# netket.operator.Heisenberg#

class netket.operator.Heisenberg#

Bases: GraphOperator

The Heisenberg hamiltonian on a lattice.

Inheritance __init__(hilbert, graph, J=1.0, sign_rule=None, *, acting_on_subspace=None)[source]#

Constructs an Heisenberg operator given a hilbert space and a graph providing the connectivity of the lattice.

Parameters:
• hilbert (AbstractHilbert) – Hilbert space the operator acts on.

• graph (AbstractGraph) – The graph upon which this hamiltonian is defined.

• J (Union[float, Sequence[float]]) – The strength of the coupling. Default is 1. Can pass a sequence of coupling strengths with coloured graphs: edges of colour n will have coupling strength J[n]

• sign_rule – If True, Marshal’s sign rule will be used. On a bipartite lattice, this corresponds to a basis change flipping the Sz direction at every odd site of the lattice. For non-bipartite lattices, the sign rule cannot be applied. Defaults to True if the lattice is bipartite, False otherwise. If a sequence of coupling strengths is passed, defaults to False and a matching sequence of sign_rule must be specified to override it

• acting_on_subspace (Union[list[int], int]) – Specifies the mapping between nodes of the graph and Hilbert space sites, so that graph node i ∈ [0, ..., graph.n_nodes - 1], corresponds to acting_on_subspace[i] ∈ [0, ..., hilbert.n_sites]. Must be a list of length graph.n_nodes. Passing a single integer start is equivalent to [start, ..., start + graph.n_nodes - 1].

Examples

Constructs a Heisenberg operator for a 1D system.

>>> import netket as nk
>>> g = nk.graph.Hypercube(length=20, n_dim=1, pbc=True)
>>> hi = nk.hilbert.Spin(s=0.5, total_sz=0, N=g.n_nodes)
>>> op = nk.operator.Heisenberg(hilbert=hi, graph=g)
>>> print(op)
Heisenberg(J=1.0, sign_rule=True; dim=20)

Attributes
H#

Returns the Conjugate-Transposed operator

J#

The coupling strength.

T#

Returns the transposed operator

acting_on#

List containing the list of the sites on which every operator acts.

Every operator self.operators[i] acts on the sites self.acting_on[i]

acting_on_subspace#

Mapping between nodes of the graph and Hilbert space sites as given in the constructor.

constant#
dtype#
graph#

The graph on which this Operator is defined

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.

mel_cutoff#

The cutoff for matrix elements. Only matrix elements such that abs(O(i,i))>mel_cutoff are considered

Type:

float

n_operators#
operators#

List of the matrices of the operators encoded in this Local Operator. Returns a copy.

size#
uses_sign_rule#
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)#

LocalOperator: Returns the complex conjugate of this operator.

copy(*, dtype=None)#

Returns a copy of the operator, while optionally changing the dtype of the operator.

Parameters:

dtype (Optional[Any]) – optional dtype

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.

get_conn_filtered(x, sections, filters)#

Finds the connected elements of the Operator using only a subset of operators. 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 (matrix) – A matrix of shape (batch_size,hilbert.size) containing the batch of quantum numbers x.

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

• filters (array) – Only operators op(filters[i]) are used to find the connected elements of x[i].

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

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

• pad (bool) – Whether to use zero-valued matrix elements in order to return all equal 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

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_pauli_strings()#

Convert to PauliStrings object

Return type:

PauliStrings

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)#

LocalOperator: Returns the transpose of this operator.