def calc_reference():
ref_orbs = [
'%s' % i for i in range(n_orbs * parms['N_x'] * parms['N_y'])
]
ref_gf_struct = op.set_operator_structure(spin_names,
ref_orbs,
off_diag=off_diag)
ref_index_converter = {(sn, o): ("loc", int(o),
"down" if sn == "dn" else "up")
for sn, o in product(spin_names, ref_orbs)}
#print ref_index_converter,ref_orbs
ref_ed = PomerolED(ref_index_converter, verbose=True)
ref_N = sum(
ops.n(sn, o) for sn, o in product(spin_names, ref_orbs))
# 2 3
# 0 1
ref_H = (parms["U"] * (ops.n('up', '0') * ops.n('dn', '0') +
ops.n('up', '1') * ops.n('dn', '1')) -
2. * parms['t1'] *
(ops.c_dag('up', '0') * ops.c('up', '1') +
ops.c_dag('up', '1') * ops.c('up', '0') +
ops.c_dag('dn', '0') * ops.c('dn', '1') +
ops.c_dag('dn', '1') * ops.c('dn', '0')) -
parms['chemical_potential'] * ref_N)
# Run the solver
ref_ed.diagonalize(ref_H)
# Compute G(i\omega)
ref_G_iw = ref_ed.G_iw(ref_gf_struct, parms['beta'], parms['n_iw'])
return ref_G_iw
def test_fundamental():
assert (op_is_fundamental(c(0, 0)) is True)
assert (op_is_fundamental(c_dag(0, 0)) is True)
assert (op_is_fundamental(c_dag(0, 0) * c(0, 0)) is False)
assert (op_is_fundamental(Operator(1.0)) is False)
assert (op_serialize_fundamental(c(0, 0)) == (False, (0, 0)))
assert (op_serialize_fundamental(c_dag(0, 0)) == (True, (0, 0)))
assert (op_serialize_fundamental(c(2, 4)) == (False, (2, 4)))
assert (op_serialize_fundamental(c_dag(4, 3)) == (True, (4, 3)))
def c_transf(s, i):
if (s, i) not in op_idx_set:
return c(s, i)
k = op_idx_map.index((s, i))
ret = Operator()
for l in range(U.shape[0]):
op_idx = op_idx_map[l]
ret += U[k, l] * c(*op_idx)
return ret
def get_fundamental_operators(op):
idx_lst = []
for term, val in op:
for has_dagger, (bl, orb) in term:
if not idx_lst.count([bl, orb]):
idx_lst.append([bl, orb])
return [c(bl, orb) for bl, orb in idx_lst]
def fundamental_operators_from_gf_struct(gf_struct):
gf_struct = fix_gf_struct_type(gf_struct)
fundamental_operators = []
for block, block_size in gf_struct:
for idx in range(block_size):
fundamental_operators.append(c(block, idx))
return fundamental_operators
def fundamental_operators_from_gf_struct(gf_struct):
""" Return a list of annihilation operators with the quantum numbers
defined in the gf_struct """
fundamental_operators = []
for block_name, indices in gf_struct:
for index in indices:
fundamental_operators.append(c(block_name, index))
return fundamental_operators
def fundamental_operators_from_gf_struct(gf_struct):
""" Return a list of annihilation operators with the quantum numbers
defined in the gf_struct """
gf_struct = fix_gf_struct_type(gf_struct)
fundamental_operators = []
for block_name, block_size in gf_struct:
for idx in range(block_size):
fundamental_operators.append(c(block_name, idx))
return fundamental_operators
def test_quadratic():
n = 10
h_loc = np.random.random((n, n))
h_loc = 0.5 * (h_loc + h_loc.T)
fund_op = [c(0, idx) for idx in range(n)]
H_loc = get_quadratic_operator(h_loc, fund_op)
h_loc_ref = quadratic_matrix_from_operator(H_loc, fund_op)
np.testing.assert_array_almost_equal(h_loc, h_loc_ref)
def convert_operator(self, O, ish=0):
""" Converts a second-quantization operator from sumk structure
to solver structure.
Parameters
----------
O : triqs.operators.Operator
Operator in sumk structure
ish : int
shell index on which the operator acts
"""
from triqs.operators import Operator, c, c_dag
T = self.transformation[ish]
sk2s = self.sumk_to_solver[ish]
O_out = Operator(0)
for monomial in O:
coefficient = monomial[-1]
new_monomial = Operator(1)
#if coefficient > 1e-10:
for single_operator in monomial[0]:
new_single_operator = Operator(0)
daggered = single_operator[0]
blockname = single_operator[1][0]
i = single_operator[1][1]
for j in range(len(T[blockname])):
if sk2s[(blockname, j)] != (None, None):
if daggered:
new_single_operator += (
T[blockname][j, i] *
c_dag(*sk2s[(blockname, j)]))
else:
new_single_operator += (
T[blockname][j, i].conjugate() *
c(*sk2s[(blockname, j)]))
new_monomial *= new_single_operator
O_out += new_monomial * coefficient
return O_out
def is_operator_composed_of_only_fundamental_operators(op,
fundamental_operators):
""" Return `True` if the operator `op` only contains operators belonging
to the fundamental operator set `fundamental_operators`, else return
`False`."""
is_fundamental = True
for term in op:
op_list, prefactor = term
d, t = list(
zip(*op_list)) # split in two lists with daggers and tuples resp
t = [tuple(x) for x in t]
for bidx, idx in t:
if c(bidx, idx) not in fundamental_operators:
is_fundamental = False
return is_fundamental
def test_single_particle_transform(verbose=False):
if verbose:
print('--> test_single_particle_transform')
h_loc = np.array([
[1.0, 0.0],
[0.0, -1.0],
])
op_imp = [c(0, 0), c(0, 1)]
H_loc = get_quadratic_operator(h_loc, op_imp)
H_loc_ref = c_dag(0, 0) * c(0, 0) - c_dag(0, 1) * c(0, 1)
assert ((H_loc - H_loc_ref).is_zero())
h_loc_ref = quadratic_matrix_from_operator(H_loc, op_imp)
np.testing.assert_array_almost_equal(h_loc, h_loc_ref)
if verbose:
print('h_loc =\n', h_loc)
print('h_loc_ref =\n', h_loc_ref)
print('H_loc =', H_loc)
T_ab = np.array([
[1., 1.],
[1., -1.],
]) / np.sqrt(2.)
Ht_loc = operator_single_particle_transform(H_loc, T_ab, op_imp)
ht_loc = quadratic_matrix_from_operator(Ht_loc, op_imp)
ht_loc_ref = np.array([
[0., 1.],
[1., 0.],
])
Ht_loc_ref = c_dag(0, 0) * c(0, 1) + c_dag(0, 1) * c(0, 0)
if verbose:
print('ht_loc =\n', ht_loc)
print('ht_loc_ref =\n', ht_loc_ref)
print('Ht_loc =', Ht_loc)
print('Ht_loc_ref =', Ht_loc_ref)
assert ((Ht_loc - Ht_loc_ref).is_zero())
def test_quartic_tensor_from_operator(verbose=False):
N = 3
fundamental_operators = [c(0, x) for x in range(N)]
shape = (N, N, N, N)
U = np.random.random(shape) + 1.j * np.random.random(shape)
U_sym = symmetrize_quartic_tensor(U)
H = operator_from_quartic_tensor(U, fundamental_operators)
U_ref = quartic_tensor_from_operator(H,
fundamental_operators,
perm_sym=True)
np.testing.assert_array_almost_equal(U_ref, U_sym)
if verbose:
print('-' * 72)
import itertools
for idxs in itertools.product(list(range(N)), repeat=4):
print(idxs, U_ref[idxs] - U_sym[idxs], U[idxs], U_ref[idxs],
U_sym[idxs])
def test_gf_struct():
orb_idxs = [0, 1, 2]
spin_idxs = ['up', 'do']
gf_struct = [[spin_idx, orb_idxs] for spin_idx in spin_idxs]
fundamental_operators = fundamental_operators_from_gf_struct(gf_struct)
fundamental_operators_ref = [
c('up', 0),
c('up', 1),
c('up', 2),
c('do', 0),
c('do', 1),
c('do', 2),
]
print(fundamental_operators)
assert (fundamental_operators == fundamental_operators_ref)
def test_sparse_matrix_representation():
up, do = 0, 1
fundamental_operators = [c(up, 0), c(do, 0)]
rep = SparseMatrixRepresentation(fundamental_operators)
# -- Test an operator
O_mat = rep.sparse_matrix(c(up, 0))
O_ref = rep.sparse_operators.c_dag[0].getH()
compare_sparse_matrices(O_mat, O_ref)
# -- Test
O_mat = rep.sparse_matrix(c(do, 0))
O_ref = rep.sparse_operators.c_dag[1].getH()
compare_sparse_matrices(O_mat, O_ref)
# -- Test expression
H_expr = c(up, 0) * c(do, 0) * c_dag(up, 0) * c_dag(do, 0)
H_mat = rep.sparse_matrix(H_expr)
c_dag0, c_dag1 = rep.sparse_operators.c_dag
c_0, c_1 = c_dag0.getH(), c_dag1.getH()
H_ref = c_0 * c_1 * c_dag0 * c_dag1
compare_sparse_matrices(H_mat, H_ref)
def run_test(t1, filename):
dptkeys = [
'verbosity', 'calculate_sigma', 'calculate_sigma1', 'calculate_sigma2'
]
parms = {
# Solver parameters
'n_iw': 100,
# Physical parameters
'U': 0.5,
't1': t1,
'beta': 10,
# DMFT loop control parameters
'calculate_sigma': True,
'calculate_sigma1': True,
'calculate_sigma2': True,
'measure_G2_iw_ph': True,
"measure_G2_n_bosonic": 10,
"measure_G2_n_fermionic": 10,
"verbosity": 4,
}
parms["N_x"] = 2
parms["N_y"] = 1
parms["N_z"] = 1
parms["ksi_delta"] = 1.0
# Chemical potential depends on the definition of H(k) that is used
parms['chemical_potential_bare'] = 0.
parms['chemical_potential'] = parms['U'] / 2. + parms[
'chemical_potential_bare']
n_orbs = 1 # Number of orbitals
off_diag = True
spin_names = ['up', 'dn'] # Outer (non-hybridizing) blocks
orb_names = ['%s' % i for i in range(n_orbs)] # Orbital indices
gf_struct = op.set_operator_structure(spin_names,
orb_names,
off_diag=off_diag)
if haspomerol:
#####
#
# Reference: 4 site cluster, calculate only G, not G2
#
#####
def calc_reference():
ref_orbs = [
'%s' % i for i in range(n_orbs * parms['N_x'] * parms['N_y'])
]
ref_gf_struct = op.set_operator_structure(spin_names,
ref_orbs,
off_diag=off_diag)
ref_index_converter = {(sn, o): ("loc", int(o),
"down" if sn == "dn" else "up")
for sn, o in product(spin_names, ref_orbs)}
#print ref_index_converter,ref_orbs
ref_ed = PomerolED(ref_index_converter, verbose=True)
ref_N = sum(
ops.n(sn, o) for sn, o in product(spin_names, ref_orbs))
# 2 3
# 0 1
ref_H = (parms["U"] * (ops.n('up', '0') * ops.n('dn', '0') +
ops.n('up', '1') * ops.n('dn', '1')) -
2. * parms['t1'] *
(ops.c_dag('up', '0') * ops.c('up', '1') +
ops.c_dag('up', '1') * ops.c('up', '0') +
ops.c_dag('dn', '0') * ops.c('dn', '1') +
ops.c_dag('dn', '1') * ops.c('dn', '0')) -
parms['chemical_potential'] * ref_N)
# Run the solver
ref_ed.diagonalize(ref_H)
# Compute G(i\omega)
ref_G_iw = ref_ed.G_iw(ref_gf_struct, parms['beta'], parms['n_iw'])
return ref_G_iw
ref_G_iw = calc_reference()
ref = ref_G_iw[ref_spin]
g2_blocks = set([("up", "up"), ("up", "dn"), ("dn", "up"),
("dn", "dn")])
index_converter = {(sn, o):
("loc", int(o), "down" if sn == "dn" else "up")
for sn, o in product(spin_names, orb_names)}
# 1 Bath degree of freedom
# Level of the bath sites
epsilon = [
-parms['chemical_potential_bare'],
]
index_converter.update({
("B%i_%s" % (k, sn), 0):
("bath" + str(k), 0, "down" if sn == "dn" else "up")
for k, sn in product(range(len(epsilon)), spin_names)
})
# Make PomerolED solver object
ed = PomerolED(index_converter, verbose=True)
N = sum(ops.n(sn, o) for sn, o in product(spin_names, orb_names))
H_loc = (parms["U"] * (ops.n('up', '0') * ops.n('dn', '0')) -
parms['chemical_potential'] * N)
# Bath Hamiltonian: levels
H_bath = sum(eps * ops.n("B%i_%s" % (k, sn), 0)
for sn, (k,
eps) in product(spin_names, enumerate(epsilon)))
# Hybridization Hamiltonian
# Bath-impurity hybridization
V = [
-2 * bath_prefactor * t1,
]
H_hyb = ops.Operator()
for k, v in enumerate(V):
H_hyb += sum(
v * ops.c_dag("B%i_%s" % (k, sn), 0) * ops.c(sn, '0') +
np.conj(v) * ops.c_dag(sn, '0') * ops.c("B%i_%s" % (k, sn), 0)
for sn in spin_names)
# Obtain bath sites from Delta and create H_ED
H_ED = H_loc + H_bath + H_hyb
# Run the solver
ed.diagonalize(H_ED)
# Compute G(i\omega)
G_iw = ed.G_iw(gf_struct, parms['beta'], parms['n_iw'])
if parms["measure_G2_iw_ph"]:
common_g2_params = {
'gf_struct': gf_struct,
'beta': parms['beta'],
'blocks': g2_blocks,
'n_iw': parms['measure_G2_n_bosonic']
}
G2_iw = ed.G2_iw_inu_inup(channel="PH",
block_order="AABB",
n_inu=parms['measure_G2_n_fermionic'],
**common_g2_params)
if mpi.is_master_node():
with ar.HDFArchive(filename, 'w') as arch:
arch["parms"] = parms
arch["G_iw"] = G_iw
arch["G2_iw"] = G2_iw
arch["ref"] = ref
else: # haspomerol is False
with ar.HDFArchive(filename, 'r') as arch:
ref = arch['ref']
G_iw = arch['G_iw']
G2_iw = arch['G2_iw']
BL = lattice.BravaisLattice(units=[
(1, 0, 0),
]) #linear lattice
kmesh = gf.MeshBrillouinZone(lattice.BrillouinZone(BL), parms["N_x"])
Hk_blocks = [gf.Gf(indices=orb_names, mesh=kmesh) for spin in spin_names]
Hk = gf.BlockGf(name_list=spin_names, block_list=Hk_blocks)
def Hk_f(k):
return -2 * parms['t1'] * (np.cos(k[0])) * np.eye(1)
for spin, _ in Hk:
for k in Hk.mesh:
Hk[spin][k] = Hk_f(k.value)
# Construct the DF2 program
X = dualfermion.Dpt(beta=parms['beta'],
gf_struct=gf_struct,
Hk=Hk,
n_iw=parms['n_iw'],
n_iw2=parms["measure_G2_n_fermionic"],
n_iW=parms["measure_G2_n_bosonic"])
for name, g0 in X.Delta:
X.Delta[name] << gf.inverse(
gf.iOmega_n +
parms['chemical_potential_bare']) * bath_prefactor**2 * 4 * t1**2
X.G2_iw << G2_iw
# Run the dual perturbation theory
X.gimp << G_iw # Load G from impurity solver
dpt_parms = {key: parms[key] for key in parms if key in dptkeys}
X.run(**dpt_parms)
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())
# Non-interacting impurity hamiltonian in matrix representation
h_0_mat = diag(eps - mu) - matrix([[0, t + a, 0, a], [t - a, 0, -a, 0],
[0, a, 0, t + a], [-a, 0, t - a, 0]])
# Bath hamiltonian in matrix representation
h_bath_mat = diag(eps_bath) - matrix([[0, t_bath, 0, 0], [t_bath, 0, 0, 0],
[0, 0, 0, t_bath], [0, 0, t_bath, 0]])
# Coupling matrix
V_mat = matrix([[1., 0., 0, 0], [0., 1., 0, 0], [0., 0., 1, 0], [0., 0., 0,
1]])
# ==== Local Hamiltonian ====
c_dag_vec = matrix([[c_dag('bl', o) for o in orb_names]])
c_vec = matrix([[c('bl', o)] for o in orb_names])
h_0 = (c_dag_vec * h_0_mat * c_vec)[0, 0]
h_int = U * n('bl', up_0) * n('bl', dn_0) + \
U * n('bl', up_1) * n('bl', dn_1) + \
U * n('bl', up_0) * n('bl', dn_1) + \
U * n('bl', up_1) * n('bl', dn_0) + \
Up * n('bl', up_0) * n('bl', up_1) + \
Up * n('bl', dn_0) * n('bl', dn_1)
h_imp = h_0 + h_int
# ==== Bath & Coupling hamiltonian ====
orb_bath_names = ['b_' + str(o) for o in orb_names]
c_dag_bath_vec = matrix([[c_dag('bl', o) for o in orb_bath_names]])
if __name__ == '__main__':
# ------------------------------------------------------------------
# -- Hubbard atom with two bath sites, Hamiltonian
beta = 10.0
V1 = 1.0
V2 = 1.0
epsilon1 = +2.30
epsilon2 = -2.30
t = 0.1
mu = 1.0
U = 2.0
up, do = 0, 1
docc = c_dag(up, 0) * c(up, 0) * c_dag(do, 0) * c(do, 0)
nA = c_dag(up, 0) * c(up, 0) + c_dag(do, 0) * c(do, 0)
nB = c_dag(up, 1) * c(up, 1) + c_dag(do, 1) * c(do, 1)
nC = c_dag(up, 2) * c(up, 2) + c_dag(do, 2) * c(do, 2)
hopA = c_dag(up, 0) * c(do, 0) + c_dag(do, 0) * c(up, 0)
hopB = c_dag(up, 1) * c(do, 1) + c_dag(do, 1) * c(up, 1)
hopC = c_dag(up, 2) * c(do, 2) + c_dag(do, 2) * c(up, 2)
H = -mu * nA + epsilon1 * nB + epsilon2 * nC + U * docc + \
V1 * (c_dag(up,0)*c(up,1) + c_dag(up,1)*c(up,0) + \
c_dag(do,0)*c(do,1) + c_dag(do,1)*c(do,0) ) + \
V2 * (c_dag(up,0)*c(up,2) + c_dag(up,2)*c(up,0) + \
c_dag(do,0)*c(do,2) + c_dag(do,2)*c(do,0) ) + \
-t * (hopA + hopB + hopC)
def test_two_particle_greens_function():
# ------------------------------------------------------------------
# -- Hubbard atom with two bath sites, Hamiltonian
beta = 2.0
V1 = 2.0
V2 = 5.0
epsilon1 = 0.00
epsilon2 = 4.00
mu = 2.0
U = 0.0
up, do = 0, 1
docc = c_dag(up, 0) * c(up, 0) * c_dag(do, 0) * c(do, 0)
nA = c_dag(up, 0) * c(up, 0) + c_dag(do, 0) * c(do, 0)
nB = c_dag(up, 1) * c(up, 1) + c_dag(do, 1) * c(do, 1)
nC = c_dag(up, 2) * c(up, 2) + c_dag(do, 2) * c(do, 2)
H = -mu * nA + epsilon1 * nB + epsilon2 * nC + U * docc + \
V1 * (c_dag(up, 0) * c(up, 1) + c_dag(up, 1) * c(up, 0) +
c_dag(do, 0) * c(do, 1) + c_dag(do, 1) * c(do, 0)) + \
V2 * (c_dag(up, 0) * c(up, 2) + c_dag(up, 2) * c(up, 0) +
c_dag(do, 0) * c(do, 2) + c_dag(do, 2) * c(do, 0))
# ------------------------------------------------------------------
# -- Exact diagonalization
fundamental_operators = [
c(up, 0), c(do, 0),
c(up, 1), c(do, 1),
c(up, 2), c(do, 2)
]
ed = TriqsExactDiagonalization(H, fundamental_operators, beta)
# ------------------------------------------------------------------
# -- single particle Green's functions
g_tau = GfImTime(name=r'$g$',
beta=beta,
statistic='Fermion',
n_points=100,
target_shape=(1, 1))
ed.set_g2_tau(g_tau[0, 0], c(up, 0), c_dag(up, 0))
# ------------------------------------------------------------------
# -- Two particle Green's functions
ntau = 10
imtime = MeshImTime(beta, 'Fermion', ntau)
prodmesh = MeshProduct(imtime, imtime, imtime)
g40_tau = Gf(name='g40_tau', mesh=prodmesh, target_shape=(1, 1, 1, 1))
g4_tau = Gf(name='g4_tau', mesh=prodmesh, target_shape=(1, 1, 1, 1))
ed.set_g40_tau_matrix(g40_tau, g_tau)
ed.set_g4_tau(g4_tau[0, 0, 0, 0], c(up, 0), c_dag(up, 0), c(up, 0),
c_dag(up, 0))
# ------------------------------------------------------------------
# -- compare
zero_outer_planes_and_equal_times(g4_tau)
zero_outer_planes_and_equal_times(g40_tau)
np.testing.assert_array_almost_equal(g4_tau.data, g40_tau.data)
from triqs.operators import c, c_dag, Operator
from triqs.atom_diag import *
from itertools import product
import numpy as np
orbs = (1, 2, 3)
spins = ("dn", "up")
# Construct a sum of the pair hopping and spin flip terms from a 3-orbital Kanamori interaction Hamiltonian
H_p = Operator()
H_J = Operator()
for o1, o2 in product(orbs, repeat=2):
if o1 == o2: continue
H_p += c_dag("dn", o1) * c_dag("up", o1) * c("up", o2) * c("dn", o2)
H_J += c_dag("dn", o1) * c("up", o1) * c_dag("up", o2) * c("dn", o2)
H = H_p + H_J
fops1 = [(s, o) for (s, o) in product(spins, orbs)]
fops2 = [(s, o) for (o, s) in product(orbs, spins)]
ad1 = AtomDiag(H, fops1)
ad2 = AtomDiag(H, fops2)
# for e1, e2 in zip(ad1.energies, ad2.energies):
# print(e1.round(5), e2.round(5))
# Check that subspace energies are identical up to ordering
assert len(ad1.energies) == len(ad2.energies)
# Non-interacting impurity hamiltonian in matrix representation
h_0_mat = diag(eps - mu) - matrix([[0, t, t], [t, 0, t], [t, t, 0]])
# Bath hamiltonian in matrix representation
h_bath_mat = diag(eps_bath) - matrix([[0, t_bath, t_bath], [t_bath, 0, t_bath],
[t_bath, t_bath, 0]])
# Coupling matrix
V_mat = matrix([[1., 1., 1.], [1., 1., 1.], [1., 1., 1.]])
# ==== Local Hamiltonian ====
c_dag_vec = {
s: matrix([[c_dag(s, o) for o in range(n_orb)]])
for s in block_names
}
c_vec = {s: matrix([[c(s, o)] for o in range(n_orb)]) for s in block_names}
h_0 = sum(c_dag_vec[s] * h_0_mat * c_vec[s] for s in block_names)[0, 0]
Umat, Upmat = U_matrix_kanamori(n_orb, U_int=U, J_hund=J)
h_int = h_int_kanamori(block_names,
range(n_orb),
Umat,
Upmat,
J,
off_diag=True)
h_imp = h_0 + h_int
# ==== Bath & Coupling hamiltonian ====
c_dag_bath_vec = {
def test_cf_G_tau_and_G_iw_nonint(verbose=False):
beta = 3.22
eps = 1.234
niw = 64
ntau = 2 * niw + 1
H = eps * c_dag(0, 0) * c(0, 0)
fundamental_operators = [c(0, 0)]
ed = TriqsExactDiagonalization(H, fundamental_operators, beta)
# ------------------------------------------------------------------
# -- Single-particle Green's functions
G_tau = GfImTime(beta=beta,
statistic='Fermion',
n_points=ntau,
target_shape=(1, 1))
G_iw = GfImFreq(beta=beta,
statistic='Fermion',
n_points=niw,
target_shape=(1, 1))
G_iw << inverse(iOmega_n - eps)
G_tau << Fourier(G_iw)
G_tau_ed = GfImTime(beta=beta,
statistic='Fermion',
n_points=ntau,
target_shape=(1, 1))
G_iw_ed = GfImFreq(beta=beta,
statistic='Fermion',
n_points=niw,
target_shape=(1, 1))
ed.set_g2_tau(G_tau_ed[0, 0], c(0, 0), c_dag(0, 0))
ed.set_g2_iwn(G_iw_ed[0, 0], c(0, 0), c_dag(0, 0))
# ------------------------------------------------------------------
# -- Compare gfs
from triqs.utility.comparison_tests import assert_gfs_are_close
assert_gfs_are_close(G_tau, G_tau_ed)
assert_gfs_are_close(G_iw, G_iw_ed)
# ------------------------------------------------------------------
# -- Plotting
if verbose:
from triqs.plot.mpl_interface import oplot, plt
subp = [3, 1, 1]
plt.subplot(*subp)
subp[-1] += 1
oplot(G_tau.real)
oplot(G_tau_ed.real)
plt.subplot(*subp)
subp[-1] += 1
diff = G_tau - G_tau_ed
oplot(diff.real)
oplot(diff.imag)
plt.subplot(*subp)
subp[-1] += 1
oplot(G_iw)
oplot(G_iw_ed)
plt.show()
n_iw = int(10 * beta) # The number of positive Matsubara frequencies
n_k = 16 # The number of k-points per dimension
block_names = ['up', 'dn'] # The spins
orb_names = [0, 1, 2] # The orbitals
idx_lst = list(range(len(block_names) * len(orb_names)))
gf_struct = [('bl', idx_lst)]
TBL = tight_binding_model(lambda_soc=SOC) # The Tight-Binding Lattice
TBL.bz = BrillouinZone(TBL.bl)
n_idx = len(idx_lst)
# ==== Local Hamiltonian ====
c_dag_vec = matrix([[c_dag('bl', idx) for idx in idx_lst]])
c_vec = matrix([[c('bl', idx)] for idx in idx_lst])
h_0_mat = TBL._hop[(0, 0, 0)]
h_0 = (c_dag_vec * h_0_mat * c_vec)[0, 0]
Umat, Upmat = U_matrix_kanamori(len(orb_names), U_int=U, J_hund=J)
op_map = {(s, o): ('bl', i)
for i, (s, o) in enumerate(product(block_names, orb_names))}
h_int = h_int_kanamori(block_names,
orb_names,
Umat,
Upmat,
J,
off_diag=True,
map_operator_structure=op_map)
h_imp = h_0 + h_int
def make_calc():
# ------------------------------------------------------------------
# -- Hamiltonian
p = ParameterCollection(
beta = 0.5,
U = 0.5,
nw = 1,
nwf = 15,
V = 1.0,
eps = 0.2,
)
p.nwf_gf = 4 * p.nwf
p.mu = 0.5*p.U
# ------------------------------------------------------------------
ca_up, cc_up = c('0', 0), c_dag('0', 0)
ca_do, cc_do = c('0', 1), c_dag('0', 1)
ca0_up, cc0_up = c('1', 0), c_dag('1', 0)
ca0_do, cc0_do = c('1', 1), c_dag('1', 1)
docc = cc_up * ca_up * cc_do * ca_do
nA = cc_up * ca_up + cc_do * ca_do
hybridiz = p.V * (cc0_up * ca_up + cc_up * ca0_up + cc0_do * ca_do + cc_do * ca0_do)
bath_lvl = p.eps * (cc0_up * ca0_up + cc0_do * ca0_do)
p.H_int = p.U * docc
p.H = -p.mu * nA + p.H_int + hybridiz + bath_lvl
# ------------------------------------------------------------------
# -- Exact diagonalization
# Conversion from TRIQS to Pomerol notation for operator indices
# TRIQS: block_name, inner_index
# Pomerol: site_label, orbital_index, spin_name
index_converter = {
('0', 0) : ('loc', 0, 'up'),
('0', 1) : ('loc', 0, 'down'),
('1', 0) : ('loc', 1, 'up'),
('1', 1) : ('loc', 1, 'down'),
}
# -- Create Exact Diagonalization instance
ed = PomerolED(index_converter, verbose=True)
ed.diagonalize(p.H) # -- Diagonalize H
p.gf_struct = [['0', [0, 1]]]
# -- Single-particle Green's functions
p.G_iw = ed.G_iw(p.gf_struct, p.beta, n_iw=p.nwf_gf)['0']
# -- Particle-particle two-particle Matsubara frequency Green's function
opt = dict(
beta=p.beta, gf_struct=p.gf_struct,
blocks=set([("0", "0")]),
n_iw=p.nw, n_inu=p.nwf)
p.G2_iw_ph = ed.G2_iw_inu_inup(channel='PH', **opt)[('0', '0')]
filename = 'data_pomerol.h5'
with HDFArchive(filename,'w') as res:
res['p'] = p
import os
os.system('tar czvf data_pomerol.tar.gz data_pomerol.h5')
os.remove('data_pomerol.h5')
# ----------------------------------------------------------------------
if __name__ == '__main__':
# ------------------------------------------------------------------
# -- Hubbard atom with two bath sites, Hamiltonian
beta = 2.0
V1 = 2.0
V2 = 5.0
epsilon1 = 0.00
epsilon2 = 4.00
mu = 2.0
U = 0.0
up, do = 0, 1
docc = c_dag(up,0) * c(up,0) * c_dag(do,0) * c(do,0)
nA = c_dag(up,0) * c(up,0) + c_dag(do,0) * c(do,0)
nB = c_dag(up,1) * c(up,1) + c_dag(do,1) * c(do,1)
nC = c_dag(up,2) * c(up,2) + c_dag(do,2) * c(do,2)
H = -mu * nA + epsilon1 * nB + epsilon2 * nC + U * docc + \
V1 * (c_dag(up,0)*c(up,1) + c_dag(up,1)*c(up,0) + \
c_dag(do,0)*c(do,1) + c_dag(do,1)*c(do,0) ) + \
V2 * (c_dag(up,0)*c(up,2) + c_dag(up,2)*c(up,0) + \
c_dag(do,0)*c(do,2) + c_dag(do,2)*c(do,0) )
# ------------------------------------------------------------------
# -- Exact diagonalization
fundamental_operators = [
c(up,0), c(do,0), c(up,1), c(do,1), c(up,2), c(do,2)]
Hk=Hk,
n_iw=parms['n_iw'],
n_iw2=parms["measure_G2_n_fermionic"],
n_iW=parms["measure_G2_n_bosonic"])
if haspomerol:
g2_blocks = set([("up", "up"), ("up", "dn"), ("dn", "up"), ("dn", "dn")])
index_converter = {(sn, o): ("loc", int(o), "down" if sn == "dn" else "up")
for sn, o in product(spin_names, orb_names)}
# Make PomerolED solver object
ed = PomerolED(index_converter, verbose=True)
N = sum(ops.n(sn, o) for sn, o in product(spin_names, orb_names))
H = (parms["Ua"] * (ops.n('up', '0') * ops.n('dn', '0')) + parms["Ub"] *
(ops.n('up', '1') * ops.n('dn', '1')) + parms['t0'] *
(ops.c_dag('up', '0') * ops.c('up', '1') + ops.c_dag('up', '1') *
ops.c('up', '0') + ops.c_dag('dn', '0') * ops.c('dn', '1') +
ops.c_dag('dn', '1') * ops.c('dn', '0')) -
parms['chemical_potential'] * N)
#####
#
# Reference: 4 site cluster, calculate only G, not G2
#
#####
def calc_reference():
ref_orbs = ['%s' % i for i in range(n_orbs * parms['N_x'])]
ref_gf_struct = op.set_operator_structure(spin_names,
ref_orbs,
off_diag=off_diag)
def test_trimer_hamiltonian():
# ------------------------------------------------------------------
# -- Hubbard atom with two bath sites, Hamiltonian
beta = 2.0
V1 = 2.0
V2 = 5.0
epsilon1 = 0.00
epsilon2 = 4.00
mu = 2.0
U = 1.0
# -- construction using triqs operators
up, do = 0, 1
docc = c_dag(up, 0) * c(up, 0) * c_dag(do, 0) * c(do, 0)
nA = c_dag(up, 0) * c(up, 0) + c_dag(do, 0) * c(do, 0)
nB = c_dag(up, 1) * c(up, 1) + c_dag(do, 1) * c(do, 1)
nC = c_dag(up, 2) * c(up, 2) + c_dag(do, 2) * c(do, 2)
H_expr = -mu * nA + epsilon1 * nB + epsilon2 * nC + U * docc + \
V1 * (c_dag(up, 0) * c(up, 1) + c_dag(up, 1) * c(up, 0) +
c_dag(do, 0) * c(do, 1) + c_dag(do, 1) * c(do, 0)) + \
V2 * (c_dag(up, 0) * c(up, 2) + c_dag(up, 2) * c(up, 0) +
c_dag(do, 0) * c(do, 2) + c_dag(do, 2) * c(do, 0))
# ------------------------------------------------------------------
fundamental_operators = [
c(up, 0), c(do, 0),
c(up, 1), c(do, 1),
c(up, 2), c(do, 2)
]
rep = SparseMatrixRepresentation(fundamental_operators)
H_mat = rep.sparse_matrix(H_expr)
# -- explicit construction
class Dummy(object):
def __init__(self):
pass
op = Dummy()
op.cdagger = rep.sparse_operators.c_dag
op.c = np.array([cdag.getH() for cdag in op.cdagger])
op.n = np.array([cdag * cop for cop, cdag in zip(op.c, op.cdagger)])
H_ref = -mu * (op.n[0] + op.n[1]) + \
epsilon1 * (op.n[2] + op.n[3]) + \
epsilon2 * (op.n[4] + op.n[5]) + \
U * op.n[0] * op.n[1] + \
V1 * (op.cdagger[0] * op.c[2] + op.cdagger[2] * op.c[0] +
op.cdagger[1] * op.c[3] + op.cdagger[3] * op.c[1]) + \
V2 * (op.cdagger[0] * op.c[4] + op.cdagger[4] * op.c[0] +
op.cdagger[1] * op.c[5] + op.cdagger[5] * op.c[1])
# ------------------------------------------------------------------
# -- compare
compare_sparse_matrices(H_mat, H_ref)
V = [ 2.0, 5.0 ] # Couplings to Bath-sites
spin_names = ['up', 'dn']
orb_names = [0]
# ==== Local Hamiltonian ====
h_0 = - mu*( n('up',0) + n('dn',0) ) - h*( n('up',0) - n('dn',0) )
h_int = U * n('up',0) * n('dn',0)
h_imp = h_0 + h_int
# ==== Bath & Coupling Hamiltonian ====
h_bath, h_coup = 0, 0
for i, E_i, V_i in zip([0, 1], E, V):
for sig in ['up','dn']:
h_bath += E_i * n(sig,'b_' + str(i))
h_coup += V_i * (c_dag(sig,0) * c(sig,'b_' + str(i)) + c_dag(sig,'b_' + str(i)) * c(sig,0))
# ==== Total impurity hamiltonian and fundamental operators ====
h_tot = h_imp + h_coup + h_bath
# ==== Green function structure ====
gf_struct = [ [s, orb_names] for s in spin_names ]
# ==== Hybridization Function ====
n_iw = int(10 * beta)
iw_mesh = MeshImFreq(beta, 'Fermion', n_iw)
Delta = BlockGf(mesh=iw_mesh, gf_struct=gf_struct)
Delta << sum([V_i*V_i * inverse(iOmega_n - E_i) for V_i,E_i in zip(V, E)]);
# ==== Non-Interacting Impurity Green function ====
G0_iw = BlockGf(mesh=iw_mesh, gf_struct=gf_struct)
spin_names = ['up', 'dn']
orb_names = [0, 1]
orb_bath_names = [0, 1, 2, 3]
# Non-interacting impurity hamiltonian in matrix representation
h_0_mat = diag(eps - mu) - matrix([[0, t], [t, 0]])
# 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]
# Non-interacting impurity hamiltonian in matrix representation
h_0_mat = diag(eps - mu) - matrix([[0, t + a, 0, a], [t - a, 0, -a, 0],
[0, a, 0, t + a], [-a, 0, t - a, 0]])
# Bath hamiltonian in matrix representation
h_bath_mat = diag(eps_bath) - matrix([[0, t_bath, 0, 0], [t_bath, 0, 0, 0],
[0, 0, 0, t_bath], [0, 0, t_bath, 0]])
# Coupling matrix
V_mat = matrix([[1., 0., 0, 0], [0., 1., 0, 0], [0., 0., 1, 0], [0., 0., 0,
1]])
# ==== Local Hamiltonian ====
c_dag_vec = matrix([[c_dag('bl', o) for o in range(n_orb)]])
c_vec = matrix([[c('bl', o)] for o in range(n_orb)])
h_0 = (c_dag_vec * h_0_mat * c_vec)[0, 0]
h_int = U * n('bl', up_0) * n('bl', dn_0) + \
U * n('bl', up_1) * n('bl', dn_1) + \
U * n('bl', up_0) * n('bl', dn_1) + \
U * n('bl', up_1) * n('bl', dn_0) + \
Up * n('bl', up_0) * n('bl', up_1) + \
Up * n('bl', dn_0) * n('bl', dn_1)
h_imp = h_0 + h_int
# ==== Bath & Coupling hamiltonian ====
c_dag_bath_vec = matrix(
[[c_dag('bl', o) for o in range(n_orb, n_orb + n_orb_bath)]])
