# Bath hamiltonian in matrix representation
h_bath_mat = diag(eps_bath)
# Coupling matrix
V_mat = matrix([[1., 1., 0.2, 0.2], [0.2, 0.2, 1, 1]])
# ==== Local Hamiltonian ====
c_dag_vec = {s: matrix([[c_dag(s, o) for o in orb_names]]) for s in spin_names}
c_vec = {s: matrix([[c(s, o)] for o in orb_names]) for s in spin_names}
h_0 = sum(c_dag_vec[s] * h_0_mat * c_vec[s] for s in spin_names)[0, 0]
h_int = h_int_kanamori(
spin_names,
orb_names,
array([[0, V - J], [V - J, 0]]), # Interaction for equal spins
array([[U, V], [V, U]]), # Interaction for opposite spins
J,
True)
h_loc = h_0 + h_int
# ==== Bath & Coupling hamiltonian ====
orb_bath_names = ['b_' + str(o) for o in orb_bath_names]
c_dag_bath_vec = {
s: matrix([[c_dag(s, o) for o in orb_bath_names]])
for s in spin_names
}
c_bath_vec = {
s: matrix([[c(s, o)] for o in orb_bath_names])
for s in spin_names
# create interaction Hamiltonian
Hint = h_int_slater(spin_names,
orb_names,
off_diag=True,
U_matrix=Umat_full_rotated)
elif h_int_type == 'kan':
mpi.report('setting up Kanamori Hamiltonian')
Umat, Upmat = U_matrix_kanamori(n_orb=n_orb, U_int=U, J_hund=J)
# create interaction Hamiltonian
if h_int_offdiag:
mpi.report('... with spin-flip and pair-hopping')
Hint = h_int_kanamori(spin_names,
orb_names,
off_diag=True,
U=Umat,
Uprime=Upmat,
J_hund=J)
else:
mpi.report('... only den den terms')
Hint = h_int_density(spin_names,
orb_names,
off_diag=True,
U=Umat,
Uprime=Upmat)
# store data obtained
if store_h5 and mpi.is_master_node():
with HDFArchive(store_h5_file, 'a') as ar:
ar['Hint'] = Hint
def test_quartic(verbose=False):
if verbose:
print('--> test_quartic')
num_orbitals = 2
num_spins = 2
U, J = 1.0, 0.2
up, do = 0, 1
spin_names = [up, do]
orb_names = list(range(num_orbitals))
U_ab, UPrime_ab = U_matrix_kanamori(n_orb=2, U_int=U, J_hund=J)
if verbose:
print('U_ab =\n', U_ab)
print('UPrime_ab =\n', UPrime_ab)
T_ab = np.array([
[1., 1.],
[1., -1.],
]) / np.sqrt(2.)
# -- Check hermitian
np.testing.assert_array_almost_equal(
np.mat(T_ab) * np.mat(T_ab).H, np.eye(2))
I = np.eye(num_spins)
T_ab_spin = np.kron(T_ab, I)
H_int = h_int_kanamori(spin_names,
orb_names,
U_ab,
UPrime_ab,
J_hund=J,
off_diag=True,
map_operator_structure=None,
H_dump=None)
op_imp = [c(up, 0), c(do, 0), c(up, 1), c(do, 1)]
Ht_int = operator_single_particle_transform(H_int, T_ab_spin, op_imp)
if verbose:
print('H_int =', H_int)
print('Ht_int =', Ht_int)
from transform_kanamori import h_int_kanamori_transformed
Ht_int_ref = h_int_kanamori_transformed([T_ab, T_ab],
spin_names,
orb_names,
U_ab,
UPrime_ab,
J_hund=J,
off_diag=True,
map_operator_structure=None,
H_dump=None)
if verbose:
print('Ht_int_ref =', Ht_int_ref)
assert ((Ht_int_ref - Ht_int).is_zero())
# Definition of a 3-orbital atom
spin_names = ('up','dn')
orb_names = [0,1,2]
# Set of fundamental operators
fops = [(sn,on) for sn, on in product(spin_names,orb_names)]
# Numbers of particles with spin up/down
N_up = n('up',0) + n('up',1) + n('up',2)
N_dn = n('dn',0) + n('dn',1) + n('dn',2)
# Construct Hubbard-Kanamori Hamiltonian
U = 3.0 * np.ones((3,3))
Uprime = 2.0 * np.ones((3,3))
J_hund = 0.5
H = h_int_kanamori(spin_names, orb_names, U, Uprime, J_hund, True)
# Add chemical potential
H += -4.0 * (N_up + N_dn)
# Add hopping terms between orbitals 0 and 1 with some complex amplitude
H += 0.1j * (c_dag('up',0) * c('up',1) - c_dag('up',1) * c('up',0))
H += 0.1j * (c_dag('dn',0) * c('dn',1) - c_dag('dn',1) * c('dn',0))
# Split H into blocks and diagonalize it using N_up and N_dn quantum numbers
ad = AtomDiag(H, fops, [N_up,N_dn])
print(ad.n_subspaces) # Number of invariant subspaces, 4 * 4 = 16
# Now split using the total number of particles, N = N_up + N_dn
ad = AtomDiag(H, fops, [N_up+N_dn])
print(ad.n_subspaces) # 7
mu = 0.0
beta, n_w, n_k = 1.0, 100, 4
U, J = 2.3, 0.4
print('U, J =', U, J)
spin_names = ['up', 'do']
#orb_names = [0, 1, 2]
orb_names = [0]
norb = 2*len(orb_names)
gf_struct = set_operator_structure(spin_names, orb_names, False) # orbital diag
U_mat, UPrime_mat = U_matrix_kanamori(n_orb=len(orb_names), U_int=U, J_hund=J)
H_int = h_int_kanamori(
spin_names, orb_names, U_mat, UPrime_mat, J_hund=J,
off_diag=False, map_operator_structure=None, H_dump=None) # orbital diag
# ------------------------------------------------------------------
# -- Tightbinding model
t = 1.0
h_loc = np.kron(np.eye(2), np.diag([0., -0.1, 0.1]))
T = - t * np.kron(np.eye(2), np.diag([0.01, 1., 1.]))
t_r = TBLattice(
units = [(1, 0, 0)],
hopping = {
# nearest neighbour hopping -t
(0,): h_loc,
(+1,): T,
orb_names = [0, 1, 2]
spin_names = ['up', 'do']
norb = len(orb_names)
gf_struct = set_operator_structure(spin_names, orb_names,
True) # orbital off diag
fundamental_operators = fundamental_operators_from_gf_struct(gf_struct)
U_ab, UPrime_ab = U_matrix_kanamori(n_orb=len(orb_names),
U_int=U,
J_hund=J)
H_int = h_int_kanamori(spin_names,
orb_names,
U_ab,
UPrime_ab,
J_hund=J,
off_diag=True,
map_operator_structure=None,
H_dump=None) # orgital offdiag
U_abcd = get_rpa_tensor(H_int, fundamental_operators)
U_c, U_s = split_quartic_tensor_in_charge_and_spin(U_abcd)
U_abcd_ref = quartic_tensor_from_charge_and_spin(U_c, U_s)
U_c_ref, U_s_ref = kanamori_charge_and_spin_quartic_interaction_tensors(
norb, U, U - 2 * J, J, J)
U_abcd_ref_2 = kanamori_quartic_tensor(norb, U, U - 2 * J, J, J)