Lattice Solver Tutorial
This tutorial using triqs lattice tools and the Hartree-Fock LatticeSolver for the case of a 2 degenerate bands with local Kanamori interactions at fixed density. At the end, we solve the same problem using the ImpuritySolver with the DMFT loop and arrive at the same solution.
We start by importing the required modules:
[1]:
import numpy as np
from triqs_hartree_fock import ImpuritySolver, LatticeSolver
from triqs_hartree_fock.utils import flatten
from triqs.gf import *
from triqs.operators import *
from triqs.operators.util import *
from h5 import HDFArchive
from triqs.lattice.tight_binding import TBLattice
from triqs.sumk import *
from triqs.lattice import *
from scipy.optimize import brentq
Warning: could not identify MPI environment!
Starting serial run at: 2022-05-02 14:09:48.986326
First we define a 2D tight binding lattice with nearest and next-nearest neighbor hopping:
[2]:
t = np.array([[1, 0.2], [0.2, 1]])
tp = np.array([[0.1, 0.05], [0.05, 0.1]])
hoppings= { (1,0) : t,
(-1,0) : t,
(0,1) : t,
(0,-1) : t,
(1,1) : tp,
(-1,-1): tp,
(1,-1) : tp,
(-1,1) : tp}
TBL = TBLattice(units=[(1, 0, 0) , (0, 1, 0)], hoppings=hoppings, orbital_positions=[(0., 0., 0.)]*2)
Next we define the single particle dispersion h0_k and the other parameters to pass to the solver
[3]:
nk = 10
beta = 40
U = 3
J = 0.5
gf_struct = [('up', 2), ('down', 2)]
N_target = 2
#create a kanamori interaction Hamiltonian
Umat, Upmat = U_matrix_kanamori(n_orb=2, U_int=U, J_hund=J)
h_int = h_int_kanamori(spin_names=['up', 'down'], orb_names=[0,1], U=Umat, Uprime=Upmat, J_hund=J, off_diag=True)
BL = BravaisLattice(units = [(1,0,0) , (0,1,0)])
BZ = BrillouinZone(BL)
mk = MeshBrZone(BZ, nk)
ekup = Gf(mesh=mk, target_shape=[2,2])
ekdn = Gf(mesh=mk, target_shape=[2,2])
ekup << TBL.fourier(mk)
ekdn << TBL.fourier(mk)
h0_k = BlockGf(name_list=['up', 'down'], block_list=(ekup, ekdn)) #single particle dispersion
Now we are ready to initialize the solver:
[4]:
S = LatticeSolver(h0_k=h0_k, gf_struct=gf_struct, beta=beta)
and finally we call the solve method at fixed density of 2 electrons per site:
[5]:
S.solve(h_int=h_int, N_target=N_target)
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╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Lattice Solver at fixed density of 2.0000
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Sigma_HF['up']:
[[ 3.25 +0.j -0.0122+0.j]
[-0.0122+0.j 3.25 +0.j]]
Sigma_HF['down']:
[[ 3.25 +0.j -0.0122+0.j]
[-0.0122+0.j 3.25 +0.j]]
mu = 3.1380
DMFT Loop with Impurity Solver:
We can arrive at the same solution of the lattice solver using a DMFT loop and impurity solver. However, it’s slower than using the lattice solver. We pass a symmetry function that forces the two spins to be equal
[6]:
SK = SumkDiscreteFromLattice(lattice=TBL, n_points=nk)
sigma = GfImFreq(beta=beta, n_points=500, target_shape=[2,2])
Sigma = BlockGf(name_list=['up', 'down'], block_list=(sigma,sigma), make_copies=True)
Gloc = Sigma.copy()
N_target = 2
mu = 0
#symmetry function forcing two spins to be equal
def make_spins_equal(Sigma):
Symmetrized_Sigma = Sigma.copy()
Symmetrized_Sigma['up'] = 0.5*(Sigma['up'] + Sigma['down'])
Symmetrized_Sigma['down'] = Symmetrized_Sigma['up']
return Symmetrized_Sigma
S_imp = ImpuritySolver(gf_struct=gf_struct, beta=beta, n_iw=500, symmetries=[make_spins_equal])
converged = False
while not converged:
for name, bl in gf_struct:
Sigma[name] << S_imp.Sigma_HF[name]
mu = brentq(lambda mu: SK(mu=mu, Sigma=Sigma).total_density().real - N_target, -5, 5)
print('mu = ', mu)
Gloc << SK(mu=mu, Sigma=Sigma)
S_imp.G0_iw << inverse(inverse(Gloc) + Sigma)
Sigma_old = S_imp.Sigma_HF.copy()
S_imp.solve(h_int=h_int)
if np.allclose(flatten(Sigma_old), flatten(S_imp.Sigma_HF), rtol=0, atol=1e-4):
converged = True
mu = -0.11655108826676398
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╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 1.3514+0.j -0.024 +0.j]
[-0.024 +0.j 1.3514+0.j]]
Sigma_HF['down']:
[[ 1.3514+0.j -0.024 +0.j]
[-0.024 +0.j 1.3514+0.j]]
mu = 1.2416047543409816
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╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 2.0346+0.j -0.0263+0.j]
[-0.0263+0.j 2.0346+0.j]]
Sigma_HF['down']:
[[ 2.0346+0.j -0.0263+0.j]
[-0.0263+0.j 2.0346+0.j]]
mu = 1.924985519971107
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╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 2.4404+0.j -0.0251+0.j]
[-0.0251+0.j 2.4404+0.j]]
Sigma_HF['down']:
[[ 2.4404+0.j -0.0251+0.j]
[-0.0251+0.j 2.4404+0.j]]
mu = 2.3307342492307934
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╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 2.6997+0.j -0.023 +0.j]
[-0.023 +0.j 2.6997+0.j]]
Sigma_HF['down']:
[[ 2.6997+0.j -0.023 +0.j]
[-0.023 +0.j 2.6997+0.j]]
mu = 2.589804077361585
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╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 2.872 +0.j -0.0208+0.j]
[-0.0208+0.j 2.872 +0.j]]
Sigma_HF['down']:
[[ 2.872 +0.j -0.0208+0.j]
[-0.0208+0.j 2.872 +0.j]]
mu = 2.761794961651811
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╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 2.989 +0.j -0.0187+0.j]
[-0.0187+0.j 2.989 +0.j]]
Sigma_HF['down']:
[[ 2.989 +0.j -0.0187+0.j]
[-0.0187+0.j 2.989 +0.j]]
mu = 2.8784151039233974
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╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 3.0693+0.j -0.017 +0.j]
[-0.017 +0.j 3.0693+0.j]]
Sigma_HF['down']:
[[ 3.0693+0.j -0.017 +0.j]
[-0.017 +0.j 3.0693+0.j]]
mu = 2.958388200331095
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╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 3.1247+0.j -0.0156+0.j]
[-0.0156+0.j 3.1247+0.j]]
Sigma_HF['down']:
[[ 3.1247+0.j -0.0156+0.j]
[-0.0156+0.j 3.1247+0.j]]
mu = 3.013551770566393
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║ ╠╦╝║║═╬╗╚═╗ ├─┤├┤
╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 3.1631+0.j -0.0147+0.j]
[-0.0147+0.j 3.1631+0.j]]
Sigma_HF['down']:
[[ 3.1631+0.j -0.0147+0.j]
[-0.0147+0.j 3.1631+0.j]]
mu = 3.051714472919698
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║ ╠╦╝║║═╬╗╚═╗ ├─┤├┤
╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 3.1897+0.j -0.0139+0.j]
[-0.0139+0.j 3.1897+0.j]]
Sigma_HF['down']:
[[ 3.1897+0.j -0.0139+0.j]
[-0.0139+0.j 3.1897+0.j]]
mu = 3.0781557418842773
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║ ╠╦╝║║═╬╗╚═╗ ├─┤├┤
╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 3.2082+0.j -0.0134+0.j]
[-0.0134+0.j 3.2082+0.j]]
Sigma_HF['down']:
[[ 3.2082+0.j -0.0134+0.j]
[-0.0134+0.j 3.2082+0.j]]
mu = 3.0964880022101333
╔╦╗╦═╗╦╔═╗ ╔═╗ ┬ ┬┌─┐
║ ╠╦╝║║═╬╗╚═╗ ├─┤├┤
╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 3.221 +0.j -0.0131+0.j]
[-0.0131+0.j 3.221 +0.j]]
Sigma_HF['down']:
[[ 3.221 +0.j -0.0131+0.j]
[-0.0131+0.j 3.221 +0.j]]
mu = 3.1092022184291164
╔╦╗╦═╗╦╔═╗ ╔═╗ ┬ ┬┌─┐
║ ╠╦╝║║═╬╗╚═╗ ├─┤├┤
╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 3.2299+0.j -0.0128+0.j]
[-0.0128+0.j 3.2299+0.j]]
Sigma_HF['down']:
[[ 3.2299+0.j -0.0128+0.j]
[-0.0128+0.j 3.2299+0.j]]
mu = 3.11802143459409
╔╦╗╦═╗╦╔═╗ ╔═╗ ┬ ┬┌─┐
║ ╠╦╝║║═╬╗╚═╗ ├─┤├┤
╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 3.236 +0.j -0.0126+0.j]
[-0.0126+0.j 3.236 +0.j]]
Sigma_HF['down']:
[[ 3.236 +0.j -0.0126+0.j]
[-0.0126+0.j 3.236 +0.j]]
mu = 3.124139484544855
╔╦╗╦═╗╦╔═╗ ╔═╗ ┬ ┬┌─┐
║ ╠╦╝║║═╬╗╚═╗ ├─┤├┤
╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 3.2403+0.j -0.0125+0.j]
[-0.0125+0.j 3.2403+0.j]]
Sigma_HF['down']:
[[ 3.2403+0.j -0.0125+0.j]
[-0.0125+0.j 3.2403+0.j]]
mu = 3.1283834605726115
╔╦╗╦═╗╦╔═╗ ╔═╗ ┬ ┬┌─┐
║ ╠╦╝║║═╬╗╚═╗ ├─┤├┤
╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 3.2433+0.j -0.0124+0.j]
[-0.0124+0.j 3.2433+0.j]]
Sigma_HF['down']:
[[ 3.2433+0.j -0.0124+0.j]
[-0.0124+0.j 3.2433+0.j]]
mu = 3.1313278331378296
╔╦╗╦═╗╦╔═╗ ╔═╗ ┬ ┬┌─┐
║ ╠╦╝║║═╬╗╚═╗ ├─┤├┤
╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 3.2453+0.j -0.0123+0.j]
[-0.0123+0.j 3.2453+0.j]]
Sigma_HF['down']:
[[ 3.2453+0.j -0.0123+0.j]
[-0.0123+0.j 3.2453+0.j]]
mu = 3.1333710379511377
╔╦╗╦═╗╦╔═╗ ╔═╗ ┬ ┬┌─┐
║ ╠╦╝║║═╬╗╚═╗ ├─┤├┤
╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 3.2468+0.j -0.0123+0.j]
[-0.0123+0.j 3.2468+0.j]]
Sigma_HF['down']:
[[ 3.2468+0.j -0.0123+0.j]
[-0.0123+0.j 3.2468+0.j]]
mu = 3.134788058674276
╔╦╗╦═╗╦╔═╗ ╔═╗ ┬ ┬┌─┐
║ ╠╦╝║║═╬╗╚═╗ ├─┤├┤
╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 3.2478+0.j -0.0123+0.j]
[-0.0123+0.j 3.2478+0.j]]
Sigma_HF['down']:
[[ 3.2478+0.j -0.0123+0.j]
[-0.0123+0.j 3.2478+0.j]]
mu = 3.135771249039664
╔╦╗╦═╗╦╔═╗ ╔═╗ ┬ ┬┌─┐
║ ╠╦╝║║═╬╗╚═╗ ├─┤├┤
╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 3.2484+0.j -0.0123+0.j]
[-0.0123+0.j 3.2484+0.j]]
Sigma_HF['down']:
[[ 3.2484+0.j -0.0123+0.j]
[-0.0123+0.j 3.2484+0.j]]
mu = 3.1364533457514936
╔╦╗╦═╗╦╔═╗ ╔═╗ ┬ ┬┌─┐
║ ╠╦╝║║═╬╗╚═╗ ├─┤├┤
╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 3.2489+0.j -0.0122+0.j]
[-0.0122+0.j 3.2489+0.j]]
Sigma_HF['down']:
[[ 3.2489+0.j -0.0122+0.j]
[-0.0122+0.j 3.2489+0.j]]
mu = 3.1369268847106087
╔╦╗╦═╗╦╔═╗ ╔═╗ ┬ ┬┌─┐
║ ╠╦╝║║═╬╗╚═╗ ├─┤├┤
╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 3.2492+0.j -0.0122+0.j]
[-0.0122+0.j 3.2492+0.j]]
Sigma_HF['down']:
[[ 3.2492+0.j -0.0122+0.j]
[-0.0122+0.j 3.2492+0.j]]
mu = 3.137254115255671
╔╦╗╦═╗╦╔═╗ ╔═╗ ┬ ┬┌─┐
║ ╠╦╝║║═╬╗╚═╗ ├─┤├┤
╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 3.2495+0.j -0.0122+0.j]
[-0.0122+0.j 3.2495+0.j]]
Sigma_HF['down']:
[[ 3.2495+0.j -0.0122+0.j]
[-0.0122+0.j 3.2495+0.j]]
mu = 3.1374808873027025
╔╦╗╦═╗╦╔═╗ ╔═╗ ┬ ┬┌─┐
║ ╠╦╝║║═╬╗╚═╗ ├─┤├┤
╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 3.2496+0.j -0.0122+0.j]
[-0.0122+0.j 3.2496+0.j]]
Sigma_HF['down']:
[[ 3.2496+0.j -0.0122+0.j]
[-0.0122+0.j 3.2496+0.j]]
mu = 3.137639326775724
╔╦╗╦═╗╦╔═╗ ╔═╗ ┬ ┬┌─┐
║ ╠╦╝║║═╬╗╚═╗ ├─┤├┤
╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 3.2497+0.j -0.0122+0.j]
[-0.0122+0.j 3.2497+0.j]]
Sigma_HF['down']:
[[ 3.2497+0.j -0.0122+0.j]
[-0.0122+0.j 3.2497+0.j]]
mu = 3.137749333580944
╔╦╗╦═╗╦╔═╗ ╔═╗ ┬ ┬┌─┐
║ ╠╦╝║║═╬╗╚═╗ ├─┤├┤
╩ ╩╚═╩╚═╝╚╚═╝ ┴ ┴└
TRIQS: Hartree-Fock solver
Running Impurity Solver
beta = 40.0000
h_int =
1.5*c_dag('down',0)*c_dag('down',1)*c('down',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',0)*c('up',1)*c('down',1) + 3*c_dag('down',0)*c_dag('up',0)*c('up',0)*c('down',0) + 2*c_dag('down',0)*c_dag('up',1)*c('up',1)*c('down',0) + 0.5*c_dag('down',0)*c_dag('up',1)*c('up',0)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',0)*c('up',1)*c('down',0) + 2*c_dag('down',1)*c_dag('up',0)*c('up',0)*c('down',1) + 3*c_dag('down',1)*c_dag('up',1)*c('up',1)*c('down',1) + 0.5*c_dag('down',1)*c_dag('up',1)*c('up',0)*c('down',0) + 1.5*c_dag('up',0)*c_dag('up',1)*c('up',1)*c('up',0)
mode: self-consistent
Including Fock terms:
True
Self Consistent Hartree-Fock converged successfully
Calculated self energy:
Sigma_HF['up']:
[[ 3.2498+0.j -0.0122+0.j]
[-0.0122+0.j 3.2498+0.j]]
Sigma_HF['down']:
[[ 3.2498+0.j -0.0122+0.j]
[-0.0122+0.j 3.2498+0.j]]