not found. Check frequency range of MaxEnt'
print('Peak at index {} with A(omega={:.3f})={:.3f}'.format(peak_index, mesh[peak_index], spectral_function[peak_index]))
# Finds band edge
edge_index = starting_index
for i in range(peak_index-direction, starting_index-direction, -direction):
# If above ratio edge_threshold of peak height, go further back to starting index
if spectral_function[i] > edge_threshold * spectral_function[peak_index]:
continue
edge_index = i
break
print('Band edge at index {} with A(omega={:.3f})={:.3f}'.format(edge_index, mesh[edge_index], spectral_function[edge_index]))
return mesh[edge_index]
[docs]
def _set_mu_to_gap_middle_with_maxent(general_params, sum_k, gf_lattice_iw, archive=None):
"""
Bundles running maxent on the total lattice GF, analyzing the spectral
function and determining the new chemical potential.
Parameters
----------
general_params : dict
general parameters as dict.
sum_k : SumkDFT object
SumkDFT object needed for original chemical potential and frequency
range of MaxEnt continuation.
gf_lattice_iw : BlockGf
trace of the lattice GF over all blocks, orbitals and spins in
Matsubara frequency.
archive : HDFArchive, optional
If given, writes spectral function (i.e. MaxEnt result) to archive.
Returns
-------
float
new chemical potential located in the middle of the gap from MaxEnt.
None if not master node or if something went wrong.
"""
if not mpi.is_master_node():
return None
if not imported_maxent:
mpi.report('WARNING: cannot find gap with MaxEnt, MaxEnt not found')
return None
# Runs MaxEnt using the Chi2Curvature analyzer
maxent_results, mesh = maxent_gf_latt._run_maxent(gf_lattice_iw, sum_k, .02, None, None, 200, 30)
mesh = np.array(mesh)
spectral_function = maxent_results['total'].get_A_out('Chi2CurvatureAnalyzer')
# Writes spectral function to archive
if archive is not None:
unpacked_results = maxent_gf_latt._unpack_maxent_results(maxent_results, mesh)
with HDFArchive(archive, 'a') as ar:
ar['DMFT_results/last_iter']['Alatt_w'] = unpacked_results
# Checks if spectral function at Fermi energy below threshold
spectral_func_threshold = general_params['beta']/np.pi * general_params['mu_gap_gb2_threshold']
if spectral_function[np.argmin(np.abs(mesh))] > spectral_func_threshold:
mpi.report('WARNING: cannot find gap with MaxEnt, spectral function not gapped at Fermi energy')
return None
# Determines band edges for conduction and valence band
edge_threshold = 0.2
conduction_edge = _determine_band_edge(mesh, spectral_function, spectral_func_threshold, False, edge_threshold)
valence_edge = _determine_band_edge(mesh, spectral_function, spectral_func_threshold, True, edge_threshold)
return sum_k.chemical_potential + (valence_edge + conduction_edge) / 2
[docs]
def set_initial_mu(general_params, sum_k, iteration_offset, archive, broadening):
"""
Handles the different ways of setting the initial chemical potential mu:
* Chemical potential set to fixed value: uses this value
* New calculation: determines mu from dichotomy method
* Resuming calculation and chemical potential not updated this iteration:
loads calculation before previous iteration.
* Resuming calculation and chemical potential is updated:
checks if the system is gapped and potentially run MaxEnt to find gap
middle. Otherwise, gets mu from dichotomy and applies mu mixing to result.
Parameters
----------
general_params : dict
general parameters as dict.
sum_k : SumkDFT object
contains system information necessary to determine the initial mu.
iteration_offset : int
the number of iterations executed in previous calculations.
archive : HDFArchive
needed to potentially load previous results and write MaxEnt results to.
Returns
-------
sum_k : SumkDFT object
the altered SumkDFT object with the initial mu set correctly.
"""
# Uses fixed_mu_value as chemical potential if parameter is given
if general_params['fixed_mu_value'] is not None:
sum_k.set_mu(general_params['fixed_mu_value'])
mpi.report('+++ Keeping the chemical potential fixed at {:.3f} eV +++'.format(general_params['fixed_mu_value']))
return sum_k
# In first iteration, determines mu and returns
if iteration_offset == 0:
sum_k.calc_mu(precision=general_params['prec_mu'], method=general_params['calc_mu_method'],
broadening=broadening)
return sum_k
# If continuing calculation and not updating mu, loads sold value
if iteration_offset % general_params['mu_update_freq'] != 0:
if mpi.is_master_node():
with HDFArchive(archive, 'r') as ar:
sum_k.chemical_potential = ar['DMFT_results/last_iter/chemical_potential_pre']
sum_k.chemical_potential = mpi.bcast(sum_k.chemical_potential)
mpi.report('Chemical potential not updated this step, '
+ 'reusing loaded one of {:.3f} eV'.format(sum_k.chemical_potential))
return sum_k
# If continuing calculation and updating mu, reads in occupation and
# chemical_potential_pre from the last run
previous_mu = None
if mpi.is_master_node():
with HDFArchive(archive, 'r') as ar:
previous_mu = ar['DMFT_results/last_iter/chemical_potential_pre']
previous_mu = mpi.bcast(previous_mu)
# Runs maxent if spectral weight too low and occupation is close to desired one
if isinstance(sum_k.mesh, MeshImFreq) and general_params['mu_gap_gb2_threshold'] is not None:
sum_k.chemical_potential = previous_mu
gf_lattice_iw, g_betahalf, occupation = _initialize_lattice_gf(sum_k, general_params)
fulfills_occupation_crit = (general_params['mu_gap_occ_deviation'] is None
or np.abs(occupation - sum_k.density_required) < general_params['mu_gap_occ_deviation'])
if -np.real(g_betahalf) < general_params['mu_gap_gb2_threshold'] and fulfills_occupation_crit:
new_mu = _set_mu_to_gap_middle_with_maxent(general_params, sum_k, gf_lattice_iw, archive)
new_mu = mpi.bcast(new_mu)
if new_mu is not None:
sum_k.chemical_potential = new_mu
mpi.report('New chemical potential in the gap: {:.3f} eV'.format(new_mu))
return sum_k
# Calculates occupation for mu mixing below
elif np.isclose(general_params['mu_mix_per_occupation_offset'], 0):
occupation = 0 # The occupation does not matter in this case
else:
_, _, occupation = _initialize_lattice_gf(sum_k, general_params)
# If system not gapped, gets chemical potential from dichotomy method
sum_k.calc_mu(precision=general_params['prec_mu'], method=general_params['calc_mu_method'],
broadening=broadening)
# Applies mu mixing to dichotomy result
sum_k.chemical_potential = _mix_chemical_potential(general_params, occupation,
sum_k.density_required,
previous_mu, sum_k.chemical_potential)
return sum_k
[docs]
def update_mu(general_params, sum_k, it, archive, broadening):
"""
Handles the different ways of updating the chemical potential mu:
* Chemical potential set to fixed value: uses this value
* Chemical potential not updated this iteration: nothing happens.
* Chemical potential is updated: checks if the system is gapped and
potentially run MaxEnt to find gap middle. Otherwise, gets mu from
dichotomy and applies mu mixing to result.
Parameters
----------
general_params : dict
general parameters as dict.
sum_k : SumkDFT object
contains system information necessary to update mu.
it : int
the number of the current iteration.
archive : HDFArchive
needed to potentially write MaxEnt results to.
Returns
-------
sum_k : SumkDFT object
the altered SumkDFT object with the updated mu.
"""
# Uses fixed_mu_value as chemical potential if parameter is given
if general_params['fixed_mu_value'] is not None:
sum_k.set_mu(general_params['fixed_mu_value'])
mpi.report('+++ Keeping the chemical potential fixed at {:.3f} eV +++'.format(general_params['fixed_mu_value']))
return sum_k
# If mu won't be updated this step, don't update it...
if it % general_params['mu_update_freq'] != 0:
mpi.report('Chemical potential not updated this step, '
+ 'reusing previous one of {:.3f} eV'.format(sum_k.chemical_potential))
return sum_k
# Runs maxent if spectral weight too low and occupation is close to desired one
# TODO: which solvers work?
if isinstance(sum_k.mesh, MeshImFreq) and general_params['mu_gap_gb2_threshold'] is not None:
gf_lattice_iw, g_betahalf, occupation = _initialize_lattice_gf(sum_k, general_params)
fulfills_occupation_crit = (general_params['mu_gap_occ_deviation'] is None
or np.abs(occupation - sum_k.density_required) < general_params['mu_gap_occ_deviation'])
if -np.real(g_betahalf) < general_params['mu_gap_gb2_threshold'] and fulfills_occupation_crit:
new_mu = _set_mu_to_gap_middle_with_maxent(general_params, sum_k, gf_lattice_iw, archive)
new_mu = mpi.bcast(new_mu)
if new_mu is not None:
sum_k.chemical_potential = new_mu
mpi.report('New chemical potential in the gap: {:.3f} eV'.format(new_mu))
return sum_k
# Calculates occupation for mu mixing below
elif np.isclose(general_params['mu_mix_per_occupation_offset'], 0):
occupation = 0 # The occupation does not matter in this case
else:
_, _, occupation = _initialize_lattice_gf(sum_k, general_params)
# If system not gapped, gets chemical potential from dichotomy method
previous_mu = sum_k.chemical_potential
sum_k.calc_mu(precision=general_params['prec_mu'], method=general_params['calc_mu_method'],
broadening=broadening)
# Applies mu mixing to dichotomy result
sum_k.chemical_potential = _mix_chemical_potential(general_params, occupation,
sum_k.density_required,
previous_mu, sum_k.chemical_potential)
mpi.barrier()
return sum_k