##########################################################################
#
# TRIQS: a Toolbox for Research in Interacting Quantum Systems
#
# Copyright (C) 2023 by A. Carta, A. Hampel
#
# TRIQS is free software: you can redistribute it and/or modify it under the
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"""
Helper utilities / standalone functions for the sumk class
"""
import numpy as np
import triqs.utility.mpi as mpi
[docs]
def compute_DC_from_density(N_tot, U, J, N_spin=None, n_orbitals=5, method='sFLL'):
"""
Computes the double counting correction using various methods.
For FLL and AMF DC the notations and equations from doi.org/10.1038/s41598-018-27731-4
are used, whereas for the Held DC the definitions from doi.org/10.1080/00018730701619647 are used.
Parameters
----------
N_tot : float
Total density of the impurity
N_spin : float , default = None
Spin density, defaults to N_tot*0.5 if not specified
U : float
U value
J : float
J value
n_orbitals : int, default = 5
Total number of orbitals
method : string, default = 'cFLL'
possibilities:
- cFLL: DC potential from Ryee for spin unpolarized DFT: (DOI: 10.1038/s41598-018-27731-4)
- sFLL: same as above for spin polarized DFT
- cAMF: around mean field
- sAMF: spin polarized around mean field
- cHeld: unpolarized Held's formula as reported in (DOI: 10.1103/PhysRevResearch.2.033088)
- sHeld: NOT IMPLEMENTED
Returns
-------
List of floats:
- DC_val: double counting potential
- E_val: double counting energy
todo:
- See whether to move this to TRIQS directly instead of dft_tools
- allow as input full density matrix to allow orbital dependent DC
"""
if N_spin is not None:
N_spin2 = N_tot-N_spin
Mag = N_spin - N_spin2
L_orbit = (n_orbitals-1)/2
if method == 'cFLL':
E_val = 0.5 * U * N_tot * (N_tot-1) - 0.5 * J * N_tot * (N_tot*0.5-1)
DC_val = U * (N_tot-0.5) - J * (N_tot*0.5-0.5)
elif method == 'sFLL':
assert N_spin is not None, "Spin density not given"
E_val = 0.5 * U * N_tot * (N_tot-1) - 0.5 * J * N_tot * (N_tot*0.5-1) - 0.25 * J * Mag**2
DC_val = U * (N_tot-0.5) - J * (N_spin-0.5)
elif method == 'cAMF':
E_val = +0.5 * U * N_tot ** 2
E_val -= 0.25*(U+2*L_orbit*J)/(2*L_orbit+1)*N_tot**2
DC_val = U * N_tot - 0.5*(U+2*L_orbit*J)/(2*L_orbit+1)*N_tot
elif method == 'sAMF':
assert N_spin is not None, "Spin density not given"
E_val = 0.5 * U * N_tot ** 2
E_val -= 0.25*(U+2*L_orbit*J)/(2*L_orbit+1)*N_tot**2
E_val -= 0.25*(U+2*L_orbit*J)/(2*L_orbit+1)*Mag**2
DC_val = U * N_tot - (U+2*L_orbit*J)/(2*L_orbit+1)*N_spin
elif method == 'cHeld':
# Valid for a Kanamori-type Hamiltonian where U'=U-2J
U_mean = (U + (n_orbitals-1)*(U-2*J)+(n_orbitals-1)*(U-3*J))/(2*n_orbitals-1)
E_val = 0.5 * U_mean * N_tot * (N_tot - 1)
DC_val = U_mean * (N_tot-0.5)
elif method == 'sHeld':
raise ValueError(f"Method sHeld not yet implemented")
else:
raise ValueError(f"DC type {method} not supported")
mpi.report(f"DC potential computed using the {method} method, V_DC = {DC_val:.6f} eV")
mpi.report(f"E_DC using the {method} method, E_DC = {E_val:.6f} eV")
if 'Held' in method:
mpi.report(f"Held method for {n_orbitals} orbitals, computed U_mean={U_mean:.6f} eV")
return DC_val, E_val