mirror of
https://github.com/triqs/dft_tools
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2060 lines
94 KiB
Python
2060 lines
94 KiB
Python
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##########################################################################
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#
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# TRIQS: a Toolbox for Research in Interacting Quantum Systems
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#
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# Copyright (C) 2011 by M. Aichhorn, L. Pourovskii, V. Vildosola
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#
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# TRIQS is free software: you can redistribute it and/or modify it under the
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# terms of the GNU General Public License as published by the Free Software
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# Foundation, either version 3 of the License, or (at your option) any later
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# version.
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#
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# TRIQS is distributed in the hope that it will be useful, but WITHOUT ANY
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# WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
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# FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
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# details.
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#
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# You should have received a copy of the GNU General Public License along with
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# TRIQS. If not, see <http://www.gnu.org/licenses/>.
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#
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##########################################################################
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from types import *
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import numpy
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import pytriqs.utility.dichotomy as dichotomy
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from pytriqs.gf import *
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import pytriqs.utility.mpi as mpi
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from pytriqs.utility.comparison_tests import assert_arrays_are_close
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from pytriqs.archive import *
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from symmetry import *
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from block_structure import BlockStructure
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from sets import Set
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from itertools import product
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from warnings import warn
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from scipy import compress
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from scipy.optimize import minimize
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class SumkDFT(object):
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"""This class provides a general SumK method for combining ab-initio code and pytriqs."""
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def __init__(self, hdf_file, h_field=0.0, use_dft_blocks=False,
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dft_data='dft_input', symmcorr_data='dft_symmcorr_input', parproj_data='dft_parproj_input',
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symmpar_data='dft_symmpar_input', bands_data='dft_bands_input', transp_data='dft_transp_input',
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misc_data='dft_misc_input'):
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r"""
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Initialises the class from data previously stored into an hdf5 archive.
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Parameters
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----------
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hdf_file : string
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Name of hdf5 containing the data.
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h_field : scalar, optional
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The value of magnetic field to add to the DFT Hamiltonian.
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The contribution -h_field*sigma is added to diagonal elements of the Hamiltonian.
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It cannot be used with the spin-orbit coupling on; namely h_field is set to 0 if self.SO=True.
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use_dft_blocks : boolean, optional
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If True, the local Green's function matrix for each spin is divided into smaller blocks
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with the block structure determined from the DFT density matrix of the corresponding correlated shell.
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Alternatively and additionally, the block structure can be analyzed using :meth:`analyse_block_structure <dft.sumk_dft.SumkDFT.analyse_block_structure>`
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and manipulated using the SumkDFT.block_structre attribute (see :class:`BlockStructure <dft.block_structure.BlockStructure>`).
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dft_data : string, optional
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Name of hdf5 subgroup in which DFT data for projector and lattice Green's function construction are stored.
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symmcorr_data : string, optional
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Name of hdf5 subgroup in which DFT data on symmetries of correlated shells
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(symmetry operations, permutaion matrices etc.) are stored.
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parproj_data : string, optional
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Name of hdf5 subgroup in which DFT data on non-normalized projectors for non-correlated
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states (used in the partial density of states calculations) are stored.
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symmpar_data : string, optional
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Name of hdf5 subgroup in which DFT data on symmetries of the non-normalized projectors
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are stored.
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bands_data : string, optional
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Name of hdf5 subgroup in which DFT data necessary for band-structure/k-resolved spectral
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function calculations (projectors, DFT Hamiltonian for a chosen path in the Brillouin zone etc.)
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are stored.
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transp_data : string, optional
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Name of hdf5 subgroup in which DFT data necessary for transport calculations are stored.
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misc_data : string, optional
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Name of hdf5 subgroup in which miscellaneous DFT data are stored.
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"""
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if not type(hdf_file) == StringType:
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mpi.report("Give a string for the hdf5 filename to read the input!")
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else:
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self.hdf_file = hdf_file
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self.dft_data = dft_data
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self.symmcorr_data = symmcorr_data
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self.parproj_data = parproj_data
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self.symmpar_data = symmpar_data
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self.bands_data = bands_data
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self.transp_data = transp_data
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self.misc_data = misc_data
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self.h_field = h_field
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# Read input from HDF:
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things_to_read = ['energy_unit', 'n_k', 'k_dep_projection', 'SP', 'SO', 'charge_below', 'density_required',
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'symm_op', 'n_shells', 'shells', 'n_corr_shells', 'corr_shells', 'use_rotations', 'rot_mat',
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'rot_mat_time_inv', 'n_reps', 'dim_reps', 'T', 'n_orbitals', 'proj_mat', 'bz_weights', 'hopping',
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'n_inequiv_shells', 'corr_to_inequiv', 'inequiv_to_corr']
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self.subgroup_present, self.value_read = self.read_input_from_hdf(
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subgrp=self.dft_data, things_to_read=things_to_read)
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if self.symm_op:
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self.symmcorr = Symmetry(hdf_file, subgroup=self.symmcorr_data)
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if self.SO and (abs(self.h_field) > 0.000001):
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self.h_field = 0.0
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mpi.report(
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"For SO, the external magnetic field is not implemented, setting it to 0!")
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self.spin_block_names = [['up', 'down'], ['ud']]
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self.n_spin_blocks = [2, 1]
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# Convert spin_block_names to indices -- if spin polarized,
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# differentiate up and down blocks
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self.spin_names_to_ind = [{}, {}]
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for iso in range(2): # SO = 0 or 1
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for isp in range(self.n_spin_blocks[iso]):
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self.spin_names_to_ind[iso][
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self.spin_block_names[iso][isp]] = isp * self.SP
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self.block_structure = BlockStructure()
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# GF structure used for the local things in the k sums
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# Most general form allowing for all hybridisation, i.e. largest
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# blocks possible
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self.gf_struct_sumk = [[(sp, range(self.corr_shells[icrsh]['dim'])) for sp in self.spin_block_names[self.corr_shells[icrsh]['SO']]]
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for icrsh in range(self.n_corr_shells)]
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# First set a standard gf_struct solver:
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self.gf_struct_solver = [dict([(sp, range(self.corr_shells[self.inequiv_to_corr[ish]]['dim']))
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for sp in self.spin_block_names[self.corr_shells[self.inequiv_to_corr[ish]]['SO']]])
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for ish in range(self.n_inequiv_shells)]
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# Set standard (identity) maps from gf_struct_sumk <->
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# gf_struct_solver
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self.sumk_to_solver = [{} for ish in range(self.n_inequiv_shells)]
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self.solver_to_sumk = [{} for ish in range(self.n_inequiv_shells)]
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self.solver_to_sumk_block = [{}
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for ish in range(self.n_inequiv_shells)]
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for ish in range(self.n_inequiv_shells):
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for block, inner_list in self.gf_struct_sumk[self.inequiv_to_corr[ish]]:
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self.solver_to_sumk_block[ish][block] = block
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for inner in inner_list:
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self.sumk_to_solver[ish][
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(block, inner)] = (block, inner)
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self.solver_to_sumk[ish][
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(block, inner)] = (block, inner)
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# assume no shells are degenerate
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self.deg_shells = [[] for ish in range(self.n_inequiv_shells)]
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self.chemical_potential = 0.0 # initialise mu
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self.init_dc() # initialise the double counting
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# Analyse the block structure and determine the smallest gf_struct
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# blocks and maps, if desired
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if use_dft_blocks:
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self.analyse_block_structure()
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################
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# hdf5 FUNCTIONS
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################
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def read_input_from_hdf(self, subgrp, things_to_read):
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r"""
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Reads data from the HDF file. Prints a warning if a requested dataset is not found.
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Parameters
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----------
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subgrp : string
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Name of hdf5 file subgroup from which the data are to be read.
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things_to_read : list of strings
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List of datasets to be read from the hdf5 file.
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Returns
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-------
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subgroup_present : boolean
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Is the subgrp is present in hdf5 file?
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value_read : boolean
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Did the reading of requested datasets succeed?
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"""
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value_read = True
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# initialise variables on all nodes to ensure mpi broadcast works at
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# the end
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for it in things_to_read:
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setattr(self, it, 0)
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subgroup_present = 0
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if mpi.is_master_node():
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ar = HDFArchive(self.hdf_file, 'r')
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if subgrp in ar:
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subgroup_present = True
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# first read the necessary things:
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for it in things_to_read:
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if it in ar[subgrp]:
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setattr(self, it, ar[subgrp][it])
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else:
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mpi.report("Loading %s failed!" % it)
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value_read = False
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else:
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if (len(things_to_read) != 0):
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mpi.report(
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"Loading failed: No %s subgroup in hdf5!" % subgrp)
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subgroup_present = False
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value_read = False
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del ar
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# now do the broadcasting:
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for it in things_to_read:
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setattr(self, it, mpi.bcast(getattr(self, it)))
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subgroup_present = mpi.bcast(subgroup_present)
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value_read = mpi.bcast(value_read)
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return subgroup_present, value_read
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def save(self, things_to_save, subgrp='user_data'):
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r"""
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Saves data from a list into the HDF file. Prints a warning if a requested data is not found in SumkDFT object.
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Parameters
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----------
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things_to_save : list of strings
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List of datasets to be saved into the hdf5 file.
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subgrp : string, optional
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Name of hdf5 file subgroup in which the data are to be stored.
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"""
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if not (mpi.is_master_node()):
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return # do nothing on nodes
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ar = HDFArchive(self.hdf_file, 'a')
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if not subgrp in ar:
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ar.create_group(subgrp)
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for it in things_to_save:
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if it in [ "gf_struct_sumk", "gf_struct_solver",
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"solver_to_sumk", "sumk_to_solver", "solver_to_sumk_block"]:
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warn("It is not recommended to save '{}' individually. Save 'block_structure' instead.".format(it))
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try:
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ar[subgrp][it] = getattr(self, it)
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except:
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mpi.report("%s not found, and so not saved." % it)
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del ar
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def load(self, things_to_load, subgrp='user_data'):
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r"""
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Loads user data from the HDF file. Raises an exeption if a requested dataset is not found.
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Parameters
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----------
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things_to_read : list of strings
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List of datasets to be read from the hdf5 file.
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subgrp : string, optional
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Name of hdf5 file subgroup from which the data are to be read.
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Returns
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-------
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list_to_return : list
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A list containing data read from hdf5.
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"""
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if not (mpi.is_master_node()):
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return # do nothing on nodes
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ar = HDFArchive(self.hdf_file, 'r')
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if not subgrp in ar:
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mpi.report("Loading %s failed!" % subgrp)
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list_to_return = []
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for it in things_to_load:
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try:
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list_to_return.append(ar[subgrp][it])
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except:
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raise ValueError, "load: %s not found, and so not loaded." % it
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del ar
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return list_to_return
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################
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# CORE FUNCTIONS
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################
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def downfold(self, ik, ish, bname, gf_to_downfold, gf_inp, shells='corr', ir=None):
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r"""
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Downfolds a block of the Green's function for a given shell and k-point using the corresponding projector matrices.
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Parameters
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----------
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ik : integer
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k-point index for which the downfolding is to be done.
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ish : integer
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Shell index of GF to be downfolded.
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- if shells='corr': ish labels all correlated shells (equivalent or not)
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- if shells='all': ish labels only representative (inequivalent) non-correlated shells
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bname : string
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Block name of the target block of the lattice Green's function.
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gf_to_downfold : Gf
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Block of the Green's function that is to be downfolded.
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gf_inp : Gf
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FIXME
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shells : string, optional
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- if shells='corr': orthonormalized projectors for correlated shells are used for the downfolding.
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- if shells='all': non-normalized projectors for all included shells are used for the downfolding.
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ir : integer, optional
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Index of equivalent site in the non-correlated shell 'ish', only used if shells='all'.
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Returns
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-------
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gf_downfolded : Gf
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Downfolded block of the lattice Green's function.
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"""
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gf_downfolded = gf_inp.copy()
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# get spin index for proj. matrices
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isp = self.spin_names_to_ind[self.SO][bname]
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n_orb = self.n_orbitals[ik, isp]
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if shells == 'corr':
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dim = self.corr_shells[ish]['dim']
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projmat = self.proj_mat[ik, isp, ish, 0:dim, 0:n_orb]
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elif shells == 'all':
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if ir is None:
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raise ValueError, "downfold: provide ir if treating all shells."
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dim = self.shells[ish]['dim']
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projmat = self.proj_mat_all[ik, isp, ish, ir, 0:dim, 0:n_orb]
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gf_downfolded.from_L_G_R(
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projmat, gf_to_downfold, projmat.conjugate().transpose())
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return gf_downfolded
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def upfold(self, ik, ish, bname, gf_to_upfold, gf_inp, shells='corr', ir=None):
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r"""
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Upfolds a block of the Green's function for a given shell and k-point using the corresponding projector matrices.
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Parameters
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----------
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ik : integer
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k-point index for which the upfolding is to be done.
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ish : integer
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Shell index of GF to be upfolded.
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- if shells='corr': ish labels all correlated shells (equivalent or not)
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- if shells='all': ish labels only representative (inequivalent) non-correlated shells
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bname : string
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Block name of the target block of the lattice Green's function.
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gf_to_upfold : Gf
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Block of the Green's function that is to be upfolded.
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gf_inp : Gf
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FIXME
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shells : string, optional
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- if shells='corr': orthonormalized projectors for correlated shells are used for the upfolding.
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- if shells='all': non-normalized projectors for all included shells are used for the upfolding.
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ir : integer, optional
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Index of equivalent site in the non-correlated shell 'ish', only used if shells='all'.
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Returns
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-------
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gf_upfolded : Gf
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Upfolded block of the lattice Green's function.
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"""
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gf_upfolded = gf_inp.copy()
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# get spin index for proj. matrices
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isp = self.spin_names_to_ind[self.SO][bname]
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n_orb = self.n_orbitals[ik, isp]
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if shells == 'corr':
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dim = self.corr_shells[ish]['dim']
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projmat = self.proj_mat[ik, isp, ish, 0:dim, 0:n_orb]
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elif shells == 'all':
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if ir is None:
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raise ValueError, "upfold: provide ir if treating all shells."
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dim = self.shells[ish]['dim']
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projmat = self.proj_mat_all[ik, isp, ish, ir, 0:dim, 0:n_orb]
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gf_upfolded.from_L_G_R(
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projmat.conjugate().transpose(), gf_to_upfold, projmat)
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return gf_upfolded
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def rotloc(self, ish, gf_to_rotate, direction, shells='corr'):
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r"""
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Rotates a block of the local Green's function from the local frame to the global frame and vice versa.
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Parameters
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----------
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ish : integer
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Shell index of GF to be rotated.
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- if shells='corr': ish labels all correlated shells (equivalent or not)
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- if shells='all': ish labels only representative (inequivalent) non-correlated shells
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gf_to_rotate : Gf
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Block of the Green's function that is to be rotated.
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direction : string
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The direction of rotation can be either
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- 'toLocal' : global -> local transformation,
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- 'toGlobal' : local -> global transformation.
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shells : string, optional
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- if shells='corr': the rotation matrix for the correlated shell 'ish' is used,
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- if shells='all': the rotation matrix for the generic (non-correlated) shell 'ish' is used.
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Returns
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-------
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gf_rotated : Gf
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Rotated block of the local Green's function.
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"""
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assert ((direction == 'toLocal') or (direction == 'toGlobal')
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), "rotloc: Give direction 'toLocal' or 'toGlobal'."
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gf_rotated = gf_to_rotate.copy()
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if shells == 'corr':
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rot_mat_time_inv = self.rot_mat_time_inv
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rot_mat = self.rot_mat
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elif shells == 'all':
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rot_mat_time_inv = self.rot_mat_all_time_inv
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rot_mat = self.rot_mat_all
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if direction == 'toGlobal':
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if (rot_mat_time_inv[ish] == 1) and self.SO:
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gf_rotated << gf_rotated.transpose()
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gf_rotated.from_L_G_R(rot_mat[ish].conjugate(
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), gf_rotated, rot_mat[ish].transpose())
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else:
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gf_rotated.from_L_G_R(rot_mat[ish], gf_rotated, rot_mat[
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ish].conjugate().transpose())
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elif direction == 'toLocal':
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if (rot_mat_time_inv[ish] == 1) and self.SO:
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gf_rotated << gf_rotated.transpose()
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gf_rotated.from_L_G_R(rot_mat[ish].transpose(
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), gf_rotated, rot_mat[ish].conjugate())
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else:
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gf_rotated.from_L_G_R(rot_mat[ish].conjugate(
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).transpose(), gf_rotated, rot_mat[ish])
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return gf_rotated
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def lattice_gf(self, ik, mu=None, iw_or_w="iw", beta=40, broadening=None, mesh=None, with_Sigma=True, with_dc=True):
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r"""
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Calculates the lattice Green function for a given k-point from the DFT Hamiltonian and the self energy.
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Parameters
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----------
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ik : integer
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k-point index.
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mu : real, optional
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Chemical potential for which the Green's function is to be calculated.
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If not provided, self.chemical_potential is used for mu.
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iw_or_w : string, optional
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- `iw_or_w` = 'iw' for a imaginary-frequency self-energy
|
|
- `iw_or_w` = 'w' for a real-frequency self-energy
|
|
|
|
beta : real, optional
|
|
Inverse temperature.
|
|
broadening : real, optional
|
|
Imaginary shift for the axis along which the real-axis GF is calculated.
|
|
If not provided, broadening will be set to double of the distance between mesh points in 'mesh'.
|
|
mesh : list, optional
|
|
Data defining mesh on which the real-axis GF will be calculated, given in the form
|
|
(om_min,om_max,n_points), where om_min is the minimum omega, om_max is the maximum omega and n_points is the number of points.
|
|
with_Sigma : boolean, optional
|
|
If True the GF will be calculated with the self-energy stored in self.Sigmaimp_(w/iw), for real/Matsubara GF, respectively.
|
|
In this case the mesh is taken from the self.Sigma_imp object.
|
|
If with_Sigma=True but self.Sigmaimp_(w/iw) is not present, with_Sigma is reset to False.
|
|
with_dc : boolean, optional
|
|
if True and with_Sigma=True, the dc correction is substracted from the self-energy before it is included into GF.
|
|
|
|
Returns
|
|
-------
|
|
G_latt : BlockGf
|
|
Lattice Green's function.
|
|
|
|
"""
|
|
if mu is None:
|
|
mu = self.chemical_potential
|
|
ntoi = self.spin_names_to_ind[self.SO]
|
|
spn = self.spin_block_names[self.SO]
|
|
if (iw_or_w != "iw") and (iw_or_w != "w"):
|
|
raise ValueError, "lattice_gf: Implemented only for Re/Im frequency functions."
|
|
if not hasattr(self, "Sigma_imp_" + iw_or_w):
|
|
with_Sigma = False
|
|
if broadening is None:
|
|
if mesh is None:
|
|
broadening = 0.01
|
|
else: # broadening = 2 * \Delta omega, where \Delta omega is the spacing of omega points
|
|
broadening = 2.0 * ((mesh[1] - mesh[0]) / (mesh[2] - 1))
|
|
|
|
# Are we including Sigma?
|
|
if with_Sigma:
|
|
Sigma_imp = getattr(self, "Sigma_imp_" + iw_or_w)
|
|
sigma_minus_dc = [s.copy() for s in Sigma_imp]
|
|
if with_dc:
|
|
sigma_minus_dc = self.add_dc(iw_or_w)
|
|
if iw_or_w == "iw":
|
|
# override beta if Sigma_iw is present
|
|
beta = Sigma_imp[0].mesh.beta
|
|
mesh = Sigma_imp[0].mesh
|
|
elif iw_or_w == "w":
|
|
mesh = Sigma_imp[0].mesh
|
|
if broadening>0 and mpi.is_master_node():
|
|
warn('lattice_gf called with Sigma and broadening > 0 (broadening = {}). You might want to explicitly set the broadening to 0.'.format(broadening))
|
|
else:
|
|
if iw_or_w == "iw":
|
|
if beta is None:
|
|
raise ValueError, "lattice_gf: Give the beta for the lattice GfReFreq."
|
|
# Default number of Matsubara frequencies
|
|
mesh = MeshImFreq(beta=beta, S='Fermion', n_max=1025)
|
|
elif iw_or_w == "w":
|
|
if mesh is None:
|
|
raise ValueError, "lattice_gf: Give the mesh=(om_min,om_max,n_points) for the lattice GfReFreq."
|
|
mesh = MeshReFreq(mesh[0], mesh[1], mesh[2])
|
|
|
|
# Check if G_latt is present
|
|
set_up_G_latt = False # Assume not
|
|
if not hasattr(self, "G_latt_" + iw_or_w):
|
|
# Need to create G_latt_(i)w
|
|
set_up_G_latt = True
|
|
else: # Check that existing GF is consistent
|
|
G_latt = getattr(self, "G_latt_" + iw_or_w)
|
|
GFsize = [gf.target_shape[0] for bname, gf in G_latt]
|
|
unchangedsize = all([self.n_orbitals[ik, ntoi[spn[isp]]] == GFsize[
|
|
isp] for isp in range(self.n_spin_blocks[self.SO])])
|
|
if not unchangedsize:
|
|
set_up_G_latt = True
|
|
if (iw_or_w == "iw") and (self.G_latt_iw.mesh.beta != beta):
|
|
set_up_G_latt = True # additional check for ImFreq
|
|
|
|
# Set up G_latt
|
|
if set_up_G_latt:
|
|
block_structure = [
|
|
range(self.n_orbitals[ik, ntoi[sp]]) for sp in spn]
|
|
gf_struct = [(spn[isp], block_structure[isp])
|
|
for isp in range(self.n_spin_blocks[self.SO])]
|
|
block_ind_list = [block for block, inner in gf_struct]
|
|
if iw_or_w == "iw":
|
|
glist = lambda: [GfImFreq(indices=inner, mesh=mesh)
|
|
for block, inner in gf_struct]
|
|
elif iw_or_w == "w":
|
|
glist = lambda: [GfReFreq(indices=inner, mesh=mesh)
|
|
for block, inner in gf_struct]
|
|
G_latt = BlockGf(name_list=block_ind_list,
|
|
block_list=glist(), make_copies=False)
|
|
G_latt.zero()
|
|
|
|
if iw_or_w == "iw":
|
|
G_latt << iOmega_n
|
|
elif iw_or_w == "w":
|
|
G_latt << Omega + 1j * broadening
|
|
|
|
idmat = [numpy.identity(
|
|
self.n_orbitals[ik, ntoi[sp]], numpy.complex_) for sp in spn]
|
|
M = copy.deepcopy(idmat)
|
|
for ibl in range(self.n_spin_blocks[self.SO]):
|
|
ind = ntoi[spn[ibl]]
|
|
n_orb = self.n_orbitals[ik, ind]
|
|
M[ibl] = self.hopping[ik, ind, 0:n_orb, 0:n_orb] - \
|
|
(idmat[ibl] * mu) - (idmat[ibl] * self.h_field * (1 - 2 * ibl))
|
|
G_latt -= M
|
|
|
|
if with_Sigma:
|
|
for icrsh in range(self.n_corr_shells):
|
|
for bname, gf in G_latt:
|
|
gf -= self.upfold(ik, icrsh, bname,
|
|
sigma_minus_dc[icrsh][bname], gf)
|
|
|
|
G_latt.invert()
|
|
setattr(self, "G_latt_" + iw_or_w, G_latt)
|
|
|
|
return G_latt
|
|
|
|
def set_Sigma(self, Sigma_imp):
|
|
self.put_Sigma(Sigma_imp)
|
|
|
|
def put_Sigma(self, Sigma_imp):
|
|
r"""
|
|
Inserts the impurity self-energies into the sumk_dft class.
|
|
|
|
Parameters
|
|
----------
|
|
Sigma_imp : list of BlockGf (Green's function) objects
|
|
List containing impurity self-energy for all inequivalent correlated shells.
|
|
Self-energies for equivalent shells are then automatically set by this function.
|
|
The self-energies can be of the real or imaginary-frequency type.
|
|
"""
|
|
|
|
assert isinstance(
|
|
Sigma_imp, list), "put_Sigma: Sigma_imp has to be a list of Sigmas for the correlated shells, even if it is of length 1!"
|
|
assert len(
|
|
Sigma_imp) == self.n_inequiv_shells, "put_Sigma: give exactly one Sigma for each inequivalent corr. shell!"
|
|
|
|
# init self.Sigma_imp_(i)w:
|
|
if all( (isinstance(gf, Gf) and isinstance (gf.mesh, MeshImFreq)) for bname, gf in Sigma_imp[0]):
|
|
# Imaginary frequency Sigma:
|
|
self.Sigma_imp_iw = [BlockGf(name_block_generator=[(block, GfImFreq(indices=inner, mesh=Sigma_imp[0].mesh))
|
|
for block, inner in self.gf_struct_sumk[icrsh]], make_copies=False)
|
|
for icrsh in range(self.n_corr_shells)]
|
|
SK_Sigma_imp = self.Sigma_imp_iw
|
|
elif all( isinstance(gf, Gf) and isinstance (gf.mesh, MeshReFreq) for bname, gf in Sigma_imp[0]):
|
|
# Real frequency Sigma:
|
|
self.Sigma_imp_w = [BlockGf(name_block_generator=[(block, GfReFreq(indices=inner, mesh=Sigma_imp[0].mesh))
|
|
for block, inner in self.gf_struct_sumk[icrsh]], make_copies=False)
|
|
for icrsh in range(self.n_corr_shells)]
|
|
SK_Sigma_imp = self.Sigma_imp_w
|
|
else:
|
|
raise ValueError, "put_Sigma: This type of Sigma is not handled."
|
|
|
|
# transform the CTQMC blocks to the full matrix:
|
|
for icrsh in range(self.n_corr_shells):
|
|
# ish is the index of the inequivalent shell corresponding to icrsh
|
|
ish = self.corr_to_inequiv[icrsh]
|
|
for block, inner in self.gf_struct_solver[ish].iteritems():
|
|
for ind1 in inner:
|
|
for ind2 in inner:
|
|
block_sumk, ind1_sumk = self.solver_to_sumk[
|
|
ish][(block, ind1)]
|
|
block_sumk, ind2_sumk = self.solver_to_sumk[
|
|
ish][(block, ind2)]
|
|
SK_Sigma_imp[icrsh][block_sumk][
|
|
ind1_sumk, ind2_sumk] << Sigma_imp[ish][block][ind1, ind2]
|
|
|
|
# rotation from local to global coordinate system:
|
|
if self.use_rotations:
|
|
for icrsh in range(self.n_corr_shells):
|
|
for bname, gf in SK_Sigma_imp[icrsh]:
|
|
gf << self.rotloc(icrsh, gf, direction='toGlobal')
|
|
|
|
def extract_G_loc(self, mu=None, iw_or_w='iw', with_Sigma=True, with_dc=True, broadening=None):
|
|
r"""
|
|
Extracts the local downfolded Green function by the Brillouin-zone integration of the lattice Green's function.
|
|
|
|
Parameters
|
|
----------
|
|
mu : real, optional
|
|
Input chemical potential. If not provided the value of self.chemical_potential is used as mu.
|
|
with_Sigma : boolean, optional
|
|
If True then the local GF is calculated with the self-energy self.Sigma_imp.
|
|
with_dc : boolean, optional
|
|
If True then the double-counting correction is subtracted from the self-energy in calculating the GF.
|
|
broadening : float, optional
|
|
Imaginary shift for the axis along which the real-axis GF is calculated.
|
|
If not provided, broadening will be set to double of the distance between mesh points in 'mesh'.
|
|
Only relevant for real-frequency GF.
|
|
|
|
Returns
|
|
-------
|
|
G_loc_inequiv : list of BlockGf (Green's function) objects
|
|
List of the local Green's functions for all inequivalent correlated shells,
|
|
rotated into the corresponding local frames.
|
|
|
|
"""
|
|
|
|
if mu is None:
|
|
mu = self.chemical_potential
|
|
|
|
if iw_or_w == "iw":
|
|
G_loc = [self.Sigma_imp_iw[icrsh].copy() for icrsh in range(
|
|
self.n_corr_shells)] # this list will be returned
|
|
beta = G_loc[0].mesh.beta
|
|
G_loc_inequiv = [BlockGf(name_block_generator=[(block, GfImFreq(indices=inner, mesh=G_loc[0].mesh)) for block, inner in self.gf_struct_solver[ish].iteritems()],
|
|
make_copies=False) for ish in range(self.n_inequiv_shells)]
|
|
elif iw_or_w == "w":
|
|
G_loc = [self.Sigma_imp_w[icrsh].copy() for icrsh in range(
|
|
self.n_corr_shells)] # this list will be returned
|
|
mesh = G_loc[0].mesh
|
|
G_loc_inequiv = [BlockGf(name_block_generator=[(block, GfReFreq(indices=inner, mesh=mesh)) for block, inner in self.gf_struct_solver[ish].iteritems()],
|
|
make_copies=False) for ish in range(self.n_inequiv_shells)]
|
|
|
|
for icrsh in range(self.n_corr_shells):
|
|
G_loc[icrsh].zero() # initialize to zero
|
|
|
|
ikarray = numpy.array(range(self.n_k))
|
|
for ik in mpi.slice_array(ikarray):
|
|
if iw_or_w == 'iw':
|
|
G_latt = self.lattice_gf(
|
|
ik=ik, mu=mu, iw_or_w=iw_or_w, with_Sigma=with_Sigma, with_dc=with_dc, beta=beta)
|
|
elif iw_or_w == 'w':
|
|
mesh_parameters = (G_loc[0].mesh.omega_min,G_loc[0].mesh.omega_max,len(G_loc[0].mesh))
|
|
G_latt = self.lattice_gf(
|
|
ik=ik, mu=mu, iw_or_w=iw_or_w, with_Sigma=with_Sigma, with_dc=with_dc, broadening=broadening, mesh=mesh_parameters)
|
|
G_latt *= self.bz_weights[ik]
|
|
|
|
for icrsh in range(self.n_corr_shells):
|
|
# init temporary storage
|
|
tmp = G_loc[icrsh].copy()
|
|
for bname, gf in tmp:
|
|
tmp[bname] << self.downfold(
|
|
ik, icrsh, bname, G_latt[bname], gf)
|
|
G_loc[icrsh] += tmp
|
|
|
|
# Collect data from mpi
|
|
for icrsh in range(self.n_corr_shells):
|
|
G_loc[icrsh] << mpi.all_reduce(
|
|
mpi.world, G_loc[icrsh], lambda x, y: x + y)
|
|
mpi.barrier()
|
|
|
|
# G_loc[:] is now the sum over k projected to the local orbitals.
|
|
# here comes the symmetrisation, if needed:
|
|
if self.symm_op != 0:
|
|
G_loc = self.symmcorr.symmetrize(G_loc)
|
|
|
|
# G_loc is rotated to the local coordinate system:
|
|
if self.use_rotations:
|
|
for icrsh in range(self.n_corr_shells):
|
|
for bname, gf in G_loc[icrsh]:
|
|
G_loc[icrsh][bname] << self.rotloc(
|
|
icrsh, gf, direction='toLocal')
|
|
|
|
# transform to CTQMC blocks:
|
|
for ish in range(self.n_inequiv_shells):
|
|
for block, inner in self.gf_struct_solver[ish].iteritems():
|
|
for ind1 in inner:
|
|
for ind2 in inner:
|
|
block_sumk, ind1_sumk = self.solver_to_sumk[
|
|
ish][(block, ind1)]
|
|
block_sumk, ind2_sumk = self.solver_to_sumk[
|
|
ish][(block, ind2)]
|
|
G_loc_inequiv[ish][block][ind1, ind2] << G_loc[
|
|
self.inequiv_to_corr[ish]][block_sumk][ind1_sumk, ind2_sumk]
|
|
|
|
# return only the inequivalent shells:
|
|
return G_loc_inequiv
|
|
|
|
def analyse_block_structure(self, threshold=0.00001, include_shells=None, dm=None, hloc=None):
|
|
r"""
|
|
Determines the block structure of local Green's functions by analysing the structure of
|
|
the corresponding density matrices and the local Hamiltonian. The resulting block structures
|
|
for correlated shells are stored in the :class:`SumkDFT.block_structure <dft.block_structure.BlockStructure>` attribute.
|
|
|
|
Parameters
|
|
----------
|
|
threshold : real, optional
|
|
If the difference between density matrix / hloc elements is below threshold,
|
|
they are considered to be equal.
|
|
include_shells : list of integers, optional
|
|
List of correlated shells to be analysed.
|
|
If include_shells is not provided all correlated shells will be analysed.
|
|
dm : list of dict, optional
|
|
List of density matrices from which block stuctures are to be analysed.
|
|
Each density matrix is a dict {block names: 2d numpy arrays}.
|
|
If not provided, dm will be calculated from the DFT Hamiltonian by a simple-point BZ integration.
|
|
hloc : list of dict, optional
|
|
List of local Hamiltonian matrices from which block stuctures are to be analysed
|
|
Each Hamiltonian is a dict {block names: 2d numpy arrays}.
|
|
If not provided, it will be calculated using eff_atomic_levels.
|
|
"""
|
|
|
|
self.gf_struct_solver = [{} for ish in range(self.n_inequiv_shells)]
|
|
self.sumk_to_solver = [{} for ish in range(self.n_inequiv_shells)]
|
|
self.solver_to_sumk = [{} for ish in range(self.n_inequiv_shells)]
|
|
self.solver_to_sumk_block = [{}
|
|
for ish in range(self.n_inequiv_shells)]
|
|
|
|
if dm is None:
|
|
dm = self.density_matrix(method='using_point_integration')
|
|
dens_mat = [dm[self.inequiv_to_corr[ish]]
|
|
for ish in range(self.n_inequiv_shells)]
|
|
if hloc is None:
|
|
hloc = self.eff_atomic_levels()
|
|
H_loc = [hloc[self.corr_to_inequiv[ish]]
|
|
for ish in range(self.n_corr_shells)]
|
|
|
|
if include_shells is None:
|
|
include_shells = range(self.n_inequiv_shells)
|
|
for ish in include_shells:
|
|
|
|
for sp in self.spin_block_names[self.corr_shells[self.inequiv_to_corr[ish]]['SO']]:
|
|
n_orb = self.corr_shells[self.inequiv_to_corr[ish]]['dim']
|
|
# gives an index list of entries larger that threshold
|
|
dmbool = (abs(dens_mat[ish][sp]) > threshold)
|
|
hlocbool = (abs(H_loc[ish][sp]) > threshold)
|
|
|
|
# Determine off-diagonal entries in upper triangular part of
|
|
# density matrix
|
|
offdiag = Set([])
|
|
for i in range(n_orb):
|
|
for j in range(i + 1, n_orb):
|
|
if dmbool[i, j] or hlocbool[i, j]:
|
|
offdiag.add((i, j))
|
|
|
|
# Determine the number of non-hybridising blocks in the gf
|
|
blocs = [[i] for i in range(n_orb)]
|
|
while len(offdiag) != 0:
|
|
pair = offdiag.pop()
|
|
for b1, b2 in product(blocs, blocs):
|
|
if (pair[0] in b1) and (pair[1] in b2):
|
|
if blocs.index(b1) != blocs.index(b2): # In separate blocks?
|
|
# Merge two blocks
|
|
b1.extend(blocs.pop(blocs.index(b2)))
|
|
break # Move on to next pair in offdiag
|
|
|
|
# Set the gf_struct for the solver accordingly
|
|
num_blocs = len(blocs)
|
|
for i in range(num_blocs):
|
|
blocs[i].sort()
|
|
self.gf_struct_solver[ish].update(
|
|
[('%s_%s' % (sp, i), range(len(blocs[i])))])
|
|
|
|
# Construct sumk_to_solver taking (sumk_block, sumk_index) --> (solver_block, solver_inner)
|
|
# and solver_to_sumk taking (solver_block, solver_inner) -->
|
|
# (sumk_block, sumk_index)
|
|
for i in range(num_blocs):
|
|
for j in range(len(blocs[i])):
|
|
block_sumk = sp
|
|
inner_sumk = blocs[i][j]
|
|
block_solv = '%s_%s' % (sp, i)
|
|
inner_solv = j
|
|
self.sumk_to_solver[ish][(block_sumk, inner_sumk)] = (
|
|
block_solv, inner_solv)
|
|
self.solver_to_sumk[ish][(block_solv, inner_solv)] = (
|
|
block_sumk, inner_sumk)
|
|
self.solver_to_sumk_block[ish][block_solv] = block_sumk
|
|
|
|
# Now calculate degeneracies of orbitals
|
|
dm = {}
|
|
for block, inner in self.gf_struct_solver[ish].iteritems():
|
|
# get dm for the blocks:
|
|
dm[block] = numpy.zeros(
|
|
[len(inner), len(inner)], numpy.complex_)
|
|
for ind1 in inner:
|
|
for ind2 in inner:
|
|
block_sumk, ind1_sumk = self.solver_to_sumk[
|
|
ish][(block, ind1)]
|
|
block_sumk, ind2_sumk = self.solver_to_sumk[
|
|
ish][(block, ind2)]
|
|
dm[block][ind1, ind2] = dens_mat[ish][
|
|
block_sumk][ind1_sumk, ind2_sumk]
|
|
|
|
for block1 in self.gf_struct_solver[ish].iterkeys():
|
|
for block2 in self.gf_struct_solver[ish].iterkeys():
|
|
if dm[block1].shape == dm[block2].shape:
|
|
if ((abs(dm[block1] - dm[block2]) < threshold).all()) and (block1 != block2):
|
|
ind1 = -1
|
|
ind2 = -2
|
|
# check if it was already there:
|
|
for n, ind in enumerate(self.deg_shells[ish]):
|
|
if block1 in ind:
|
|
ind1 = n
|
|
if block2 in ind:
|
|
ind2 = n
|
|
if (ind1 < 0) and (ind2 >= 0):
|
|
self.deg_shells[ish][ind2].append(block1)
|
|
elif (ind1 >= 0) and (ind2 < 0):
|
|
self.deg_shells[ish][ind1].append(block2)
|
|
elif (ind1 < 0) and (ind2 < 0):
|
|
self.deg_shells[ish].append([block1, block2])
|
|
|
|
def _get_hermitian_quantity_from_gf(self, G):
|
|
""" Convert G to a Hermitian quantity
|
|
|
|
For G(tau) and G(iw), G(tau) is returned.
|
|
For G(t) and G(w), the spectral function is returned.
|
|
|
|
Parameters
|
|
----------
|
|
G : list of BlockGf of GfImFreq, GfImTime, GfReFreq or GfReTime
|
|
the input Green's function
|
|
|
|
Returns
|
|
-------
|
|
gf : list of BlockGf of GfImTime or GfReFreq
|
|
the output G(tau) or A(w)
|
|
"""
|
|
# make a GfImTime from the supplied GfImFreq
|
|
if all(isinstance(g_sh._first(), GfImFreq) for g_sh in G):
|
|
gf = [BlockGf(name_block_generator = [(name, GfImTime(beta=block.mesh.beta,
|
|
indices=block.indices,n_points=len(block.mesh)+1)) for name, block in g_sh],
|
|
make_copies=False) for g_sh in G]
|
|
for ish in range(len(gf)):
|
|
for name, g in gf[ish]:
|
|
g.set_from_inverse_fourier(G[ish][name])
|
|
# keep a GfImTime from the supplied GfImTime
|
|
elif all(isinstance(g_sh._first(), GfImTime) for g_sh in G):
|
|
gf = G
|
|
# make a spectral function from the supplied GfReFreq
|
|
elif all(isinstance(g_sh._first(), GfReFreq) for g_sh in G):
|
|
gf = [g_sh.copy() for g_sh in G]
|
|
for ish in range(len(gf)):
|
|
for name, g in gf[ish]:
|
|
g << 1.0j*(g-g.conjugate().transpose())/2.0/numpy.pi
|
|
elif all(isinstance(g_sh._first(), GfReTime) for g_sh in G):
|
|
def get_delta_from_mesh(mesh):
|
|
w0 = None
|
|
for w in mesh:
|
|
if w0 is None:
|
|
w0 = w
|
|
else:
|
|
return w-w0
|
|
gf = [BlockGf(name_block_generator = [(name, GfReFreq(
|
|
window=(-numpy.pi*(len(block.mesh)-1) / (len(block.mesh)*get_delta_from_mesh(block.mesh)),
|
|
numpy.pi*(len(block.mesh)-1) / (len(block.mesh)*get_delta_from_mesh(block.mesh))),
|
|
n_points=len(block.mesh), indices=block.indices)) for name, block in g_sh], make_copies=False)
|
|
for g_sh in G]
|
|
|
|
for ish in range(len(gf)):
|
|
for name, g in gf[ish]:
|
|
g.set_from_fourier(G[ish][name])
|
|
g << 1.0j*(g-g.conjugate().transpose())/2.0/numpy.pi
|
|
else:
|
|
raise Exception("G must be a list of BlockGf of either GfImFreq, GfImTime, GfReFreq or GfReTime")
|
|
return gf
|
|
|
|
|
|
|
|
def analyse_block_structure_from_gf(self, G, threshold=1.e-5, include_shells=None, analyse_deg_shells = True):
|
|
r"""
|
|
Determines the block structure of local Green's functions by analysing
|
|
the structure of the corresponding non-interacting Green's function.
|
|
The resulting block structures for correlated shells are
|
|
stored in the :class:`SumkDFT.block_structure <dft.block_structure.BlockStructure>`
|
|
attribute.
|
|
|
|
This is a safer alternative to analyse_block_structure, because
|
|
the full non-interacting Green's function is taken into account
|
|
and not just the density matrix and Hloc.
|
|
|
|
Parameters
|
|
----------
|
|
G : list of BlockGf of GfImFreq, GfImTime, GfReFreq or GfReTime
|
|
the non-interacting Green's function for each inequivalent correlated shell
|
|
threshold : real, optional
|
|
If the difference between matrix elements is below threshold,
|
|
they are considered to be equal.
|
|
include_shells : list of integers, optional
|
|
List of correlated shells to be analysed.
|
|
If include_shells is not provided all correlated shells will be analysed.
|
|
analyse_deg_shells : bool
|
|
Whether to call the analyse_deg_shells function
|
|
after having finished the block structure analysis
|
|
|
|
Returns
|
|
-------
|
|
G : list of BlockGf of GfImFreq or GfImTime
|
|
the Green's function transformed into the new block structure
|
|
"""
|
|
|
|
gf = self._get_hermitian_quantity_from_gf(G)
|
|
|
|
# initialize the variables
|
|
self.gf_struct_solver = [{} for ish in range(self.n_inequiv_shells)]
|
|
self.sumk_to_solver = [{} for ish in range(self.n_inequiv_shells)]
|
|
self.solver_to_sumk = [{} for ish in range(self.n_inequiv_shells)]
|
|
self.solver_to_sumk_block = [{}
|
|
for ish in range(self.n_inequiv_shells)]
|
|
|
|
# the maximum value of each matrix element of each block and shell
|
|
max_gf = [{name:numpy.max(numpy.abs(g.data),0) for name, g in gf[ish]} for ish in range(self.n_inequiv_shells)]
|
|
|
|
if include_shells is None:
|
|
# include all shells
|
|
include_shells = range(self.n_inequiv_shells)
|
|
|
|
for ish in include_shells:
|
|
for sp in self.spin_block_names[self.corr_shells[self.inequiv_to_corr[ish]]['SO']]:
|
|
n_orb = self.corr_shells[self.inequiv_to_corr[ish]]['dim']
|
|
|
|
# gives an index list of entries larger that threshold
|
|
maxgf_bool = (abs(max_gf[ish][sp]) > threshold)
|
|
|
|
# Determine off-diagonal entries in upper triangular part of the
|
|
# Green's function
|
|
offdiag = Set([])
|
|
for i in range(n_orb):
|
|
for j in range(i + 1, n_orb):
|
|
if maxgf_bool[i, j]:
|
|
offdiag.add((i, j))
|
|
|
|
# Determine the number of non-hybridising blocks in the gf
|
|
blocs = [[i] for i in range(n_orb)]
|
|
while len(offdiag) != 0:
|
|
pair = offdiag.pop()
|
|
for b1, b2 in product(blocs, blocs):
|
|
if (pair[0] in b1) and (pair[1] in b2):
|
|
if blocs.index(b1) != blocs.index(b2): # In separate blocks?
|
|
# Merge two blocks
|
|
b1.extend(blocs.pop(blocs.index(b2)))
|
|
break # Move on to next pair in offdiag
|
|
|
|
# Set the gf_struct for the solver accordingly
|
|
num_blocs = len(blocs)
|
|
for i in range(num_blocs):
|
|
blocs[i].sort()
|
|
self.gf_struct_solver[ish].update(
|
|
[('%s_%s' % (sp, i), range(len(blocs[i])))])
|
|
|
|
# Construct sumk_to_solver taking (sumk_block, sumk_index) --> (solver_block, solver_inner)
|
|
# and solver_to_sumk taking (solver_block, solver_inner) -->
|
|
# (sumk_block, sumk_index)
|
|
for i in range(num_blocs):
|
|
for j in range(len(blocs[i])):
|
|
block_sumk = sp
|
|
inner_sumk = blocs[i][j]
|
|
block_solv = '%s_%s' % (sp, i)
|
|
inner_solv = j
|
|
self.sumk_to_solver[ish][(block_sumk, inner_sumk)] = (
|
|
block_solv, inner_solv)
|
|
self.solver_to_sumk[ish][(block_solv, inner_solv)] = (
|
|
block_sumk, inner_sumk)
|
|
self.solver_to_sumk_block[ish][block_solv] = block_sumk
|
|
|
|
# transform G to the new structure
|
|
full_structure = BlockStructure.full_structure(
|
|
[{sp:range(self.corr_shells[self.inequiv_to_corr[ish]]['dim'])
|
|
for sp in self.spin_block_names[self.corr_shells[self.inequiv_to_corr[ish]]['SO']]}
|
|
for ish in range(self.n_inequiv_shells)],None)
|
|
G_transformed = [
|
|
self.block_structure.convert_gf(G[ish],
|
|
full_structure, ish, mesh=G[ish].mesh.copy(), show_warnings=threshold,
|
|
gf_function=type(G[ish]._first()))
|
|
for ish in range(self.n_inequiv_shells)]
|
|
|
|
if analyse_deg_shells:
|
|
self.analyse_deg_shells(G_transformed, threshold, include_shells)
|
|
return G_transformed
|
|
|
|
def analyse_deg_shells(self, G, threshold=1.e-5, include_shells=None):
|
|
r"""
|
|
Determines the degenerate shells of local Green's functions by analysing
|
|
the structure of the corresponding non-interacting Green's function.
|
|
The results are stored in the
|
|
:class:`SumkDFT.block_structure <dft.block_structure.BlockStructure>`
|
|
attribute.
|
|
|
|
Due to the implementation and numerics, the maximum difference between
|
|
two matrix elements that are detected as equal can be a bit higher
|
|
(e.g. a factor of two) than the actual threshold.
|
|
|
|
Parameters
|
|
----------
|
|
G : list of BlockGf of GfImFreq or GfImTime
|
|
the non-interacting Green's function for each inequivalent correlated shell
|
|
threshold : real, optional
|
|
If the difference between matrix elements is below threshold,
|
|
they are considered to be equal.
|
|
include_shells : list of integers, optional
|
|
List of correlated shells to be analysed.
|
|
If include_shells is not provided all correlated shells will be analysed.
|
|
"""
|
|
|
|
# initialize
|
|
self.deg_shells = [[] for ish in range(self.n_inequiv_shells)]
|
|
|
|
# helper function
|
|
def null(A, eps=1e-15):
|
|
""" Calculate the null-space of matrix A """
|
|
u, s, vh = numpy.linalg.svd(A)
|
|
null_mask = (s <= eps)
|
|
null_space = compress(null_mask, vh, axis=0)
|
|
return null_space.conjugate().transpose()
|
|
|
|
gf = self._get_hermitian_quantity_from_gf(G)
|
|
|
|
if include_shells is None:
|
|
# include all shells
|
|
include_shells = range(self.n_inequiv_shells)
|
|
|
|
# We consider two blocks equal, if their Green's functions obey
|
|
# maybe_conjugate1( v1^dagger G1 v1 ) = maybe_conjugate2( v2^dagger G2 v2 )
|
|
# where maybe_conjugate is a function that conjugates the Green's
|
|
# function if the flag 'conjugate' is set and the v are unitary
|
|
# matrices
|
|
#
|
|
# for each pair of blocks, we check whether there is a transformation
|
|
# maybe_conjugate( T G1 T^dagger ) = G2
|
|
# where our goal is to find T
|
|
# we just try whether there is such a T with and without conjugation
|
|
for ish in include_shells:
|
|
for block1 in self.gf_struct_solver[ish].iterkeys():
|
|
for block2 in self.gf_struct_solver[ish].iterkeys():
|
|
if block1==block2: continue
|
|
|
|
# check if the blocks are already present in the deg_shells
|
|
ind1 = -1
|
|
ind2 = -2
|
|
for n, ind in enumerate(self.deg_shells[ish]):
|
|
if block1 in ind:
|
|
ind1 = n
|
|
v1 = ind[block1]
|
|
if block2 in ind:
|
|
ind2 = n
|
|
v2 = ind[block2]
|
|
|
|
# if both are already present, go to the next pair of blocks
|
|
if ind1 >= 0 and ind2 >= 0:
|
|
continue
|
|
|
|
gf1 = gf[ish][block1]
|
|
gf2 = gf[ish][block2]
|
|
|
|
# the two blocks have to have the same shape
|
|
if gf1.target_shape != gf2.target_shape:
|
|
continue
|
|
|
|
# Instead of directly comparing the two blocks, we
|
|
# compare its eigenvalues. As G(tau) is Hermitian,
|
|
# they are real and the eigenvector matrix is unitary.
|
|
# Thus, if the eigenvalues are equal we can transform
|
|
# one block to make it equal to the other (at least
|
|
# for tau=0).
|
|
|
|
e1 = numpy.linalg.eigvalsh(gf1.data[0])
|
|
e2 = numpy.linalg.eigvalsh(gf2.data[0])
|
|
if numpy.any(abs(e1-e2) > threshold): continue
|
|
|
|
for conjugate in [False,True]:
|
|
if conjugate:
|
|
gf2 = gf2.conjugate()
|
|
|
|
# we want T gf1 T^dagger = gf2
|
|
# while for a given tau, T could be calculated
|
|
# by diagonalizing gf1 and gf2, this does not
|
|
# work for all taus simultaneously because of
|
|
# numerical imprecisions
|
|
|
|
# rather, we rewrite the equation to
|
|
# T gf1 = gf2 T
|
|
# which is the Sylvester equation.
|
|
# For that equation, one can use the Kronecker
|
|
# product to get a linear problem, which consists
|
|
# of finding the null space of M vec T = 0.
|
|
|
|
M = numpy.kron(numpy.eye(*gf1.target_shape),gf2.data[0])-numpy.kron(gf1.data[0].transpose(),numpy.eye(*gf1.target_shape))
|
|
N = null(M, threshold)
|
|
|
|
# now we get the intersection of the null spaces
|
|
# of all values of tau
|
|
for i in range(1,len(gf1.data)):
|
|
M = numpy.kron(numpy.eye(*gf1.target_shape),gf2.data[i])-numpy.kron(gf1.data[i].transpose(),numpy.eye(*gf1.target_shape))
|
|
# transform M into current null space
|
|
M = numpy.dot(M, N)
|
|
N = numpy.dot(N, null(M, threshold))
|
|
if numpy.size(N) == 0:
|
|
break
|
|
|
|
# no intersection of the null spaces -> no symmetry
|
|
if numpy.size(N) == 0: continue
|
|
|
|
# reshape N: it then has the indices matrix, matrix, number of basis vectors of the null space
|
|
N = N.reshape(gf1.target_shape[0], gf1.target_shape[1], -1).transpose([1, 0, 2])
|
|
|
|
"""
|
|
any matrix in the null space can now be constructed as
|
|
M = 0
|
|
for i in range(N.shape[-1]):
|
|
M += y[i]*N[:,:,i]
|
|
with coefficients (complex numbers) y[i].
|
|
|
|
We want to get a set of coefficients y so that M is unitary.
|
|
Unitary means M M^dagger = 1.
|
|
Thus,
|
|
sum y[i] N[:,:,i] y[j].conjugate() N[:,:,j].conjugate().transpose() = eye.
|
|
The object N[:,:,i] N[:,:,j] is a four-index object which we call Z.
|
|
"""
|
|
Z = numpy.einsum('aci,bcj->abij', N, N.conjugate())
|
|
|
|
"""
|
|
function chi2
|
|
This function takes a real parameter vector y and reinterprets it as complex.
|
|
Then, it calculates the chi2 of
|
|
sum y[i] N[:,:,i] y[j].conjugate() N[:,:,j].conjugate().transpose() - eye.
|
|
"""
|
|
def chi2(y):
|
|
# reinterpret y as complex number
|
|
y = y.view(numpy.complex_)
|
|
ret = 0.0
|
|
for a in range(Z.shape[0]):
|
|
for b in range(Z.shape[1]):
|
|
ret += numpy.abs(numpy.dot(y, numpy.dot(Z[a, b], y.conjugate()))
|
|
- (1.0 if a == b else 0.0))**2
|
|
return ret
|
|
|
|
# use the minimization routine from scipy
|
|
res = minimize(chi2, numpy.ones(2 * N.shape[-1]))
|
|
|
|
# if the minimization fails, there is probably no symmetry
|
|
if not res.success: continue
|
|
# check if the minimization returned zero within the tolerance
|
|
if res.fun > threshold: continue
|
|
|
|
# reinterpret the solution as a complex number
|
|
y = res.x.view(numpy.complex_)
|
|
|
|
# reconstruct the T matrix
|
|
T = numpy.zeros(N.shape[:-1], dtype=numpy.complex_)
|
|
for i in range(len(y)):
|
|
T += N[:, :, i] * y[i]
|
|
|
|
# transform gf1 using T
|
|
G_transformed = gf1.copy()
|
|
G_transformed.from_L_G_R(T, gf1, T.conjugate().transpose())
|
|
|
|
# it does not make sense to check the tails for an
|
|
# absolute error because it will usually not hold;
|
|
# we could just check the relative error
|
|
# (here, we ignore it, reasoning that if the data
|
|
# is the same, the tails have to coincide as well)
|
|
try:
|
|
assert_arrays_are_close(G_transformed.data, gf2.data, threshold)
|
|
except (RuntimeError, AssertionError):
|
|
# the symmetry does not hold
|
|
continue
|
|
|
|
# Now that we have found a valid T, we have to
|
|
# rewrite it to match the convention that
|
|
# C1(v1^dagger G1 v1) = C2(v2^dagger G2 v2),
|
|
# where C conjugates if the flag is True
|
|
|
|
# For each group of degenerate shells, the list
|
|
# SK.deg_shells[ish] contains a dict. The keys
|
|
# of the dict are the block names, the values
|
|
# are tuples. The first entry of the tuple is
|
|
# the transformation matrix v, the second entry
|
|
# is the conjugation flag
|
|
|
|
# the second block is already present
|
|
# set v1 and C1 so that they are compatible with
|
|
# C(T gf1 T^dagger) = gf2
|
|
# and with
|
|
# C1(v1^dagger G1 v1) = C2(v2^dagger G2 v2)
|
|
if (ind1 < 0) and (ind2 >= 0):
|
|
if conjugate:
|
|
self.deg_shells[ish][ind2][block1] = numpy.dot(T.conjugate().transpose(), v2[0].conjugate()), not v2[1]
|
|
else:
|
|
self.deg_shells[ish][ind2][block1] = numpy.dot(T.conjugate().transpose(), v2[0]), v2[1]
|
|
# the first block is already present
|
|
# set v2 and C2 so that they are compatible with
|
|
# C(T gf1 T^dagger) = gf2
|
|
# and with
|
|
# C1(v1^dagger G1 v1) = C2(v2^dagger G2 v2)
|
|
elif (ind1 >= 0) and (ind2 < 0):
|
|
if conjugate:
|
|
self.deg_shells[ish][ind1][block2] = numpy.dot(T.conjugate(), v1[0].conjugate()), not v1[1]
|
|
else:
|
|
self.deg_shells[ish][ind1][block2] = numpy.dot(T, v1[0]), v1[1]
|
|
# the blocks are not already present
|
|
# we arbitrarily choose v1=eye and C1=False and
|
|
# set v2 and C2 so that they are compatible with
|
|
# C(T gf1 T^dagger) = gf2
|
|
# and with
|
|
# C1(v1^dagger G1 v1) = C2(v2^dagger G2 v2)
|
|
elif (ind1 < 0) and (ind2 < 0):
|
|
d = dict()
|
|
d[block1] = numpy.eye(*gf1.target_shape), False
|
|
if conjugate:
|
|
d[block2] = T.conjugate(), True
|
|
else:
|
|
d[block2] = T, False
|
|
self.deg_shells[ish].append(d)
|
|
|
|
# a block was found, break out of the loop
|
|
break
|
|
|
|
|
|
def density_matrix(self, method='using_gf', beta=40.0):
|
|
"""Calculate density matrices in one of two ways.
|
|
|
|
Parameters
|
|
----------
|
|
method : string, optional
|
|
|
|
- if 'using_gf': First get lattice gf (g_loc is not set up), then density matrix.
|
|
It is useful for Hubbard I, and very quick.
|
|
No assumption on the hopping structure is made (ie diagonal or not).
|
|
- if 'using_point_integration': Only works for diagonal hopping matrix (true in wien2k).
|
|
|
|
beta : float, optional
|
|
Inverse temperature.
|
|
|
|
Returns
|
|
-------
|
|
dens_mat : list of dicts
|
|
Density matrix for each spin in each correlated shell.
|
|
"""
|
|
dens_mat = [{} for icrsh in range(self.n_corr_shells)]
|
|
for icrsh in range(self.n_corr_shells):
|
|
for sp in self.spin_block_names[self.corr_shells[icrsh]['SO']]:
|
|
dens_mat[icrsh][sp] = numpy.zeros(
|
|
[self.corr_shells[icrsh]['dim'], self.corr_shells[icrsh]['dim']], numpy.complex_)
|
|
|
|
ikarray = numpy.array(range(self.n_k))
|
|
for ik in mpi.slice_array(ikarray):
|
|
|
|
if method == "using_gf":
|
|
|
|
G_latt_iw = self.lattice_gf(
|
|
ik=ik, mu=self.chemical_potential, iw_or_w="iw", beta=beta)
|
|
G_latt_iw *= self.bz_weights[ik]
|
|
dm = G_latt_iw.density()
|
|
MMat = [dm[sp] for sp in self.spin_block_names[self.SO]]
|
|
|
|
elif method == "using_point_integration":
|
|
|
|
ntoi = self.spin_names_to_ind[self.SO]
|
|
spn = self.spin_block_names[self.SO]
|
|
dims = {sp:self.n_orbitals[ik, ntoi[sp]] for sp in spn}
|
|
MMat = [numpy.zeros([dims[sp], dims[sp]], numpy.complex_) for sp in spn]
|
|
|
|
for isp, sp in enumerate(spn):
|
|
ind = ntoi[sp]
|
|
for inu in range(self.n_orbitals[ik, ind]):
|
|
# only works for diagonal hopping matrix (true in
|
|
# wien2k)
|
|
if (self.hopping[ik, ind, inu, inu] - self.h_field * (1 - 2 * isp)) < 0.0:
|
|
MMat[isp][inu, inu] = 1.0
|
|
else:
|
|
MMat[isp][inu, inu] = 0.0
|
|
|
|
else:
|
|
raise ValueError, "density_matrix: the method '%s' is not supported." % method
|
|
|
|
for icrsh in range(self.n_corr_shells):
|
|
for isp, sp in enumerate(self.spin_block_names[self.corr_shells[icrsh]['SO']]):
|
|
ind = self.spin_names_to_ind[
|
|
self.corr_shells[icrsh]['SO']][sp]
|
|
dim = self.corr_shells[icrsh]['dim']
|
|
n_orb = self.n_orbitals[ik, ind]
|
|
projmat = self.proj_mat[ik, ind, icrsh, 0:dim, 0:n_orb]
|
|
if method == "using_gf":
|
|
dens_mat[icrsh][sp] += numpy.dot(numpy.dot(projmat, MMat[isp]),
|
|
projmat.transpose().conjugate())
|
|
elif method == "using_point_integration":
|
|
dens_mat[icrsh][sp] += self.bz_weights[ik] * numpy.dot(numpy.dot(projmat, MMat[isp]),
|
|
projmat.transpose().conjugate())
|
|
|
|
# get data from nodes:
|
|
for icrsh in range(self.n_corr_shells):
|
|
for sp in dens_mat[icrsh]:
|
|
dens_mat[icrsh][sp] = mpi.all_reduce(
|
|
mpi.world, dens_mat[icrsh][sp], lambda x, y: x + y)
|
|
mpi.barrier()
|
|
|
|
if self.symm_op != 0:
|
|
dens_mat = self.symmcorr.symmetrize(dens_mat)
|
|
|
|
# Rotate to local coordinate system:
|
|
if self.use_rotations:
|
|
for icrsh in range(self.n_corr_shells):
|
|
for sp in dens_mat[icrsh]:
|
|
if self.rot_mat_time_inv[icrsh] == 1:
|
|
dens_mat[icrsh][sp] = dens_mat[icrsh][sp].conjugate()
|
|
dens_mat[icrsh][sp] = numpy.dot(numpy.dot(self.rot_mat[icrsh].conjugate().transpose(), dens_mat[icrsh][sp]),
|
|
self.rot_mat[icrsh])
|
|
|
|
return dens_mat
|
|
|
|
# For simple dft input, get crystal field splittings.
|
|
def eff_atomic_levels(self):
|
|
r"""
|
|
Calculates the effective local Hamiltonian required as an input for
|
|
the Hubbard I Solver.
|
|
The local Hamiltonian (effective atomic levels) is calculated by
|
|
projecting the on-site Bloch Hamiltonian:
|
|
|
|
.. math:: H^{loc}_{m m'} = \sum_{k} P_{m \nu}(k) H_{\nu\nu'}(k) P^{*}_{\nu' m'}(k),
|
|
|
|
where
|
|
|
|
.. math:: H_{\nu\nu'}(k) = [\epsilon_{\nu k} - h_{z} \sigma_{z}] \delta_{\nu\nu'}.
|
|
|
|
Parameters
|
|
----------
|
|
None
|
|
|
|
Returns
|
|
-------
|
|
eff_atlevels : gf_struct_sumk like
|
|
Effective local Hamiltonian :math:`H^{loc}_{m m'}` for each
|
|
inequivalent correlated shell.
|
|
|
|
"""
|
|
|
|
# define matrices for inequivalent shells:
|
|
eff_atlevels = [{} for ish in range(self.n_inequiv_shells)]
|
|
for ish in range(self.n_inequiv_shells):
|
|
for sp in self.spin_block_names[self.corr_shells[self.inequiv_to_corr[ish]]['SO']]:
|
|
eff_atlevels[ish][sp] = numpy.identity(
|
|
self.corr_shells[self.inequiv_to_corr[ish]]['dim'], numpy.complex_)
|
|
eff_atlevels[ish][sp] *= -self.chemical_potential
|
|
eff_atlevels[ish][
|
|
sp] -= self.dc_imp[self.inequiv_to_corr[ish]][sp]
|
|
|
|
# sum over k:
|
|
if not hasattr(self, "Hsumk"):
|
|
# calculate the sum over k. Does not depend on mu, so do it only
|
|
# once:
|
|
self.Hsumk = [{} for icrsh in range(self.n_corr_shells)]
|
|
for icrsh in range(self.n_corr_shells):
|
|
dim = self.corr_shells[icrsh]['dim']
|
|
for sp in self.spin_block_names[self.corr_shells[icrsh]['SO']]:
|
|
self.Hsumk[icrsh][sp] = numpy.zeros(
|
|
[dim, dim], numpy.complex_)
|
|
for isp, sp in enumerate(self.spin_block_names[self.corr_shells[icrsh]['SO']]):
|
|
ind = self.spin_names_to_ind[
|
|
self.corr_shells[icrsh]['SO']][sp]
|
|
for ik in range(self.n_k):
|
|
n_orb = self.n_orbitals[ik, ind]
|
|
MMat = numpy.identity(n_orb, numpy.complex_)
|
|
MMat = self.hopping[
|
|
ik, ind, 0:n_orb, 0:n_orb] - (1 - 2 * isp) * self.h_field * MMat
|
|
projmat = self.proj_mat[ik, ind, icrsh, 0:dim, 0:n_orb]
|
|
self.Hsumk[icrsh][sp] += self.bz_weights[ik] * numpy.dot(numpy.dot(projmat, MMat),
|
|
projmat.conjugate().transpose())
|
|
# symmetrisation:
|
|
if self.symm_op != 0:
|
|
self.Hsumk = self.symmcorr.symmetrize(self.Hsumk)
|
|
|
|
# Rotate to local coordinate system:
|
|
if self.use_rotations:
|
|
for icrsh in range(self.n_corr_shells):
|
|
for sp in self.Hsumk[icrsh]:
|
|
if self.rot_mat_time_inv[icrsh] == 1:
|
|
self.Hsumk[icrsh][sp] = self.Hsumk[
|
|
icrsh][sp].conjugate()
|
|
self.Hsumk[icrsh][sp] = numpy.dot(numpy.dot(self.rot_mat[icrsh].conjugate().transpose(), self.Hsumk[icrsh][sp]),
|
|
self.rot_mat[icrsh])
|
|
|
|
# add to matrix:
|
|
for ish in range(self.n_inequiv_shells):
|
|
for sp in eff_atlevels[ish]:
|
|
eff_atlevels[ish][
|
|
sp] += self.Hsumk[self.inequiv_to_corr[ish]][sp]
|
|
|
|
return eff_atlevels
|
|
|
|
def init_dc(self):
|
|
r"""
|
|
Initializes the double counting terms.
|
|
|
|
Parameters
|
|
----------
|
|
None
|
|
|
|
"""
|
|
self.dc_imp = [{} for icrsh in range(self.n_corr_shells)]
|
|
for icrsh in range(self.n_corr_shells):
|
|
dim = self.corr_shells[icrsh]['dim']
|
|
spn = self.spin_block_names[self.corr_shells[icrsh]['SO']]
|
|
for sp in spn:
|
|
self.dc_imp[icrsh][sp] = numpy.zeros([dim, dim], numpy.float_)
|
|
self.dc_energ = [0.0 for icrsh in range(self.n_corr_shells)]
|
|
|
|
def set_dc(self, dc_imp, dc_energ):
|
|
r"""
|
|
Sets double counting corrections to given values.
|
|
|
|
Parameters
|
|
----------
|
|
dc_imp : gf_struct_sumk like
|
|
Double-counting self-energy term.
|
|
dc_energ : list of floats
|
|
Double-counting energy corrections for each correlated shell.
|
|
|
|
"""
|
|
|
|
self.dc_imp = dc_imp
|
|
self.dc_energ = dc_energ
|
|
|
|
def calc_dc(self, dens_mat, orb=0, U_interact=None, J_hund=None, use_dc_formula=0, use_dc_value=None):
|
|
r"""
|
|
Calculates and sets the double counting corrections.
|
|
|
|
If 'use_dc_value' is provided the double-counting term is uniformly initialized
|
|
with this constant and 'U_interact' and 'J_hund' are ignored.
|
|
|
|
If 'use_dc_value' is None the correction is evaluated according to
|
|
one of the following formulae:
|
|
|
|
* use_dc_formula = 0: fully-localised limit (FLL)
|
|
* use_dc_formula = 1: Held's formula, i.e. mean-field formula for the Kanamori
|
|
type of the interaction Hamiltonian
|
|
* use_dc_formula = 2: around mean-field (AMF)
|
|
|
|
Note that FLL and AMF formulae were derived assuming a full Slater-type interaction
|
|
term and should be thus used accordingly. For the Kanamori-type interaction
|
|
one should use formula 1.
|
|
|
|
The double-counting self-energy term is stored in `self.dc_imp` and the energy
|
|
correction in `self.dc_energ`.
|
|
|
|
Parameters
|
|
----------
|
|
dens_mat : gf_struct_solver like
|
|
Density matrix for the specified correlated shell.
|
|
orb : int, optional
|
|
Index of an inequivalent shell.
|
|
U_interact : float, optional
|
|
Value of interaction parameter `U`.
|
|
J_hund : float, optional
|
|
Value of interaction parameter `J`.
|
|
use_dc_formula : int, optional
|
|
Type of double-counting correction (see description).
|
|
use_dc_value : float, optional
|
|
Value of the double-counting correction. If specified
|
|
`U_interact`, `J_hund` and `use_dc_formula` are ignored.
|
|
|
|
"""
|
|
|
|
for icrsh in range(self.n_corr_shells):
|
|
|
|
# ish is the index of the inequivalent shell corresponding to icrsh
|
|
ish = self.corr_to_inequiv[icrsh]
|
|
if ish != orb:
|
|
continue # ignore this orbital
|
|
# *(1+self.corr_shells[icrsh]['SO'])
|
|
dim = self.corr_shells[icrsh]['dim']
|
|
spn = self.spin_block_names[self.corr_shells[icrsh]['SO']]
|
|
|
|
Ncr = {sp: 0.0 for sp in spn}
|
|
for block, inner in self.gf_struct_solver[ish].iteritems():
|
|
bl = self.solver_to_sumk_block[ish][block]
|
|
Ncr[bl] += dens_mat[block].real.trace()
|
|
Ncrtot = sum(Ncr.itervalues())
|
|
for sp in spn:
|
|
self.dc_imp[icrsh][sp] = numpy.identity(dim, numpy.float_)
|
|
if self.SP == 0: # average the densities if there is no SP:
|
|
Ncr[sp] = Ncrtot / len(spn)
|
|
# correction for SO: we have only one block in this case, but
|
|
# in DC we need N/2
|
|
elif self.SP == 1 and self.SO == 1:
|
|
Ncr[sp] = Ncrtot / 2.0
|
|
|
|
if use_dc_value is None:
|
|
|
|
if U_interact is None and J_hund is None:
|
|
raise ValueError, "set_dc: either provide U_interact and J_hund or set use_dc_value to dc value."
|
|
|
|
if use_dc_formula == 0: # FLL
|
|
|
|
self.dc_energ[icrsh] = U_interact / \
|
|
2.0 * Ncrtot * (Ncrtot - 1.0)
|
|
for sp in spn:
|
|
Uav = U_interact * (Ncrtot - 0.5) - \
|
|
J_hund * (Ncr[sp] - 0.5)
|
|
self.dc_imp[icrsh][sp] *= Uav
|
|
self.dc_energ[icrsh] -= J_hund / \
|
|
2.0 * (Ncr[sp]) * (Ncr[sp] - 1.0)
|
|
mpi.report(
|
|
"DC for shell %(icrsh)i and block %(sp)s = %(Uav)f" % locals())
|
|
|
|
elif use_dc_formula == 1: # Held's formula, with U_interact the interorbital onsite interaction
|
|
|
|
self.dc_energ[icrsh] = (U_interact + (dim - 1) * (U_interact - 2.0 * J_hund) + (
|
|
dim - 1) * (U_interact - 3.0 * J_hund)) / (2 * dim - 1) / 2.0 * Ncrtot * (Ncrtot - 1.0)
|
|
for sp in spn:
|
|
Uav = (U_interact + (dim - 1) * (U_interact - 2.0 * J_hund) + (dim - 1)
|
|
* (U_interact - 3.0 * J_hund)) / (2 * dim - 1) * (Ncrtot - 0.5)
|
|
self.dc_imp[icrsh][sp] *= Uav
|
|
mpi.report(
|
|
"DC for shell %(icrsh)i and block %(sp)s = %(Uav)f" % locals())
|
|
|
|
elif use_dc_formula == 2: # AMF
|
|
|
|
self.dc_energ[icrsh] = 0.5 * U_interact * Ncrtot * Ncrtot
|
|
for sp in spn:
|
|
Uav = U_interact * \
|
|
(Ncrtot - Ncr[sp] / dim) - \
|
|
J_hund * (Ncr[sp] - Ncr[sp] / dim)
|
|
self.dc_imp[icrsh][sp] *= Uav
|
|
self.dc_energ[
|
|
icrsh] -= (U_interact + (dim - 1) * J_hund) / dim * 0.5 * Ncr[sp] * Ncr[sp]
|
|
mpi.report(
|
|
"DC for shell %(icrsh)i and block %(sp)s = %(Uav)f" % locals())
|
|
|
|
mpi.report("DC energy for shell %s = %s" %
|
|
(icrsh, self.dc_energ[icrsh]))
|
|
|
|
else: # use value provided for user to determine dc_energ and dc_imp
|
|
|
|
self.dc_energ[icrsh] = use_dc_value * Ncrtot
|
|
for sp in spn:
|
|
self.dc_imp[icrsh][sp] *= use_dc_value
|
|
|
|
mpi.report(
|
|
"DC for shell %(icrsh)i = %(use_dc_value)f" % locals())
|
|
mpi.report("DC energy = %s" % self.dc_energ[icrsh])
|
|
|
|
def add_dc(self, iw_or_w="iw"):
|
|
r"""
|
|
Subtracts the double counting term from the impurity self energy.
|
|
|
|
Parameters
|
|
----------
|
|
iw_or_w : string, optional
|
|
|
|
- `iw_or_w` = 'iw' for a imaginary-frequency self-energy
|
|
- `iw_or_w` = 'w' for a real-frequency self-energy
|
|
|
|
Returns
|
|
-------
|
|
sigma_minus_dc : gf_struct_sumk like
|
|
Self-energy with a subtracted double-counting term.
|
|
|
|
"""
|
|
|
|
# Be careful: Sigma_imp is already in the global coordinate system!!
|
|
sigma_minus_dc = [s.copy()
|
|
for s in getattr(self, "Sigma_imp_" + iw_or_w)]
|
|
for icrsh in range(self.n_corr_shells):
|
|
for bname, gf in sigma_minus_dc[icrsh]:
|
|
# Transform dc_imp to global coordinate system
|
|
dccont = numpy.dot(self.rot_mat[icrsh], numpy.dot(self.dc_imp[icrsh][
|
|
bname], self.rot_mat[icrsh].conjugate().transpose()))
|
|
sigma_minus_dc[icrsh][bname] -= dccont
|
|
|
|
return sigma_minus_dc
|
|
|
|
def symm_deg_gf(self, gf_to_symm, orb):
|
|
r"""
|
|
Averages a GF over degenerate shells.
|
|
|
|
Degenerate shells of an inequivalent correlated shell are defined by
|
|
`self.deg_shells`. This function enforces corresponding degeneracies
|
|
in the input GF.
|
|
|
|
Parameters
|
|
----------
|
|
gf_to_symm : gf_struct_solver like
|
|
Input and output GF (i.e., it gets overwritten)
|
|
orb : int
|
|
Index of an inequivalent shell.
|
|
|
|
"""
|
|
|
|
# when reading block_structures written with older versions from
|
|
# an h5 file, self.deg_shells might be None
|
|
if self.deg_shells is None: return
|
|
|
|
for degsh in self.deg_shells[orb]:
|
|
# ss will hold the averaged orbitals in the basis where the
|
|
# blocks are all equal
|
|
# i.e. maybe_conjugate(v^dagger gf v)
|
|
ss = None
|
|
n_deg = len(degsh)
|
|
for key in degsh:
|
|
if ss is None:
|
|
ss = gf_to_symm[key].copy()
|
|
ss.zero()
|
|
helper = ss.copy()
|
|
# get the transformation matrix
|
|
if isinstance(degsh, dict):
|
|
v, C = degsh[key]
|
|
else:
|
|
# for backward compatibility, allow degsh to be a list
|
|
v = numpy.eye(*ss.target_shape)
|
|
C = False
|
|
# the helper is in the basis where the blocks are all equal
|
|
helper.from_L_G_R(v.conjugate().transpose(), gf_to_symm[key], v)
|
|
if C:
|
|
helper << helper.transpose()
|
|
# average over all shells
|
|
ss += helper / (1.0 * n_deg)
|
|
# now put back the averaged gf to all shells
|
|
for key in degsh:
|
|
if isinstance(degsh, dict):
|
|
v, C = degsh[key]
|
|
else:
|
|
# for backward compatibility, allow degsh to be a list
|
|
v = numpy.eye(*ss.target_shape)
|
|
C = False
|
|
if C:
|
|
gf_to_symm[key].from_L_G_R(v, ss.transpose(), v.conjugate().transpose())
|
|
else:
|
|
gf_to_symm[key].from_L_G_R(v, ss, v.conjugate().transpose())
|
|
|
|
def total_density(self, mu=None, iw_or_w="iw", with_Sigma=True, with_dc=True, broadening=None):
|
|
r"""
|
|
Calculates the total charge within the energy window for a given chemical potential.
|
|
The chemical potential is either given by parameter `mu` or, if it is not specified,
|
|
taken from `self.chemical_potential`.
|
|
|
|
The total charge is calculated from the trace of the GF in the Bloch basis.
|
|
By default, a full interacting GF is used. To use the non-interacting GF, set
|
|
parameter `with_Sigma = False`.
|
|
|
|
The number of bands within the energy windows generally depends on `k`. The trace is
|
|
therefore calculated separately for each `k`-point.
|
|
|
|
Since in general n_orbitals depends on k, the calculation is done in the following order:
|
|
|
|
.. math:: n_{tot} = \sum_{k} n(k),
|
|
|
|
with
|
|
|
|
.. math:: n(k) = Tr G_{\nu\nu'}(k, i\omega_{n}).
|
|
|
|
The calculation is done in the global coordinate system, if distinction is made between local/global.
|
|
|
|
Parameters
|
|
----------
|
|
mu : float, optional
|
|
Input chemical potential. If not specified, `self.chemical_potential` is used instead.
|
|
iw_or_w : string, optional
|
|
- `iw_or_w` = 'iw' for a imaginary-frequency self-energy
|
|
- `iw_or_w` = 'w' for a real-frequency self-energy
|
|
with_Sigma : boolean, optional
|
|
If `True` the full interacing GF is evaluated, otherwise the self-energy is not
|
|
included and the charge would correspond to a non-interacting system.
|
|
with_dc : boolean, optional
|
|
Whether or not to subtract the double-counting term from the self-energy.
|
|
broadening : float, optional
|
|
Imaginary shift for the axis along which the real-axis GF is calculated.
|
|
If not provided, broadening will be set to double of the distance between mesh points in 'mesh'.
|
|
Only relevant for real-frequency GF.
|
|
|
|
Returns
|
|
-------
|
|
dens : float
|
|
Total charge :math:`n_{tot}`.
|
|
|
|
"""
|
|
|
|
if mu is None:
|
|
mu = self.chemical_potential
|
|
dens = 0.0
|
|
ikarray = numpy.array(range(self.n_k))
|
|
for ik in mpi.slice_array(ikarray):
|
|
G_latt = self.lattice_gf(
|
|
ik=ik, mu=mu, iw_or_w=iw_or_w, with_Sigma=with_Sigma, with_dc=with_dc, broadening=broadening)
|
|
dens += self.bz_weights[ik] * G_latt.total_density()
|
|
# collect data from mpi:
|
|
dens = mpi.all_reduce(mpi.world, dens, lambda x, y: x + y)
|
|
mpi.barrier()
|
|
|
|
return dens
|
|
|
|
def set_mu(self, mu):
|
|
r"""
|
|
Sets a new chemical potential.
|
|
|
|
Parameters
|
|
----------
|
|
mu : float
|
|
New value of the chemical potential.
|
|
|
|
"""
|
|
self.chemical_potential = mu
|
|
|
|
def calc_mu(self, precision=0.01, iw_or_w='iw', broadening=None, delta=0.5):
|
|
r"""
|
|
Searches for the chemical potential that gives the DFT total charge.
|
|
A simple bisection method is used.
|
|
|
|
Parameters
|
|
----------
|
|
precision : float, optional
|
|
A desired precision of the resulting total charge.
|
|
iw_or_w : string, optional
|
|
- `iw_or_w` = 'iw' for a imaginary-frequency self-energy
|
|
- `iw_or_w` = 'w' for a real-frequency self-energy
|
|
broadening : float, optional
|
|
Imaginary shift for the axis along which the real-axis GF is calculated.
|
|
If not provided, broadening will be set to double of the distance between mesh points in 'mesh'.
|
|
Only relevant for real-frequency GF.
|
|
|
|
Returns
|
|
-------
|
|
mu : float
|
|
Value of the chemical potential giving the DFT total charge
|
|
within specified precision.
|
|
|
|
"""
|
|
F = lambda mu: self.total_density(
|
|
mu=mu, iw_or_w=iw_or_w, broadening=broadening)
|
|
density = self.density_required - self.charge_below
|
|
|
|
self.chemical_potential = dichotomy.dichotomy(function=F,
|
|
x_init=self.chemical_potential, y_value=density,
|
|
precision_on_y=precision, delta_x=delta, max_loops=100,
|
|
x_name="Chemical Potential", y_name="Total Density",
|
|
verbosity=3)[0]
|
|
|
|
return self.chemical_potential
|
|
|
|
def calc_density_correction(self, filename=None, dm_type='wien2k'):
|
|
r"""
|
|
Calculates the charge density correction and stores it into a file.
|
|
|
|
The charge density correction is needed for charge-self-consistent DFT+DMFT calculations.
|
|
It represents a density matrix of the interacting system defined in Bloch basis
|
|
and it is calculated from the sum over Matsubara frequecies of the full GF,
|
|
|
|
..math:: N_{\nu\nu'}(k) = \sum_{i\omega_{n}} G_{\nu\nu'}(k, i\omega_{n})
|
|
|
|
The density matrix for every `k`-point is stored into a file.
|
|
|
|
Parameters
|
|
----------
|
|
filename : string
|
|
Name of the file to store the charge density correction.
|
|
|
|
Returns
|
|
-------
|
|
(deltaN, dens) : tuple
|
|
Returns a tuple containing the density matrix `deltaN` and
|
|
the corresponing total charge `dens`.
|
|
|
|
"""
|
|
assert dm_type in ('vasp', 'wien2k'), "'dm_type' must be either 'vasp' or 'wienk'"
|
|
|
|
if filename is None:
|
|
if dm_type == 'wien2k':
|
|
filename = 'dens_mat.dat'
|
|
elif dm_type == 'vasp':
|
|
filename = 'GAMMA'
|
|
|
|
assert type(filename) == StringType, ("calc_density_correction: "
|
|
"filename has to be a string!")
|
|
|
|
ntoi = self.spin_names_to_ind[self.SO]
|
|
spn = self.spin_block_names[self.SO]
|
|
dens = {sp: 0.0 for sp in spn}
|
|
band_en_correction = 0.0
|
|
|
|
# Fetch Fermi weights and energy window band indices
|
|
if dm_type == 'vasp':
|
|
fermi_weights = 0
|
|
band_window = 0
|
|
if mpi.is_master_node():
|
|
ar = HDFArchive(self.hdf_file,'r')
|
|
fermi_weights = ar['dft_misc_input']['dft_fermi_weights']
|
|
band_window = ar['dft_misc_input']['band_window']
|
|
del ar
|
|
fermi_weights = mpi.bcast(fermi_weights)
|
|
band_window = mpi.bcast(band_window)
|
|
|
|
# Convert Fermi weights to a density matrix
|
|
dens_mat_dft = {}
|
|
for sp in spn:
|
|
dens_mat_dft[sp] = [fermi_weights[ik, ntoi[sp], :].astype(numpy.complex_) for ik in xrange(self.n_k)]
|
|
|
|
|
|
# Set up deltaN:
|
|
deltaN = {}
|
|
for sp in spn:
|
|
deltaN[sp] = [numpy.zeros([self.n_orbitals[ik, ntoi[sp]], self.n_orbitals[
|
|
ik, ntoi[sp]]], numpy.complex_) for ik in range(self.n_k)]
|
|
|
|
ikarray = numpy.array(range(self.n_k))
|
|
for ik in mpi.slice_array(ikarray):
|
|
G_latt_iw = self.lattice_gf(
|
|
ik=ik, mu=self.chemical_potential, iw_or_w="iw")
|
|
for bname, gf in G_latt_iw:
|
|
deltaN[bname][ik] = G_latt_iw[bname].density()
|
|
|
|
dens[bname] += self.bz_weights[ik] * G_latt_iw[bname].total_density()
|
|
if dm_type == 'vasp':
|
|
# In 'vasp'-mode subtract the DFT density matrix
|
|
nb = self.n_orbitals[ik, ntoi[bname]]
|
|
diag_inds = numpy.diag_indices(nb)
|
|
deltaN[bname][ik][diag_inds] -= dens_mat_dft[bname][ik][:nb]
|
|
dens[bname] -= self.bz_weights[ik] * dens_mat_dft[bname][ik].sum().real
|
|
isp = ntoi[bname]
|
|
b1, b2 = band_window[isp][ik, :2]
|
|
nb = b2 - b1 + 1
|
|
assert nb == self.n_orbitals[ik, ntoi[bname]], "Number of bands is inconsistent at ik = %s"%(ik)
|
|
band_en_correction += numpy.dot(deltaN[bname][ik], self.hopping[ik, isp, :nb, :nb]).trace().real * self.bz_weights[ik]
|
|
|
|
# mpi reduce:
|
|
for bname in deltaN:
|
|
for ik in range(self.n_k):
|
|
deltaN[bname][ik] = mpi.all_reduce(
|
|
mpi.world, deltaN[bname][ik], lambda x, y: x + y)
|
|
dens[bname] = mpi.all_reduce(
|
|
mpi.world, dens[bname], lambda x, y: x + y)
|
|
mpi.barrier()
|
|
band_en_correction = mpi.all_reduce(mpi.world, band_en_correction, lambda x,y : x+y)
|
|
|
|
# now save to file:
|
|
if dm_type == 'wien2k':
|
|
if mpi.is_master_node():
|
|
if self.SP == 0:
|
|
f = open(filename, 'w')
|
|
else:
|
|
f = open(filename + 'up', 'w')
|
|
f1 = open(filename + 'dn', 'w')
|
|
# write chemical potential (in Rydberg):
|
|
f.write("%.14f\n" % (self.chemical_potential / self.energy_unit))
|
|
if self.SP != 0:
|
|
f1.write("%.14f\n" %
|
|
(self.chemical_potential / self.energy_unit))
|
|
# write beta in rydberg-1
|
|
f.write("%.14f\n" % (G_latt_iw.mesh.beta * self.energy_unit))
|
|
if self.SP != 0:
|
|
f1.write("%.14f\n" % (G_latt_iw.mesh.beta * self.energy_unit))
|
|
|
|
if self.SP == 0: # no spin-polarization
|
|
|
|
for ik in range(self.n_k):
|
|
f.write("%s\n" % self.n_orbitals[ik, 0])
|
|
for inu in range(self.n_orbitals[ik, 0]):
|
|
for imu in range(self.n_orbitals[ik, 0]):
|
|
valre = (deltaN['up'][ik][
|
|
inu, imu].real + deltaN['down'][ik][inu, imu].real) / 2.0
|
|
valim = (deltaN['up'][ik][
|
|
inu, imu].imag + deltaN['down'][ik][inu, imu].imag) / 2.0
|
|
f.write("%.14f %.14f " % (valre, valim))
|
|
f.write("\n")
|
|
f.write("\n")
|
|
f.close()
|
|
|
|
elif self.SP == 1: # with spin-polarization
|
|
|
|
# dict of filename: (spin index, block_name)
|
|
if self.SO == 0:
|
|
to_write = {f: (0, 'up'), f1: (1, 'down')}
|
|
if self.SO == 1:
|
|
to_write = {f: (0, 'ud'), f1: (0, 'ud')}
|
|
for fout in to_write.iterkeys():
|
|
isp, sp = to_write[fout]
|
|
for ik in range(self.n_k):
|
|
fout.write("%s\n" % self.n_orbitals[ik, isp])
|
|
for inu in range(self.n_orbitals[ik, isp]):
|
|
for imu in range(self.n_orbitals[ik, isp]):
|
|
fout.write("%.14f %.14f " % (deltaN[sp][ik][
|
|
inu, imu].real, deltaN[sp][ik][inu, imu].imag))
|
|
fout.write("\n")
|
|
fout.write("\n")
|
|
fout.close()
|
|
elif dm_type == 'vasp':
|
|
assert self.SP == 0, "Spin-polarized density matrix is not implemented"
|
|
|
|
if mpi.is_master_node():
|
|
with open(filename, 'w') as f:
|
|
f.write(" %i -1 ! Number of k-points, default number of bands\n"%(self.n_k))
|
|
for ik in xrange(self.n_k):
|
|
ib1 = band_window[0][ik, 0]
|
|
ib2 = band_window[0][ik, 1]
|
|
f.write(" %i %i %i\n"%(ik + 1, ib1, ib2))
|
|
for inu in xrange(self.n_orbitals[ik, 0]):
|
|
for imu in xrange(self.n_orbitals[ik, 0]):
|
|
valre = (deltaN['up'][ik][inu, imu].real + deltaN['down'][ik][inu, imu].real) / 2.0
|
|
valim = (deltaN['up'][ik][inu, imu].imag + deltaN['down'][ik][inu, imu].imag) / 2.0
|
|
f.write(" %.14f %.14f"%(valre, valim))
|
|
f.write("\n")
|
|
else:
|
|
raise NotImplementedError("Unknown density matrix type: '%s'"%(dm_type))
|
|
|
|
res = deltaN, dens
|
|
|
|
if dm_type == 'vasp':
|
|
res += (band_en_correction,)
|
|
|
|
return res
|
|
|
|
|
|
################
|
|
# FIXME LEAVE UNDOCUMENTED
|
|
################
|
|
|
|
def calc_dc_for_density(self, orb, dc_init, dens_mat, density=None, precision=0.01):
|
|
"""Searches for DC in order to fulfill charge neutrality.
|
|
If density is given, then DC is set such that the LOCAL charge of orbital
|
|
orb coincides with the given density."""
|
|
|
|
def F(dc):
|
|
self.calc_dc(dens_mat=dens_mat, U_interact=0,
|
|
J_hund=0, orb=orb, use_dc_value=dc)
|
|
if dens_req is None:
|
|
return self.total_density(mu=mu)
|
|
else:
|
|
return self.extract_G_loc()[orb].total_density()
|
|
|
|
if density is None:
|
|
density = self.density_required - self.charge_below
|
|
|
|
dc = dichotomy.dichotomy(function=F,
|
|
x_init=dc_init, y_value=density,
|
|
precision_on_y=precision, delta_x=0.5,
|
|
max_loops=100, x_name="Double Counting", y_name="Total Density",
|
|
verbosity=3)[0]
|
|
|
|
return dc
|
|
|
|
def check_projectors(self):
|
|
"""Calculated the density matrix from projectors (DM = P Pdagger) to check that it is correct and
|
|
specifically that it matches DFT."""
|
|
dens_mat = [numpy.zeros([self.corr_shells[icrsh]['dim'], self.corr_shells[icrsh]['dim']], numpy.complex_)
|
|
for icrsh in range(self.n_corr_shells)]
|
|
|
|
for ik in range(self.n_k):
|
|
for icrsh in range(self.n_corr_shells):
|
|
dim = self.corr_shells[icrsh]['dim']
|
|
n_orb = self.n_orbitals[ik, 0]
|
|
projmat = self.proj_mat[ik, 0, icrsh, 0:dim, 0:n_orb]
|
|
dens_mat[icrsh][
|
|
:, :] += numpy.dot(projmat, projmat.transpose().conjugate()) * self.bz_weights[ik]
|
|
|
|
if self.symm_op != 0:
|
|
dens_mat = self.symmcorr.symmetrize(dens_mat)
|
|
|
|
# Rotate to local coordinate system:
|
|
if self.use_rotations:
|
|
for icrsh in range(self.n_corr_shells):
|
|
if self.rot_mat_time_inv[icrsh] == 1:
|
|
dens_mat[icrsh] = dens_mat[icrsh].conjugate()
|
|
dens_mat[icrsh] = numpy.dot(numpy.dot(self.rot_mat[icrsh].conjugate().transpose(), dens_mat[icrsh]),
|
|
self.rot_mat[icrsh])
|
|
|
|
return dens_mat
|
|
|
|
def sorts_of_atoms(self, shells):
|
|
"""
|
|
Determine the number of inequivalent sorts.
|
|
"""
|
|
sortlst = [shells[i]['sort'] for i in range(len(shells))]
|
|
n_sorts = len(set(sortlst))
|
|
return n_sorts
|
|
|
|
def number_of_atoms(self, shells):
|
|
"""
|
|
Determine the number of inequivalent atoms.
|
|
"""
|
|
atomlst = [shells[i]['atom'] for i in range(len(shells))]
|
|
n_atoms = len(set(atomlst))
|
|
return n_atoms
|
|
|
|
# The following methods are here to ensure backward-compatibility
|
|
# after introducing the block_structure class
|
|
def __get_gf_struct_sumk(self):
|
|
return self.block_structure.gf_struct_sumk
|
|
def __set_gf_struct_sumk(self,value):
|
|
self.block_structure.gf_struct_sumk = value
|
|
gf_struct_sumk = property(__get_gf_struct_sumk,__set_gf_struct_sumk)
|
|
|
|
def __get_gf_struct_solver(self):
|
|
return self.block_structure.gf_struct_solver
|
|
def __set_gf_struct_solver(self,value):
|
|
self.block_structure.gf_struct_solver = value
|
|
gf_struct_solver = property(__get_gf_struct_solver,__set_gf_struct_solver)
|
|
|
|
def __get_solver_to_sumk(self):
|
|
return self.block_structure.solver_to_sumk
|
|
def __set_solver_to_sumk(self,value):
|
|
self.block_structure.solver_to_sumk = value
|
|
solver_to_sumk = property(__get_solver_to_sumk,__set_solver_to_sumk)
|
|
|
|
def __get_sumk_to_solver(self):
|
|
return self.block_structure.sumk_to_solver
|
|
def __set_sumk_to_solver(self,value):
|
|
self.block_structure.sumk_to_solver = value
|
|
sumk_to_solver = property(__get_sumk_to_solver,__set_sumk_to_solver)
|
|
|
|
def __get_solver_to_sumk_block(self):
|
|
return self.block_structure.solver_to_sumk_block
|
|
def __set_solver_to_sumk_block(self,value):
|
|
self.block_structure.solver_to_sumk_block = value
|
|
solver_to_sumk_block = property(__get_solver_to_sumk_block,__set_solver_to_sumk_block)
|
|
|
|
def __get_deg_shells(self):
|
|
return self.block_structure.deg_shells
|
|
def __set_deg_shells(self,value):
|
|
self.block_structure.deg_shells = value
|
|
deg_shells = property(__get_deg_shells,__set_deg_shells)
|