# netket.experimental.operator.FermionOperator2nd#

class netket.experimental.operator.FermionOperator2nd#

A fermionic operator in $$2^{nd}$$ quantization.

Inheritance __init__(hilbert, terms, weights=None, constant=0.0, dtype=None)[source]#

Constructs a fermion operator given the single terms (set of creation/annihilation operators) in second quantization formalism.

This class can be initialized in the following form: FermionOperator2nd(hilbert, terms, weights …). The term contains pairs of (idx, dagger), where idx ∈ range(hilbert.size) (it identifies an orbital) and dagger is a True/False flag determining if the operator is a creation or destruction operator. A term of the form $$\hat{a}_1^\dagger \hat{a}_2$$ would take the form ((1,1), (2,0)), where (1,1) represents $$\hat{a}_1^\dagger$$ and (2,0) represents $$\hat{a}_2$$. To split up per spin, use the creation and annihilation operators to build the operator.

Parameters:
Returns:

A FermionOperator2nd object.

Example

Constructs the fermionic hamiltonian in $$2^{nd}$$ quantization $$(0.5-0.5j)*(a_0^\dagger a_1) + (0.5+0.5j)*(a_2^\dagger a_1)$$.

>>> import netket.experimental as nkx
>>> terms, weights = (((0,1),(1,0)),((2,1),(1,0))), (0.5-0.5j,0.5+0.5j)
>>> hi = nkx.hilbert.SpinOrbitalFermions(3)
>>> op = nkx.operator.FermionOperator2nd(hi, terms, weights)
>>> op
FermionOperator2nd(hilbert=SpinOrbitalFermions(n_orbitals=3), n_operators=2, dtype=complex128)
>>> terms = ("0^ 1", "2^ 1")
>>> op = nkx.operator.FermionOperator2nd(hi, terms, weights)
>>> op
FermionOperator2nd(hilbert=SpinOrbitalFermions(n_orbitals=3), n_operators=2, dtype=complex128)
>>> op.hilbert
SpinOrbitalFermions(n_orbitals=3)
>>> op.hilbert.size
3

Attributes
H#

Returns the Conjugate-Transposed operator

T#

Returns the transposed operator

dtype#

The dtype of the operator’s matrix elements ⟨σ|Ô|σ’⟩.

hilbert#

The hilbert space associated to this operator.

is_hermitian#

Returns true if this operator is hermitian.

max_conn_size#

The maximum number of non zero ⟨x|O|x’⟩ for every x.

Methods
__call__(v)#

Call self as a function.

Return type:

ndarray

Parameters:

v (ndarray) –

apply(v)#
Return type:

ndarray

Parameters:

v (ndarray) –

collect()#

Returns a guaranteed concrete instance of an operator.

As some operations on operators return lazy wrappers (such as transpose, hermitian conjugate…), this is used to obtain a guaranteed non-lazy operator.

Return type:

AbstractOperator

conj(*, concrete=False)#
Return type:

AbstractOperator

conjugate(*, concrete=False)[source]#

Returns the complex conjugate of this operator.

copy(*, dtype=None)[source]#
Parameters:

dtype (Optional[Any]) –

static from_openfermion(hilbert, of_fermion_operator=None, *, n_orbitals=None, convert_spin_blocks=False)[source]#

Converts an openfermion FermionOperator into a netket FermionOperator2nd.

The hilbert first argument can be dropped, see __init__ for details and default value. Warning: convention of openfermion.hamiltonians is different from ours: instead of strong spin components as subsequent hilbert state outputs (i.e. the 1/2 spin components of spin-orbit i are stored in locations (2*i, 2*i+1)), we concatenate blocks of definite spin (i.e. locations (i, n_orbitals+i)).

Parameters:
Return type:

FermionOperator2nd

get_conn(x)#

Finds the connected elements of the Operator. Starting from a given quantum number x, it finds all other quantum numbers x’ such that the matrix element $$O(x,x')$$ is different from zero. In general there will be several different connected states x’ satisfying this condition, and they are denoted here $$x'(k)$$, for $$k=0,1...N_{\mathrm{connected}}$$.

Parameters:

x (ndarray) – An array of shape (hilbert.size, ) containing the quantum numbers x.

Returns:

The connected states x’ of shape (N_connected,hilbert.size) array: An array containing the matrix elements $$O(x,x')$$ associated to each x’.

Return type:

matrix

Raises:

ValueError – If the given quantum number is not compatible with the hilbert space.

Finds the connected elements of the Operator.

Starting from a given quantum number x, it finds all other quantum numbers x’ such that the matrix element $$O(x,x')$$ is different from zero. In general there will be several different connected states x’ satisfying this condition, and they are denoted here $$x'(k)$$, for $$k=0,1...N_{\mathrm{connected}}$$.

This is a batched version, where x is a matrix of shape (batch_size,hilbert.size).

Parameters:
• x – A matrix of shape (batch_size,hilbert.size) containing the batch of quantum numbers x.

• sections – An array of size (batch_size) useful to unflatten the output of this function. See numpy.split for the meaning of sections.

Returns:

The connected states x’, flattened together in a single matrix. array: An array containing the matrix elements $$O(x,x')$$ associated to each x’.

Return type:

matrix

Finds the connected elements of the Operator.

Starting from a batch of quantum numbers $$x={x_1, ... x_n}$$ of size $$B \times M$$ where $$B$$ size of the batch and $$M$$ size of the hilbert space, finds all states $$y_i^1, ..., y_i^K$$ connected to every $$x_i$$.

Returns a matrix of size $$B \times K_{max} \times M$$ where $$K_{max}$$ is the maximum number of connections for every $$y_i$$.

Parameters:

x (ndarray) – A N-tensor of shape $$(...,hilbert.size)$$ containing the batch/batches of quantum numbers $$x$$.

Returns:

The connected states x’, in a N+1-tensor and an N-tensor containing the matrix elements $$O(x,x')$$ associated to each x’ for every batch.

Return type:

(x_primes, mels)

n_conn(x, out=None)#

Return the number of states connected to x.

Parameters:
• x (matrix) – A matrix of shape (batch_size,hilbert.size) containing the batch of quantum numbers x.

• out (array) – If None an output array is allocated.

Returns:

The number of connected states x’ for each x[i].

Return type:

array

operator_string()[source]#

Return a readable string describing all the operator terms

Return type:

str

to_dense()#

Returns the dense matrix representation of the operator. Note that, in general, the size of the matrix is exponential in the number of quantum numbers, and this operation should thus only be performed for low-dimensional Hilbert spaces or sufficiently sparse operators.

This method requires an indexable Hilbert space.

Return type:

ndarray

Returns:

The dense matrix representation of the operator as a Numpy array.

to_linear_operator()#
to_qobj()#

Convert the operator to a qutip’s Qobj.

Returns:

A qutip.Qobj object.

to_sparse()#

Returns the sparse matrix representation of the operator. Note that, in general, the size of the matrix is exponential in the number of quantum numbers, and this operation should thus only be performed for low-dimensional Hilbert spaces or sufficiently sparse operators.

This method requires an indexable Hilbert space.

Return type:

csr_matrix

Returns:

The sparse matrix representation of the operator.

transpose(*, concrete=False)#

Returns the transpose of this operator.

Parameters:

concrete – if True returns a concrete operator and not a lazy wrapper

Return type:

AbstractOperator

Returns:

if concrete is not True, self or a lazy wrapper; the transposed operator otherwise