4. Model#

Use the Model classes to define the system to be calculated. We provide the Holstein model renormalizer.model.HolsteinModel and the spin-boson model SpinBosonModel out of box, yet any arbitrary model with sum-of-product Hamiltonian can be constructed by using the most general Model class. These classes require BasisSet and Op as input.

4.1. The Model classes#

class renormalizer.model.Model(basis: List[BasisSet], ham_terms: List[Op], dipole: Optional[Dict] = None, output_ordering: Optional[List[BasisSet]] = None)[source]#

The most general model that supports any Hamiltonian in sum-of-product form. Base class for HolsteinModel and SpinBosonModel.

Parameters:

basis (list of BasisSet) – Local basis for each site of the MPS. The order determines the DoF order in the MPS.
ham_terms (list of Op) – Terms of the system Hamiltonian in sum-of-product form. Identities can be omitted in the operators. All terms must be included, without assuming Hermitian or something else.
dipole (dict) – Contains the transition dipole matrix element. The key is the dof name.
output_ordering (list of BasisSet) – The ordering of the local basis for output. Default is the same with basis.

check_operator_terms(terms: List[Op])[source]#

Check and clean operator terms in the input terms. Errors will be raised if the type of operator is not Op or the operator contains DoF not defined in self.basis. Operators with factor = 0 are discarded.

Parameters:: terms (list of Op) – The terms to check.
Returns:: new_terms – Operator list with 0-factor terms discarded.
Return type:: list of Op

copy()[source]#

dofs#: list of DoF names.

e_dofs#: list of electronic DoF names.

get_mpos(key: str, fun: Callable)[source]#

Get MPOs related to the model, such as MPOs to calculate electronic occupations \({a^\dagger_i a_i}\). The purpose of the function is to avoid repeated MPO construction.

Parameters:

key (str) – Name of the MPOs. In principle other hashable types are also OK.
fun (callable) – The function to generate MPOs if the MPOs have not been constructed before. The function should accept only one argument which is the model and return a list of Mpo.

Returns:

mpos – The required MPOs.

Return type:

list of Mpo

n_dofs#: Number of total DoFs.

n_edofs#: Number of total electronic DoFs.

n_vdofs#: Number of total vibrational DoFs.

nsite#: Number of sites in the MPS/MPO to be constructed. Length of self.basis.

to_dict() → Dict[source]#

Convert the object into a dict that contains only objects of Python primitive types or NumPy types. This is primarily for dump purposes.

Returns:: info_dict – The information of the model in a dict.
Return type:: dict

v_dofs#: list of vibrational DoF names.

class renormalizer.model.HolsteinModel(mol_list: List[Mol], j_matrix: Union[Quantity, ndarray], scheme: int = 2, periodic: bool = False)[source]#

Interface for convenient Holstein model construction. The Hamiltonian of the Holstein model:

\[\hat H = \sum_{ij} J_{ij} a^\dagger_i a_j + \sum_{i\lambda} \omega_{i\lambda} b^\dagger_{i\lambda} b_{i\lambda} + \sum_{i\lambda} g_{i\lambda} \omega_{i\lambda} a^\dagger_i a_i (b^\dagger_{i\lambda} + b_{i\lambda})\]

Parameters:

mol_list (list of Mol) – Information for the molecules contains in the system. See the Mol class for more details.
j_matrix (np.ndarray or Quantity.) – \(J_{ij}\) in the Holstein Hamiltonian. When Quantity is used as input, the system is taken to have homogeneous nearest-neighbour interaction \(J_{ij}=J \delta_{i, j+1} + J \delta_{i, j-1}\). For the boundary condition, see the periodic option.
scheme (int) –
The scheme of the basis for the model. Historically four numbers are permitted: 1, 2, 3, 4. Now 1, 2 and 3 are equivalent, and the bases are arranged as:

\[[\rm{e}_{0}, \rm{ph}_{0,0}, \rm{ph}_{0,1}, \cdots, \rm{e}_{1}, \rm{ph}_{1,0}, \rm{ph}_{1,1}, \cdots ]\]

And when scheme is set to 4, all electronic DoF is contained in one BasisSet using BasisMultiElectronVac and the bases are arranged as:

\[[\rm{ph}_{0,0}, \rm{ph}_{0,1}, \cdots, \rm{ph}_{n/2, 0}, \rm{ph}_{n/2, 1}, \cdots, \rm{e}_{0, 1, \cdots, n} \rm{ph}_{n/2+1, 0}, \rm{ph}_{n/2+1, 1}, \cdots]\]
periodic (bool) – Whether use periodical boundary condition when constructing j_matrix from Quantity. Default is False.

basis: List[BasisSet]#

check_operator_terms(terms: List[Op])#

Parameters:: terms (list of Op) – The terms to check.
Returns:: new_terms – Operator list with 0-factor terms discarded.
Return type:: list of Op

copy()[source]#

dofs#: list of DoF names.

e_dofs#: list of electronic DoF names.

get_mpos(key: str, fun: Callable)#

Get MPOs related to the model, such as MPOs to calculate electronic occupations \({a^\dagger_i a_i}\). The purpose of the function is to avoid repeated MPO construction.

Parameters:

key (str) – Name of the MPOs. In principle other hashable types are also OK.
fun (callable) – The function to generate MPOs if the MPOs have not been constructed before. The function should accept only one argument which is the model and return a list of Mpo.

Returns:

mpos – The required MPOs.

Return type:

list of Mpo

property gs_zpe: float#: Ground state zero-point energy \(\sum_{i, j} \frac{1}{2}\omega_{i, j}\).

ham_terms: List[Op]#

property j_constant#

Extract electronic coupling constant from self.j_matrix. Useful in transport model when \(J\) is a constant.. If J is actually not a constant, a value error will be raised.

Returns:: j constant – J constant extracted from self.j_matrix.
Return type:: float

n_dofs#: Number of total DoFs.

n_edofs#: Number of total electronic DoFs.

n_vdofs#: Number of total vibrational DoFs.

nsite#: Number of sites in the MPS/MPO to be constructed. Length of self.basis.

qn_size: int#

switch_scheme(scheme: int) → HolsteinModel[source]#

Switch the scheme of the current model.

Parameters:: scheme (int) – The target scheme.
Returns:: new_model – The new model with the specified scheme.
Return type:: HolsteinModel

to_dict() → Dict#

Convert the object into a dict that contains only objects of Python primitive types or NumPy types. This is primarily for dump purposes.

Returns:: info_dict – The information of the model in a dict.
Return type:: dict

v_dofs#: list of vibrational DoF names.

class renormalizer.model.SpinBosonModel(epsilon: Quantity, delta: Quantity, ph_list: List[Phonon], dipole: Optional[float] = None)[source]#

Spin-Boson model

\[\hat{H} = \epsilon \sigma_z + \Delta \sigma_x + \frac{1}{2} \sum_i(p_i^2+\omega^2_i q_i^2) + \sigma_z \sum_i c_i q_i\]

basis: List[BasisSet]#

check_operator_terms(terms: List[Op])#

Parameters:: terms (list of Op) – The terms to check.
Returns:: new_terms – Operator list with 0-factor terms discarded.
Return type:: list of Op

copy()#

dofs#: list of DoF names.

e_dofs#: list of electronic DoF names.

get_mpos(key: str, fun: Callable)#

Get MPOs related to the model, such as MPOs to calculate electronic occupations \({a^\dagger_i a_i}\). The purpose of the function is to avoid repeated MPO construction.

Parameters:

key (str) – Name of the MPOs. In principle other hashable types are also OK.
fun (callable) – The function to generate MPOs if the MPOs have not been constructed before. The function should accept only one argument which is the model and return a list of Mpo.

Returns:

mpos – The required MPOs.

Return type:

list of Mpo

ham_terms: List[Op]#

n_dofs#: Number of total DoFs.

n_edofs#: Number of total electronic DoFs.

n_vdofs#: Number of total vibrational DoFs.

nsite#: Number of sites in the MPS/MPO to be constructed. Length of self.basis.

qn_size: int#

to_dict() → Dict#

Convert the object into a dict that contains only objects of Python primitive types or NumPy types. This is primarily for dump purposes.

Returns:: info_dict – The information of the model in a dict.
Return type:: dict

v_dofs#: list of vibrational DoF names.

class renormalizer.model.TI1DModel(basis: List[BasisSet], local_ham_terms: List[Op], nonlocal_ham_terms: List[Op], ncell: int)[source]#

Translational invariant one dimensional model with periodic boundary condition. The Hamiltonian should take the form:

\[\hat H = \sum_i(\hat h_i + \sum_j \hat h_{i,j})\]

where \(\hat h_i\) is the local Hamiltonian acting on one single unit cell and \(\hat h_{i,j}\) represents the \(j\) th interaction between the \(i\) th cell and other unit cells.

Yet doesn’t support setting transition dipoles.

Parameters:

basis (list of BasisSet) – Local basis of each site for a single unit cell of the system. The full basis set is constructed by repeating the basis ncell times. To distinguish between different DoFs at different unit cells, the DoF names in the unit cell are transformed to a two-element-tuple of the form ("cell0", dof), where dof is the original DoF name and the "cell0" indicates the cell ID.
local_ham_terms (list of Op) – Terms of the system local Hamiltonian \(\hat h_i\) in sum-of-product form. DoF names should be consistent with the basis argument.
nonlocal_ham_terms (list of Op) – Terms of system nonlocal Hamiltonian \(\hat h_{i,j}\). To indicate the IDs of the unit cells that are involved in the nonlocal interaction, the DoF name in basis should be transformed to a two-element-tuple, in which the first element is an integer indicating its distance from \(i\), and the second element is the original DoF name. For example, if one unit cell contains one electron DoF with name e, a nearest neighbour hopping interaction should take the form Op(r"a^\dagger a", [(0, "e"), (1, "e")]) (with its Hermite conjugation being another term). The definition is not unique in that Op(r"a^\dagger a", [(1, "e"), (2, "e")]) produces equivalent output.
ncell (int) – Number of unit cells in the system.

basis: List[BasisSet]#

check_operator_terms(terms: List[Op])#

Parameters:: terms (list of Op) – The terms to check.
Returns:: new_terms – Operator list with 0-factor terms discarded.
Return type:: list of Op

copy()#

dofs#: list of DoF names.

e_dofs#: list of electronic DoF names.

get_mpos(key: str, fun: Callable)#

Get MPOs related to the model, such as MPOs to calculate electronic occupations \({a^\dagger_i a_i}\). The purpose of the function is to avoid repeated MPO construction.

Parameters:

key (str) – Name of the MPOs. In principle other hashable types are also OK.
fun (callable) – The function to generate MPOs if the MPOs have not been constructed before. The function should accept only one argument which is the model and return a list of Mpo.

Returns:

mpos – The required MPOs.

Return type:

list of Mpo

ham_terms: List[Op]#

n_dofs#: Number of total DoFs.

n_edofs#: Number of total electronic DoFs.

n_vdofs#: Number of total vibrational DoFs.

nsite#: Number of sites in the MPS/MPO to be constructed. Length of self.basis.

qn_size: int#

to_dict() → Dict#

Convert the object into a dict that contains only objects of Python primitive types or NumPy types. This is primarily for dump purposes.

Returns:: info_dict – The information of the model in a dict.
Return type:: dict

v_dofs#: list of vibrational DoF names.

4.2. Basis Functions#

class renormalizer.model.basis.BasisSet(dof, nbas: int, sigmaqn: List)[source]#

the parent class for local basis set

Parameters:

dof_name – The name(s) of the DoF(s) contained in the basis set. For basis containing only one DoF, the type could be anything that can be hashed. For basis containing multiple DoFs, the type should be a list or tuple of anything that can be hashed.
nbas (int) – number of dimension of the basis set
sigmaqn (List) – the quantum number of each basis. Could be an integer or tuple of integers

copy(new_dof)[source]#

Return a copy of the basis set with new DoF name specified in the argument.

Parameters:: new_dof – New DoF name.
Returns:: new_basis – A copy of the basis with new DoF name.
Return type:: Basis

property dofs#

Names of the DoFs contained in the basis. Returns a tuple even if the basis contains only one DoF.

Returns:: dof names – A tuple of DoF names.
Return type:: tuple

is_electron = False#: If the basis set represent electronic DoF.

is_phonon = False#: If the basis set represent vibrational DoF.

is_spin = False#: If the basis set represent spin DoF.

multi_dof = False#: If the basis set contains multiple DoFs.

op_mat(op: Op)[source]#

Matrix representation under the basis set of the input operator. The factor is included.

Parameters:: op (Op) – The operator. For basis set with only one DoF, :class:str is also acceptable.
Returns:: mat – Matrix representation of op.
Return type:: np.ndarray

class renormalizer.model.basis.BasisSHO(dof, omega, nbas, x0=0.0, dvr=False, general_xp_power=False)[source]#

simple harmonic oscillator basis set

Parameters:

dof – The name of the DoF contained in the basis set. The type could be anything that can be hashed.
omega (float) – the frequency of the oscillator.
nbas (int) – number of dimension of the basis set (highest occupation number of the harmonic oscillator)
x0 (float) – the origin of the harmonic oscillator. Default = 0.
dvr (bool) – whether to use discrete variable representation. Default = False.
general_xp_power (bool) – whether calculate \(x\) and \(x^2\) (or \(p\) and \(p^2\)) through general expression for \(x`power or :math:`p\) power. This is not efficient because \(x\) and \(x^2\) (or \(p\) and \(p^2\)) have been hard-coded already. The option is only used for testing.

copy(new_dof)[source]#

Return a copy of the basis set with new DoF name specified in the argument.

Parameters:: new_dof – New DoF name.
Returns:: new_basis – A copy of the basis with new DoF name.
Return type:: Basis

property dofs#

Names of the DoFs contained in the basis. Returns a tuple even if the basis contains only one DoF.

Returns:: dof names – A tuple of DoF names.
Return type:: tuple

is_electron = False#: If the basis set represent electronic DoF.

is_phonon = True#: If the basis set represent vibrational DoF.

is_spin = False#: If the basis set represent spin DoF.

multi_dof = False#: If the basis set contains multiple DoFs.

op_mat(op: Union[Op, str])[source]#

Matrix representation under the basis set of the input operator. The factor is included.

Parameters:: op (Op) – The operator. For basis set with only one DoF, :class:str is also acceptable.
Returns:: mat – Matrix representation of op.
Return type:: np.ndarray

sigmaqn: ndarray#

class renormalizer.model.basis.BasisHopsBoson(dof, nbas)[source]#

Bosonic like basis but with uncommon ladder operator, used in Hierarchy of Pure States method

\[\begin{split}\tilde{b}^\dagger | n \rangle = (n+1) | n+1\rangle \\ \tilde{b} | n \rangle = | n-1\rangle\end{split}\]

Parameters:

dof – The name of the DoF contained in the basis set. The type could be anything that can be hashed.
nbas (int) – number of dimension of the basis set (the highest occupation number)

copy(new_dof)[source]#

Return a copy of the basis set with new DoF name specified in the argument.

Parameters:: new_dof – New DoF name.
Returns:: new_basis – A copy of the basis with new DoF name.
Return type:: Basis

property dofs#

Names of the DoFs contained in the basis. Returns a tuple even if the basis contains only one DoF.

Returns:: dof names – A tuple of DoF names.
Return type:: tuple

is_electron = False#: If the basis set represent electronic DoF.

is_phonon = True#: If the basis set represent vibrational DoF.

is_spin = False#: If the basis set represent spin DoF.

multi_dof = False#: If the basis set contains multiple DoFs.

op_mat(op: Union[Op, str])[source]#

Matrix representation under the basis set of the input operator. The factor is included.

Parameters:: op (Op) – The operator. For basis set with only one DoF, :class:str is also acceptable.
Returns:: mat – Matrix representation of op.
Return type:: np.ndarray

sigmaqn: ndarray#

class renormalizer.model.basis.BasisSineDVR(dof, nbas, xi, xf, endpoint=False, quadrature=False, dvr=False)[source]#

Sine DVR basis (particle-in-a-box) for vibrational, angular, and dissociative modes. See Phys. Rep. 324, 1–105 (2000).

\[\psi_j(x) = \sqrt{\frac{2}{L}} \sin(j\pi(x-x_0)/L) \, \textrm{for} \, x_0 \le x \le x_{N+1}, L = x_{N+1} - x_0\]

the grid points are at

\[x_\alpha = x_0 + \alpha \frac{L}{N+1}\]

Operators supported:: \[I, x, x^1, x^2, x^\textrm{moment}, dx, dx^2, p, p^2, x dx, x^2 p^2, x^2 dx, x p^2, x^3 p^2\]

Parameters:

dof (str, int) – The name of the DoF contained in the basis set. The type could be anything that can be hashed.
nbas (int) – Number of grid points.
xi (float) – The leftmost grid point of the coordinate.
xf (float) – The rightmost grid point of the coordinate.
endpoint (bool, optional) – If endpoint=False, \(x_0=x_i, x_{N+1}=x_f\); otherwise \(x_1=x_i, x_{N}=x_f\).

copy(new_dof)[source]#

Return a copy of the basis set with new DoF name specified in the argument.

Parameters:: new_dof – New DoF name.
Returns:: new_basis – A copy of the basis with new DoF name.
Return type:: Basis

property dofs#

Names of the DoFs contained in the basis. Returns a tuple even if the basis contains only one DoF.

Returns:: dof names – A tuple of DoF names.
Return type:: tuple

property eigenfunc#

is_electron = False#: If the basis set represent electronic DoF.

is_phonon = True#: If the basis set represent vibrational DoF.

is_spin = False#: If the basis set represent spin DoF.

multi_dof = False#: If the basis set contains multiple DoFs.

op_mat(op: Union[Op, str])[source]#

Matrix representation under the basis set of the input operator. The factor is included.

Parameters:: op (Op) – The operator. For basis set with only one DoF, :class:str is also acceptable.
Returns:: mat – Matrix representation of op.
Return type:: np.ndarray

quad(expr)[source]#

sigmaqn: ndarray#

class renormalizer.model.basis.BasisSimpleElectron(dof, sigmaqn=None)[source]#

The basis set for simple electron DoF, two state with 0: unoccupied, 1: occupied

Parameters:: dof (any hashable object) – The name of the DoF contained in the basis set.

Examples

>>> b = BasisSimpleElectron(0)
>>> b
BasisSimpleElectron(dof: 0, nbas: 2, qn: [[0], [1]])
>>> b.op_mat(r"a^\dagger")
array([[0., 0.],
       [1., 0.]])

copy(new_dof)[source]#

Return a copy of the basis set with new DoF name specified in the argument.

Parameters:: new_dof – New DoF name.
Returns:: new_basis – A copy of the basis with new DoF name.
Return type:: Basis

property dofs#

Names of the DoFs contained in the basis. Returns a tuple even if the basis contains only one DoF.

Returns:: dof names – A tuple of DoF names.
Return type:: tuple

is_electron = True#: If the basis set represent electronic DoF.

is_phonon = False#: If the basis set represent vibrational DoF.

is_spin = False#: If the basis set represent spin DoF.

multi_dof = False#: If the basis set contains multiple DoFs.

op_mat(op)[source]#

Matrix representation under the basis set of the input operator. The factor is included.

Parameters:: op (Op) – The operator. For basis set with only one DoF, :class:str is also acceptable.
Returns:: mat – Matrix representation of op.
Return type:: np.ndarray

sigmaqn: ndarray#

class renormalizer.model.basis.BasisMultiElectron(dof, sigmaqn: List)[source]#

The basis set for multi electronic state on one single site, The basis order is [dof_names[0], dof_names[1], dof_names[2], …].

Parameters:

dof (a list or tuple of hashable objects.) – The names of the DoFs contained in the basis set.
sigmaqn (list of int or list of containers of int) – The quantum number of each basis

copy(new_dof)[source]#

Return a copy of the basis set with new DoF name specified in the argument.

Parameters:: new_dof – New DoF name.
Returns:: new_basis – A copy of the basis with new DoF name.
Return type:: Basis

property dofs#

Names of the DoFs contained in the basis. Returns a tuple even if the basis contains only one DoF.

Returns:: dof names – A tuple of DoF names.
Return type:: tuple

is_electron = True#: If the basis set represent electronic DoF.

is_phonon = False#: If the basis set represent vibrational DoF.

is_spin = False#: If the basis set represent spin DoF.

multi_dof = True#: If the basis set contains multiple DoFs.

op_mat(op: Op)[source]#

Matrix representation under the basis set of the input operator. The factor is included.

Parameters:: op (Op) – The operator. For basis set with only one DoF, :class:str is also acceptable.
Returns:: mat – Matrix representation of op.
Return type:: np.ndarray

sigmaqn: ndarray#

class renormalizer.model.basis.BasisMultiElectronVac(dof)[source]#

Another basis set for multi electronic state on one single site. Vacuum state is included. The basis order is [vacuum, dof_names[0], dof_names[1],…]. sigma qn is [0, 1, 1, 1, …]

Parameters:: dof (a list or tuple of hashable objects.) – The names of the DoFs contained in the basis set.

copy(new_dof)[source]#

Return a copy of the basis set with new DoF name specified in the argument.

Parameters:: new_dof – New DoF name.
Returns:: new_basis – A copy of the basis with new DoF name.
Return type:: Basis

property dofs#

Names of the DoFs contained in the basis. Returns a tuple even if the basis contains only one DoF.

Returns:: dof names – A tuple of DoF names.
Return type:: tuple

is_electron = True#: If the basis set represent electronic DoF.

is_phonon = False#: If the basis set represent vibrational DoF.

is_spin = False#: If the basis set represent spin DoF.

multi_dof = True#: If the basis set contains multiple DoFs.

op_mat(op: Op)[source]#

Matrix representation under the basis set of the input operator. The factor is included.

Parameters:: op (Op) – The operator. For basis set with only one DoF, :class:str is also acceptable.
Returns:: mat – Matrix representation of op.
Return type:: np.ndarray

sigmaqn: ndarray#

class renormalizer.model.basis.BasisHalfSpin(dof, sigmaqn: Optional[List] = None)[source]#

The basis the for 1/2 spin DoF

Parameters:

dof (any hashable object such as integer) – The name of the DoF contained in the basis set.
sigmaqn (list of int or list of containers of int) – The quantum number of each basis

Examples

>>> b = BasisHalfSpin(0)
>>> b
BasisHalfSpin(dof: 0, nbas: 2)
>>> b.op_mat("X")
array([[0., 1.],
       [1., 0.]])
>>> -1 * b.op_mat("iY") @ b.op_mat("iY")  # convenient for real Hamiltonian
array([[1., 0.],
       [0., 1.]])

copy(new_dof)[source]#

Return a copy of the basis set with new DoF name specified in the argument.

Parameters:: new_dof – New DoF name.
Returns:: new_basis – A copy of the basis with new DoF name.
Return type:: Basis

property dofs#

Names of the DoFs contained in the basis. Returns a tuple even if the basis contains only one DoF.

Returns:: dof names – A tuple of DoF names.
Return type:: tuple

is_electron = False#: If the basis set represent electronic DoF.

is_phonon = False#: If the basis set represent vibrational DoF.

is_spin = True#: If the basis set represent spin DoF.

multi_dof = False#: If the basis set contains multiple DoFs.

op_mat(op: Union[Op, str])[source]#

Matrix representation under the basis set of the input operator. The factor is included.

Parameters:: op (Op) – The operator. For basis set with only one DoF, :class:str is also acceptable.
Returns:: mat – Matrix representation of op.
Return type:: np.ndarray

sigmaqn: ndarray#

4.3. The Operator Class#

class renormalizer.model.Op(symbol: str, dof, factor: Union[float, Quantity] = 1.0, qn: Optional[Union[List, int]] = None)[source]#

The operator class. The class can be considered as a symbolic way to express quantum operator such as \(a^\dagger_i a_j\), \(b^\dagger_i + b_i\) or \(g \omega a^\dagger_i a_i (b^\dagger_{ij} + b_{ij})\). Additions and multiplications between operators are supported.

Parameters:

symbol (str) – The string of the operator, such as "a", "a^\dagger a" and r"b^\dagger + b". The supported operators are defined in BasisSet. For complex symbols consisting of multiplication of simple symbols, separate the simple symbols with a single space.
dof (hashable object or list of hashable objects.) – The name of the DoF related to the operator. For simple symbol such as "a", the type could be any hashable object. int, str and tuple of them are recommended types for dof. For complex symbol such as "a^\dagger a", the type should be a list of hashable objects, with each element representing one of the simple symbol contained in the complex symbol. Using a single hashable object is also supported for complex symbol, in which case every symbol is assumed to share the same DoF name.
factor (float or complex or Quantity) – The prefactor of the operator.
qn (int or list of int or list of containers of int.) – The quantum number of the symbol. For simple symbol the qn should be an int. For complex symbol quantum number of each simple symbol contained in the complex symbol should be provided in a list. If qn is set to None, then the quantum number for every symbol is set to 0 except for``”a^dagger”`` and "a", whose quantum number are set to 1 and -1 respectively. For multiple quantum numbers list of containers of int should be provided without any abbreviation.

Notes

Symbols connected by plus "+" such as r"b^\dagger + b" is considered as a simple symbol, because this class is designed to deal with operator multiplication rather than addition.

Warning

If you wish to specify the DoF of each symbol in a complex symbol, you should use a list instead of tuple. Because in the latter case the tuple is recognized as a single DoF and the DoFs of all simple symbols are set to the tuple.

Examples

>>> from renormalizer.model import Op
>>> Op(r"a^\dagger a", ['site0', "site1"], 2., qn=[1, -1])
Op('a^\\dagger a', ['site0', 'site1'], 2.0, [[1], [-1]])
>>> x = Op("X", 0, 0.5)
>>> 3 * x
Op('X', [0], 1.5)
>>> y = Op("Y", 1, 0.2)
>>> x + y
[Op('X', [0], 0.5), Op('Y', [1], 0.2)]
>>> x - y
[Op('X', [0], 0.5), Op('Y', [1], -0.2)]
>>> x * y
Op('X Y', [0, 1], 0.1)
>>> x * (x + y)
[Op('X X', [0, 0], 0.25), Op('X Y', [0, 1], 0.1)]
>>> (y + x) * x
[Op('Y X', [1, 0], 0.1), Op('X X', [0, 0], 0.25)]
>>> (y + x) * (x + y)
[Op('Y X', [1, 0], 0.1), Op('Y Y', [1, 1], 0.04000000000000001), Op('X X', [0, 0], 0.25), Op('X Y', [0, 1], 0.1)]

property factor#: The factor of the operator.

classmethod identity(dof, qn_size=1, factor=1.0)[source]#: Construct identity operator.

property is_identity: bool#: Whether the operator is the identity operator. Note the factor is not taken into account.

classmethod product(op_list: List[Op])[source]#

Construct a new operator as a multiplication product of operators.

Parameters:: op_list (list of Op) – The operators to be multiplied.
Returns:: op – The product operator.
Return type:: Op

property qn: ndarray#: Total quantum number of the operator. Sum of self.qn_list. Quantum number of "a^\dagger" and "a" is taken to be 1 and -1 if not specified.

property qn_size: int#: Size of the quantum numbers

same_term(other) → bool[source]#

Judge whether two operators are the same term and can be added together.

Parameters:: other (Op) – The other operator to compare
Returns:: result – Whether the two terms
Return type:: bool

Examples

>>> from renormalizer.model import Op
>>> op1 = Op("X", 0, 0.1)
>>> op2 = Op("X", 0, 0.2)
>>> op1.same_term(op2)
True
>>> op3 = Op("Y", 1)
>>> op1.same_term(op3)
False

split_elementary(dof_to_siteidx) → Tuple[List[Op], Union[float, complex]][source]#

Construct elementary operators according to site index. “elementary operator” means that in the operator every DoF is on the same MPS site.

Parameters:

dof_to_siteidx (dict) – Mapping from DoF name to MPS site index.

Returns:

elementary_operators (list of Op) – A list of elementary operators. The order is determined by the DoF name. Factors are set to 1.
factor (float or class:complex) – Factor of the operator.

Examples

>>> from renormalizer.model import Op
>>> op = Op("X Y", [3, 2], 0.5) * Op("Y X", [2, 3], 3.0) * Op("Z Z", [2, 2], 1.0)
>>> op.split_elementary({2:0, 3:1})
([Op('Y Y Z Z', [2, 2, 2, 2], 1.0), Op('X X', [3, 3], 1.0)], 1.5)

squeeze_identity()[source]#

Remove all the identity terms in Op.

Returns:: op – The operator with identity term removed
Return type:: Op

Examples

>>> from renormalizer.model import Op
>>> op = Op("X I Y I", [0, 1, 2, 3], 0.5)
>>> op.squeeze_identity()
Op('X Y', [0, 2], 0.5)
>>> op = Op("I", 0, -0.5)
>>> op.squeeze_identity()
Op('I', [0], -0.5)

to_tuple() → Tuple[source]#

Convert the operator into a tuple. The fields are the symbol, the DoFs the factor and the quantum number list. The converted tuple can be hashed.

Returns:: op_tuple – The converted tuple.
Return type:: tuple