# netket.operator.Heisenberg¶

class netket.operator.Heisenberg(hilbert, graph, J=1, sign_rule=None)[source]

Bases: netket.operator._graph_operator.GraphOperator

The Heisenberg hamiltonian on a lattice.

__init__(hilbert, graph, J=1, sign_rule=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 (float) – The strength of the coupling. Default is 1.

• sign_rule – If enabled, 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.

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, sign_rule=True; dim=20)

Attributes
H

Returns the Conjugate-Transposed operator

Return type

AbstractOperator

J

The coupling strength.

Return type

float

T

Returns the transposed operator

Return type

AbstractOperator

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]

Return type
constant
Return type

Number

dtype
Return type

Any

graph

The graph on which this Operator is defined

Return type

AbstractGraph

hilbert

The hilbert space associated to this operator.

Return type

AbstractHilbert

is_hermitian

Returns true if this operator is hermitian.

Return type

bool

max_conn_size

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

Return type

int

mel_cutoff

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

Type

float

Return type

float

n_operators
Return type

int

operators

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

Return type
size
Return type

int

uses_sign_rule
Methods
__call__(v)

Call self as a function.

Return type

ndarray

Parameters

v (numpy.ndarray) –

apply(v)
Return type

ndarray

Parameters

v (numpy.ndarray) –

collect()

Returns a guranteed concrete instancce of an operator.

As some operations on operators return lazy wrapperes (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 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 (array) – 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

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

get_conn_flattened(x, sections, pad=False)

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

get_conn_padded(x)

Finds the connected elements of the Operator. Starting from a batch of quantum numbers x={x_1, … x_n} of size B x 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 x Kmax x M where Kmax 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. mels: A N-tensor containing the matrix elements $$O(x,x')$$

associated to each x’ for every batch.

Return type

x_primes

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_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 tranpose of this operator.