def sz_integer(bitstring):
"""Computes the total number of occupied up sites
minus the total number of occupied down sites."""
n_sites = len(bitstring) // 2
n_up = len([site for site in range(n_sites)
if bitstring[up_index(site)] == '1'])
n_down = len([site for site in range(n_sites)
if bitstring[down_index(site)] == '1'])
return n_up - n_down
def uccsd_singlet_get_packed_amplitudes(single_amplitudes, double_amplitudes,
n_qubits, n_electrons):
"""Convert amplitudes for use with singlet UCCSD
The output list contains only those amplitudes that are relevant to
singlet UCCSD, in an order suitable for use with the function
`uccsd_singlet_generator`.
Args:
single_amplitudes(ndarray): [NxN] array storing single excitation
amplitudes corresponding to t[i,j] * (a_i^\dagger a_j - H.C.)
double_amplitudes(ndarray): [NxNxNxN] array storing double
excitation amplitudes corresponding to
t[i,j,k,l] * (a_i^\dagger a_j a_k^\dagger a_l - H.C.)
n_qubits(int): Number of spin-orbitals used to represent the system,
which also corresponds to number of qubits in a non-compact map.
n_electrons(int): Number of electrons in the physical system.
Returns:
packed_amplitudes(list): List storing the unique single
and double excitation amplitudes for a singlet UCCSD operator.
The ordering lists unique single excitations before double
excitations.
"""
n_spatial_orbitals = n_qubits // 2
n_occupied = int(numpy.ceil(n_electrons / 2))
n_virtual = n_spatial_orbitals - n_occupied
singles = []
doubles_1 = []
doubles_2 = []
# Get singles and doubles amplitudes associated with one
# spatial occupied-virtual pair
for p, q in itertools.product(range(n_virtual), range(n_occupied)):
# Get indices of spatial orbitals
virtual_spatial = n_occupied + p
occupied_spatial = q
# Get indices of spin orbitals
virtual_up = up_index(virtual_spatial)
virtual_down = down_index(virtual_spatial)
occupied_up = up_index(occupied_spatial)
occupied_down = down_index(occupied_spatial)
# Get singles amplitude
# Just get up amplitude, since down should be the same
singles.append(single_amplitudes[virtual_up, occupied_up])
# Get doubles amplitude
doubles_1.append(double_amplitudes[virtual_up, occupied_up,
virtual_down, occupied_down])
# Get doubles amplitudes associated with two spatial occupied-virtual pairs
for (p, q), (r, s) in itertools.combinations(
itertools.product(range(n_virtual), range(n_occupied)), 2):
# Get indices of spatial orbitals
virtual_spatial_1 = n_occupied + p
occupied_spatial_1 = q
virtual_spatial_2 = n_occupied + r
occupied_spatial_2 = s
# Get indices of spin orbitals
# Just get up amplitudes, since down and cross terms should be the same
virtual_1_up = up_index(virtual_spatial_1)
occupied_1_up = up_index(occupied_spatial_1)
virtual_2_up = up_index(virtual_spatial_2)
occupied_2_up = up_index(occupied_spatial_2)
# Get amplitude
doubles_2.append(double_amplitudes[virtual_1_up, occupied_1_up,
virtual_2_up, occupied_2_up])
return singles + doubles_1 + doubles_2
def jw_sz_indices(sz_value, n_qubits, n_electrons=None):
r"""Return the indices of basis vectors with fixed Sz under JW encoding.
The returned indices label computational basis vectors which lie within
the corresponding eigenspace of the Sz operator,
.. math::
\begin{align}
S^{z} = \frac{1}{2}\sum_{i = 1}^{n}(n_{i, \alpha} - n_{i, \beta})
\end{align}
Args:
sz_value(float): Desired Sz value. Should be an integer or
half-integer.
n_qubits(int): Number of qubits defining the total state
n_electrons(int, optional): Number of particles to restrict the
operator to, if such a restriction is desired
Returns:
indices(list): The list of indices
"""
if n_qubits % 2 != 0:
raise ValueError('Number of qubits must be even')
if not (2. * sz_value).is_integer():
raise ValueError('Sz value must be an integer or half-integer')
n_sites = n_qubits // 2
sz_integer = int(2. * sz_value)
indices = []
if n_electrons is not None:
# Particle number is fixed, so the number of spin-up electrons
# (as well as the number of spin-down electrons) is fixed
if ((n_electrons + sz_integer) % 2 != 0 or
n_electrons < abs(sz_integer)):
raise ValueError('The specified particle number and sz value are '
'incompatible.')
num_up = (n_electrons + sz_integer) // 2
num_down = n_electrons - num_up
up_occupations = itertools.combinations(range(n_sites), num_up)
down_occupations = list(
itertools.combinations(range(n_sites), num_down))
# Each arrangement of up spins can be paired with an arrangement
# of down spins
for up_occupation in up_occupations:
up_occupation = [up_index(index) for index in up_occupation]
for down_occupation in down_occupations:
down_occupation = [down_index(index)
for index in down_occupation]
occupation = up_occupation + down_occupation
indices.append(sum(2 ** (n_qubits - 1 - k)
for k in occupation))
else:
# Particle number is not fixed
if sz_integer < 0:
# There are more down spins than up spins
more_map = down_index
less_map = up_index
else:
# There are at least as many up spins as down spins
more_map = up_index
less_map = down_index
for n in range(abs(sz_integer), n_sites + 1):
# Choose n of the 'more' spin and n - abs(sz_integer) of the
# 'less' spin
more_occupations = itertools.combinations(range(n_sites), n)
less_occupations = list(itertools.combinations(
range(n_sites), n - abs(sz_integer)))
# Each arrangement of the 'more' spins can be paired with an
# arrangement of the 'less' spin
for more_occupation in more_occupations:
more_occupation = [more_map(index)
for index in more_occupation]
for less_occupation in less_occupations:
less_occupation = [less_map(index)
for index in less_occupation]
occupation = more_occupation + less_occupation
indices.append(sum(2 ** (n_qubits - 1 - k)
for k in occupation))
return indices
def uccsd_singlet_generator(packed_amplitudes, n_qubits, n_electrons):
"""Create a singlet UCCSD generator for a system with n_electrons
This function generates a FermionOperator for a UCCSD generator designed
to act on a single reference state consisting of n_qubits spin orbitals
and n_electrons electrons, that is a spin singlet operator, meaning it
conserves spin.
Args:
packed_amplitudes(ndarray): Compact array storing the unique single
and double excitation amplitudes for a singlet UCCSD operator.
The ordering lists unique single excitations before double
excitations.
n_qubits(int): Number of spin-orbitals used to represent the system,
which also corresponds to number of qubits in a non-compact map.
n_electrons(int): Number of electrons in the physical system.
Returns:
generator(FermionOperator): Generator of the UCCSD operator that
builds the UCCSD wavefunction.
"""
if n_qubits % 2 != 0:
raise ValueError('The total number of spin-orbitals should be even.')
n_spatial_orbitals = n_qubits // 2
n_occupied = int(numpy.ceil(n_electrons / 2))
n_virtual = n_spatial_orbitals - n_occupied
# Unpack amplitudes
n_single_amplitudes = n_occupied * n_virtual
# Single amplitudes
t1 = packed_amplitudes[:n_single_amplitudes]
# Double amplitudes associated with one spatial occupied-virtual pair
t2_1 = packed_amplitudes[n_single_amplitudes:2 * n_single_amplitudes]
# Double amplitudes associated with two spatial occupied-virtual pairs
t2_2 = packed_amplitudes[2 * n_single_amplitudes:]
# Initialize operator
generator = FermionOperator()
# Generate all spin-conserving single and double excitations derived
# from one spatial occupied-virtual pair
for i, (p, q) in enumerate(
itertools.product(range(n_virtual), range(n_occupied))):
# Get indices of spatial orbitals
virtual_spatial = n_occupied + p
occupied_spatial = q
# Get indices of spin orbitals
virtual_up = up_index(virtual_spatial)
virtual_down = down_index(virtual_spatial)
occupied_up = up_index(occupied_spatial)
occupied_down = down_index(occupied_spatial)
# Generate single excitations
coeff = t1[i]
# Spin up excitation
generator += FermionOperator(((virtual_up, 1), (occupied_up, 0)),
coeff)
generator += FermionOperator(((occupied_up, 1), (virtual_up, 0)),
-coeff)
# Spin down excitation
generator += FermionOperator(((virtual_down, 1), (occupied_down, 0)),
coeff)
generator += FermionOperator(((occupied_down, 1), (virtual_down, 0)),
-coeff)
# Generate double excitation
coeff = t2_1[i]
generator += FermionOperator(((virtual_up, 1), (occupied_up, 0),
(virtual_down, 1), (occupied_down, 0)),
coeff)
generator += FermionOperator(((occupied_down, 1), (virtual_down, 0),
(occupied_up, 1), (virtual_up, 0)),
-coeff)
# Generate all spin-conserving double excitations derived
# from two spatial occupied-virtual pairs
for i, ((p, q), (r, s)) in enumerate(
itertools.combinations(
itertools.product(range(n_virtual), range(n_occupied)), 2)):
# Get indices of spatial orbitals
virtual_spatial_1 = n_occupied + p
occupied_spatial_1 = q
virtual_spatial_2 = n_occupied + r
occupied_spatial_2 = s
# Generate double excitations
coeff = t2_2[i]
spin_index_functions = [up_index, down_index]
for s in range(2):
# Get the functions which map a spatial orbital index to a
# spin orbital index
this_index = spin_index_functions[s]
other_index = spin_index_functions[1 - s]
# Get indices of spin orbitals
virtual_1_this = this_index(virtual_spatial_1)
occupied_1_this = this_index(occupied_spatial_1)
virtual_2_other = other_index(virtual_spatial_2)
occupied_2_other = other_index(occupied_spatial_2)
virtual_2_this = this_index(virtual_spatial_2)
occupied_2_this = this_index(occupied_spatial_2)
# p -> q is this spin and s -> r is the other spin
generator += FermionOperator(
((virtual_1_this, 1), (occupied_1_this, 0),
(virtual_2_other, 1), (occupied_2_other, 0)), coeff)
generator += FermionOperator(
((occupied_2_other, 1), (virtual_2_other, 0),
(occupied_1_this, 1), (virtual_1_this, 0)), -coeff)
# Both are this spin
generator += FermionOperator(
((virtual_1_this, 1), (occupied_1_this, 0),
(virtual_2_this, 1), (occupied_2_this, 0)), coeff)
generator += FermionOperator(
((occupied_2_this, 1), (virtual_2_this, 0),
(occupied_1_this, 1), (virtual_1_this, 0)), -coeff)
return generator
def fermi_hubbard(x_dimension,
y_dimension,
tunneling,
coulomb,
chemical_potential=0.,
magnetic_field=0.,
periodic=True,
spinless=False,
particle_hole_symmetry=False):
"""Return symbolic representation of a Fermi-Hubbard Hamiltonian.
The idea of this model is that some fermions move around on a grid and the
energy of the model depends on where the fermions are.
The Hamiltonians of this model live on a grid of dimensions
`x_dimension` x `y_dimension`.
The grid can have periodic boundary conditions or not.
In the standard Fermi-Hubbard model (which we call the "spinful" model),
there is room for an "up" fermion and a "down" fermion at each site on the
grid. In this model, there are a total of `2N` spin-orbitals,
where `N = x_dimension * y_dimension` is the number of sites.
In the spinless model, there is only one spin-orbital per site
for a total of `N`.
The Hamiltonian for the spinful model has the form
.. math::
\\begin{align}
H = &- t \sum_{\langle i,j \\rangle} \sum_{\sigma}
(a^\dagger_{i, \sigma} a_{j, \sigma} +
a^\dagger_{j, \sigma} a_{i, \sigma})
+ U \sum_{i} a^\dagger_{i, \\uparrow} a_{i, \\uparrow}
a^\dagger_{j, \downarrow} a_{j, \downarrow}
\\\\
&- \mu \sum_i \sum_{\sigma} a^\dagger_{i, \sigma} a_{i, \sigma}
- h \sum_i (a^\dagger_{i, \\uparrow} a_{i, \\uparrow} -
a^\dagger_{i, \downarrow} a_{i, \downarrow})
\\end{align}
where
- The indices :math:`\langle i, j \\rangle` run over pairs
:math:`i` and :math:`j` of sites that are connected to each other
in the grid
- :math:`\sigma \in \\{\\uparrow, \downarrow\\}` is the spin
- :math:`t` is the tunneling amplitude
- :math:`U` is the Coulomb potential
- :math:`\mu` is the chemical potential
- :math:`h` is the magnetic field
One can also construct the Hamiltonian for the spinless model, which
has the form
.. math::
H = - t \sum_{k=1}^{N-1} (a_k^\dagger a_{k + 1} + a_{k+1}^\dagger a_k)
+ U \sum_{k=1}^{N-1} a_k^\dagger a_k a_{k+1}^\dagger a_{k+1}
- \mu \sum_{k=1}^N a_k^\dagger a_k.
Args:
x_dimension (int): The width of the grid.
y_dimension (int): The height of the grid.
tunneling (float): The tunneling amplitude :math:`t`.
coulomb (float): The attractive local interaction strength :math:`U`.
chemical_potential (float, optional): The chemical potential
:math:`\mu` at each site. Default value is 0.
magnetic_field (float, optional): The magnetic field :math:`h`
at each site. Default value is 0. Ignored for the spinless case.
periodic (bool, optional): If True, add periodic boundary conditions.
Default is True.
spinless (bool, optional): If True, return a spinless Fermi-Hubbard
model. Default is False.
particle_hole_symmetry (bool, optional): If False, the repulsion
term corresponds to:
.. math::
U \sum_{k=1}^{N-1} a_k^\dagger a_k a_{k+1}^\dagger a_{k+1}
If True, the repulsion term is replaced by:
.. math::
U \sum_{k=1}^{N-1} (a_k^\dagger a_k - \\frac12)
(a_{k+1}^\dagger a_{k+1} - \\frac12)
which is unchanged under a particle-hole transformation.
Default is False
Returns:
hubbard_model: An instance of the FermionOperator class.
"""
tunneling = float(tunneling)
coulomb = float(coulomb)
chemical_potential = float(chemical_potential)
magnetic_field = float(magnetic_field)
# Initialize fermion operator class.
n_sites = x_dimension * y_dimension
n_spin_orbitals = n_sites
if not spinless:
n_spin_orbitals *= 2
hubbard_model = FermionOperator.zero()
# Select particle-hole symmetry
if particle_hole_symmetry:
coulomb_shift = FermionOperator((), 0.5)
else:
coulomb_shift = FermionOperator.zero()
# Loop through sites and add terms.
for site in range(n_sites):
# Add chemical potential to the spinless case. The magnetic field
# doesn't contribute.
if spinless and chemical_potential:
hubbard_model += number_operator(n_spin_orbitals, site,
-chemical_potential)
# With spin, add the chemical potential and magnetic field terms.
elif not spinless:
hubbard_model += number_operator(
n_spin_orbitals, up_index(site),
-chemical_potential - magnetic_field)
hubbard_model += number_operator(
n_spin_orbitals, down_index(site),
-chemical_potential + magnetic_field)
# Add local pair interaction terms.
operator_1 = number_operator(n_spin_orbitals,
up_index(site)) - coulomb_shift
operator_2 = number_operator(n_spin_orbitals,
down_index(site)) - coulomb_shift
hubbard_model += coulomb * operator_1 * operator_2
# Index coupled orbitals.
right_neighbor = site + 1
bottom_neighbor = site + x_dimension
# Account for periodic boundaries.
if periodic:
if (x_dimension > 2) and ((site + 1) % x_dimension == 0):
right_neighbor -= x_dimension
if (y_dimension > 2) and (site + x_dimension + 1 > n_sites):
bottom_neighbor -= x_dimension * y_dimension
# Add transition to neighbor on right.
if (right_neighbor) % x_dimension or (periodic and x_dimension > 2):
if spinless:
# Add Coulomb term.
operator_1 = number_operator(n_spin_orbitals, site,
1.0) - coulomb_shift
operator_2 = number_operator(n_spin_orbitals, right_neighbor,
1.0) - coulomb_shift
hubbard_model += coulomb * operator_1 * operator_2
# Add hopping term.
operators = ((site, 1), (right_neighbor, 0))
else:
# Add hopping term.
operators = ((up_index(site), 1), (up_index(right_neighbor),
0))
hopping_term = FermionOperator(operators, -tunneling)
hubbard_model += hopping_term
hubbard_model += hermitian_conjugated(hopping_term)
operators = ((down_index(site), 1),
(down_index(right_neighbor), 0))
hopping_term = FermionOperator(operators, -tunneling)
hubbard_model += hopping_term
hubbard_model += hermitian_conjugated(hopping_term)
# Add transition to neighbor below.
if site + x_dimension + 1 <= n_sites or (periodic and y_dimension > 2):
if spinless:
# Add Coulomb term.
operator_1 = number_operator(n_spin_orbitals,
site) - coulomb_shift
operator_2 = number_operator(n_spin_orbitals,
bottom_neighbor) - coulomb_shift
hubbard_model += coulomb * operator_1 * operator_2
# Add hopping term.
operators = ((site, 1), (bottom_neighbor, 0))
else:
# Add hopping term.
operators = ((up_index(site), 1), (up_index(bottom_neighbor),
0))
hopping_term = FermionOperator(operators, -tunneling)
hubbard_model += hopping_term
hubbard_model += hermitian_conjugated(hopping_term)
operators = ((down_index(site), 1),
(down_index(bottom_neighbor), 0))
hopping_term = FermionOperator(operators, -tunneling)
hubbard_model += hopping_term
hubbard_model += hermitian_conjugated(hopping_term)
return hubbard_model
def mean_field_dwave(x_dimension,
y_dimension,
tunneling,
sc_gap,
chemical_potential=0.,
periodic=True):
r"""Return symbolic representation of a BCS mean-field d-wave Hamiltonian.
The Hamiltonians of this model live on a grid of dimensions
`x_dimension` x `y_dimension`.
The grid can have periodic boundary conditions or not.
Each site on the grid can have an "up" fermion and a "down" fermion.
Therefore, there are a total of `2N` spin-orbitals,
where `N = x_dimension * y_dimension` is the number of sites.
The Hamiltonian for this model has the form
.. math::
\begin{align}
H = &- t \sum_{\langle i,j \rangle} \sum_\sigma
(a^\dagger_{i, \sigma} a_{j, \sigma} +
a^\dagger_{j, \sigma} a_{i, \sigma})
- \mu \sum_i \sum_{\sigma} a^\dagger_{i, \sigma} a_{i, \sigma}
\\
&- \sum_{\langle i,j \rangle} \Delta_{ij}
(a^\dagger_{i, \uparrow} a^\dagger_{j, \downarrow} -
a^\dagger_{i, \downarrow} a^\dagger_{j, \uparrow} +
a_{j, \downarrow} a_{i, \uparrow} -
a_{j, \uparrow} a_{i, \downarrow})
\end{align}
where
- The indices :math:`\langle i, j \rangle` run over pairs
:math:`i` and :math:`j` of sites that are connected to each other
in the grid
- :math:`\sigma \in \{\uparrow, \downarrow\}` is the spin
- :math:`t` is the tunneling amplitude
- :math:`\Delta_{ij}` is equal to :math:`+\Delta/2` for
horizontal edges and :math:`-\Delta/2` for vertical edges,
where :math:`\Delta` is the superconducting gap.
- :math:`\mu` is the chemical potential
Args:
x_dimension (int): The width of the grid.
y_dimension (int): The height of the grid.
tunneling (float): The tunneling amplitude :math:`t`.
sc_gap (float): The superconducting gap :math:`\Delta`
chemical_potential (float, optional): The chemical potential
:math:`\mu` at each site. Default value is 0.
periodic (bool, optional): If True, add periodic boundary conditions.
Default is True.
Returns:
mean_field_dwave_model: An instance of the FermionOperator class.
"""
# Initialize fermion operator class.
n_sites = x_dimension * y_dimension
n_spin_orbitals = 2 * n_sites
mean_field_dwave_model = FermionOperator()
# Loop through sites and add terms.
for site in range(n_sites):
# Add chemical potential
mean_field_dwave_model += number_operator(n_spin_orbitals,
up_index(site),
-chemical_potential)
mean_field_dwave_model += number_operator(n_spin_orbitals,
down_index(site),
-chemical_potential)
# Index coupled orbitals.
right_neighbor = site + 1
bottom_neighbor = site + x_dimension
# Account for periodic boundaries.
if periodic:
if (x_dimension > 2) and ((site + 1) % x_dimension == 0):
right_neighbor -= x_dimension
if (y_dimension > 2) and (site + x_dimension + 1 > n_sites):
bottom_neighbor -= x_dimension * y_dimension
# Add transition to neighbor on right
if (site + 1) % x_dimension or (periodic and x_dimension > 2):
# Add spin-up hopping term.
operators = ((up_index(site), 1), (up_index(right_neighbor), 0))
hopping_term = FermionOperator(operators, -tunneling)
mean_field_dwave_model += hopping_term
mean_field_dwave_model += hermitian_conjugated(hopping_term)
# Add spin-down hopping term
operators = ((down_index(site), 1), (down_index(right_neighbor),
0))
hopping_term = FermionOperator(operators, -tunneling)
mean_field_dwave_model += hopping_term
mean_field_dwave_model += hermitian_conjugated(hopping_term)
# Add pairing term
operators = ((up_index(site), 1), (down_index(right_neighbor), 1))
pairing_term = FermionOperator(operators, sc_gap / 2.)
operators = ((down_index(site), 1), (up_index(right_neighbor), 1))
pairing_term += FermionOperator(operators, -sc_gap / 2.)
mean_field_dwave_model -= pairing_term
mean_field_dwave_model -= hermitian_conjugated(pairing_term)
# Add transition to neighbor below.
if site + x_dimension + 1 <= n_sites or (periodic and y_dimension > 2):
# Add spin-up hopping term.
operators = ((up_index(site), 1), (up_index(bottom_neighbor), 0))
hopping_term = FermionOperator(operators, -tunneling)
mean_field_dwave_model += hopping_term
mean_field_dwave_model += hermitian_conjugated(hopping_term)
# Add spin-down hopping term
operators = ((down_index(site), 1), (down_index(bottom_neighbor),
0))
hopping_term = FermionOperator(operators, -tunneling)
mean_field_dwave_model += hopping_term
mean_field_dwave_model += hermitian_conjugated(hopping_term)
# Add pairing term
operators = ((up_index(site), 1), (down_index(bottom_neighbor), 1))
pairing_term = FermionOperator(operators, -sc_gap / 2.)
operators = ((down_index(site), 1), (up_index(bottom_neighbor), 1))
pairing_term += FermionOperator(operators, sc_gap / 2.)
mean_field_dwave_model -= pairing_term
mean_field_dwave_model -= hermitian_conjugated(pairing_term)
# Return.
return mean_field_dwave_model
def _hubbard(parity,
x_dimension,
y_dimension,
tunneling,
local_interaction,
coulomb,
chemical_potential=0.,
magnetic_field=0.,
periodic=True,
spinless=False,
particle_hole_symmetry=False):
r"""Returns a generic Hubbard-style Hamiltonian, to be used
by the bose_hubbard and fermi_hubbard wrapper functions.
See the corresponding bose_hubbard and fermi_hubbard for descriptions
of the various arguments.
Args:
parity (int): parity=-1 returns the output as an instance
of the FermionOperator class. Alternatively, parity=1
returns the output as an instance of the BosonOperator
class
Returns:
hubbard_model: An instance of the FermionOperator or
BosonOperator class.
"""
tunneling = float(tunneling)
local_interaction = float(local_interaction)
coulomb = float(coulomb)
chemical_potential = float(chemical_potential)
magnetic_field = float(magnetic_field)
if parity == -1:
Op = FermionOperator
elif parity == 1:
Op = BosonOperator
# Initialize operator class.
n_sites = x_dimension * y_dimension
n_spin_orbitals = n_sites
if not spinless:
n_spin_orbitals *= 2
hubbard_model = Op.zero()
# Select particle-hole symmetry
if particle_hole_symmetry:
coulomb_shift = Op((), 0.5)
else:
coulomb_shift = Op.zero()
# Loop through sites and add terms.
for site in range(n_sites):
# Add chemical potential to the spinless case. The magnetic field
# doesn't contribute.
if spinless and chemical_potential:
hubbard_model += number_operator(n_spin_orbitals,
site,
-chemical_potential,
parity=parity)
# no Pauli-Exclusion principle, spinless particles interact on-site
if parity == 1:
int_op = number_operator(
n_sites, site, local_interaction, parity=1) \
* (number_operator(n_sites, site, parity=1)
- Op.identity())
hubbard_model += int_op
# With spin, add the chemical potential and magnetic field terms.
elif not spinless:
hubbard_model += number_operator(n_spin_orbitals,
up_index(site),
-chemical_potential -
magnetic_field,
parity=parity)
hubbard_model += number_operator(n_spin_orbitals,
down_index(site),
-chemical_potential +
magnetic_field,
parity=parity)
# Add local pair interaction terms.
operator_1 = number_operator(
n_spin_orbitals, up_index(site), parity=parity) \
- coulomb_shift
operator_2 = number_operator(
n_spin_orbitals, down_index(site), parity=parity) \
- coulomb_shift
hubbard_model += coulomb * operator_1 * operator_2
# Index coupled orbitals.
right_neighbor = site + 1
bottom_neighbor = site + x_dimension
# Account for periodic boundaries.
if periodic:
if (x_dimension > 2) and ((site + 1) % x_dimension == 0):
right_neighbor -= x_dimension
if (y_dimension > 2) and (site + x_dimension + 1 > n_sites):
bottom_neighbor -= x_dimension * y_dimension
# Add transition to neighbor on right.
if (right_neighbor) % x_dimension or (periodic and x_dimension > 2):
if spinless:
# Add Coulomb term.
operator_1 = number_operator(
n_spin_orbitals, site, 1.0, parity=parity) - coulomb_shift
operator_2 = number_operator(
n_spin_orbitals, right_neighbor, 1.0, parity=parity) \
- coulomb_shift
hubbard_model += coulomb * operator_1 * operator_2
# Add hopping term.
operators = ((site, 1), (right_neighbor, 0))
else:
# Add hopping term.
operators = ((up_index(site), 1), (up_index(right_neighbor),
0))
hopping_term = Op(operators, -tunneling)
hubbard_model += hopping_term
hubbard_model += hermitian_conjugated(hopping_term)
operators = ((down_index(site), 1),
(down_index(right_neighbor), 0))
hopping_term = Op(operators, -tunneling)
hubbard_model += hopping_term
hubbard_model += hermitian_conjugated(hopping_term)
# Add transition to neighbor below.
if site + x_dimension + 1 <= n_sites or (periodic and y_dimension > 2):
if spinless:
# Add Coulomb term.
operator_1 = number_operator(
n_spin_orbitals, site, parity=parity) - coulomb_shift
operator_2 = number_operator(
n_spin_orbitals, bottom_neighbor, parity=parity) \
- coulomb_shift
hubbard_model += coulomb * operator_1 * operator_2
# Add hopping term.
operators = ((site, 1), (bottom_neighbor, 0))
else:
# Add hopping term.
operators = ((up_index(site), 1), (up_index(bottom_neighbor),
0))
hopping_term = Op(operators, -tunneling)
hubbard_model += hopping_term
hubbard_model += hermitian_conjugated(hopping_term)
operators = ((down_index(site), 1),
(down_index(bottom_neighbor), 0))
hopping_term = Op(operators, -tunneling)
hubbard_model += hopping_term
hubbard_model += hermitian_conjugated(hopping_term)
return hubbard_model
def test_up_down(self):
self.assertTrue(numpy.isclose(down_index(2), 5))
self.assertTrue(numpy.isclose(down_index(5), 11))