netket.hilbert.SpinOrbitalFermions

class netket.hilbert.SpinOrbitalFermions

Bases: HomogeneousHilbert

Hilbert space for 2nd quantization fermions with spin s distributed among n_orbital orbitals.

The number of fermions can be fixed globally or fixed on a per spin projection.

Note

This class is simply a convenient wrapper that creates a Fock or TensorHilbert of Fock spaces with occupation numbers 0 or 1. It is mainly useful to avoid needing to specify the n_max=1 each time, and adds convenient functions such as _get_index and _spin_index, which allow one to index the correct TensorHilbert corresponding to the right spin projection.

Inheritance

__init__(n_orbitals, s=None, *, n_fermions=None, n_fermions_per_spin=None, constraint=None)

Constructs the hilbert space for spin-s fermions on n_orbitals.

Samples of this hilbert space represent occupation numbers \((0,1)\) of the orbitals. The number of fermions may be fixed to n_fermions. If the spin is different from 0 or None, n_fermions can also be a list to fix the number of fermions per spin component. Using this class, one can generate a tensor product of fermionic hilbert spaces that distinguish particles with different spin.

Parameters:

• n_orbitals (int) – number of orbitals we store occupation numbers for. If the number of fermions per spin is conserved, the different spin configurations are not counted as orbitals and are handled differently.

• s (float | None) – spin of the fermions.

• n_fermions (int | None) – (optional) fixed number of fermions per spin (conserved). If specified, the total number of fermions is conserved. If the fermions have a spin, the number of fermions per spin subsector is not conserved.

• n_fermions_per_spin (tuple[int, ...] | None) – (optional) list of the fixed number of fermions for every spin subsector. This automatically enforces a global population constraint. Only one between n_fermions and n_fermions_per_spin can be specified. The length of the iterable should be \(2S+1\).

• constraint (DiscreteHilbertConstraint | None) – An extra constraint for the Hilbert space, defined according to the constraint API (see netket.hilbert.constraint.DiscreteHilbertConstraint for more details).

Returns:

A SpinOrbitalFermions object

Attributes

constrained

The hilbert space does not contain prod(hilbert.shape) basis states.

Typical constraints are population constraints (such as fixed number of bosons, fixed magnetization…) which ensure that only a subset of the total unconstrained space is populated.

Typically, objects defined in the constrained space cannot be converted to QuTiP or other formats.

constraint

The constraint inflicted upon this poor Hilbert space.

Instead of all possible combinations of local states, only those satisfying the constraint are part of this Hilbert space.

is_finite

Whether the local hilbert space is finite.

is_indexable

Whether the space can be indexed with an integer

local_size

Size of the local degrees of freedom that make the total hilbert space.

local_states

A list of discrete local quantum numbers. If the local states are infinitely many, None is returned.

n_fermions

The total number of fermions. None if unspecified.

n_fermions_per_spin

Tuple identifying the per-subsector population constraint.

This tuple has length 1 for spinless fermions and length 2s+1 for spinful fermions. Every element is an integer or None, where None means no constraint on the number of fermions in that subsector.

n_orbitals

The number of orbitals (for each spin subsector if relevant).

n_spin_subsectors

Total number of spin subsectors. If spin is None, this is 1.

n_states

The total dimension of the many-body Hilbert space. Throws an exception iff the space is not indexable.

shape

The size of the hilbert space on every site.

size

The total number number of degrees of freedom.

spin

Returns the spin of the fermions

Methods

all_states()

Returns all valid states of the Hilbert space.

Throws an exception if the space is not indexable.

Return type:

ndarray

Returns:

A (n_states x size) batch of states. this corresponds to the pre-allocated array if it was passed.

local_indices_to_states(x, dtype=None)

Converts a tensor of integers to the corresponding local_values in this hilbert space.

Equivalent to

The input last dimension must match the size of this Hilbert space. This function can be jax-jitted.

Parameters:

x (Union[ndarray, Array]) – a tensor with integer dtype and whose last dimension matches the size of this Hilbert space.

Returns:

a tensor with the same shape as the input, and values corresponding to the local_state indexed by the input tensor x.

numbers_to_states(numbers)

Returns the quantum numbers corresponding to the n-th basis state for input n.

n is an array of integer indices such that numbers[k]=Index(states[k]). Throws an exception iff the space is not indexable.

This function validates the inputs by checking that the numbers provided are smaller than the Hilbert space size, and throws an error if that condition is not met. When called from within a jax.jit context, this uses {func}`equinox.error_if` to throw runtime errors.

Parameters:

numbers (numpy.array) – Batch of input numbers to be converted into arrays of quantum numbers.

Return type:

Array

ptrace(sites)

Returns the hilbert space without the selected sites.

Not all hilbert spaces support this operation.

Parameters:

sites (int | Sequence[int]) – a site or list of sites to trace away

Return type:

Optional[Self]

Returns:

The partially-traced hilbert space. The type of the resulting hilbert space might be different from the starting one.

random_state(key=None, size=None, dtype=None)

Generates either a single or a batch of uniformly distributed random states. Runs as random_state(self, key, size=None, dtype=np.float32) by default.

Parameters:

• key (Any) – rng state from a jax-style functional generator.

• size (int | None) – If provided, returns a batch of configurations of the form (size, N) if size is an integer or (*size, N) if it is a tuple and where \(N\) is the Hilbert space size. By default, a single random configuration with shape (#,) is returned.

• dtype – DType of the resulting vector.

Return type:

Array

Returns:

A state or batch of states sampled from the uniform distribution on the hilbert space.

Example

>>> import netket, jax
>>> hi = netket.hilbert.Qubit(N=2)
>>> k1, k2 = jax.random.split(jax.random.PRNGKey(1))
>>> print(hi.random_state(key=k1))
[0 0]
>>> print(hi.random_state(key=k2, size=2))
[[0 0]
 [0 0]]

size_at_index(i)

Size of the local degrees of freedom for the i-th variable.

Parameters:

i (int) – The index of the desired site

Return type:

int

Returns:

The number of degrees of freedom at that site

states()

Returns an iterator over all valid configurations of the Hilbert space. Throws an exception iff the space is not indexable. Iterating over all states with this method is typically inefficient, and `all_states` should be preferred.

Return type:

Iterator[ndarray]

states_at_index(i)

A list of discrete local quantum numbers at the site i.

If the local states are infinitely many, None is returned.

Parameters:

i (int) – The index of the desired site.

Returns:

A list of values or None if there are infinitely many.

states_to_local_indices(x)

Returns a tensor with the same shape of x, where all local values are converted to indices in the range 0…self.shape[i]. This function is guaranteed to be jax-jittable.

For the Fock space this returns x, but for other hilbert spaces such as Spin this returns an array of indices.

Warning

This function is experimental. Use at your own risk.

Parameters:

x (Union[ndarray, Array]) – a tensor containing samples from this hilbert space

Returns:

a tensor containing integer indices into the local hilbert

states_to_numbers(states)

Returns the basis state number corresponding to given quantum states.

The states are given in a batch, such that states[k] has shape (hilbert.size). Throws an exception iff the space is not indexable.

Parameters:

states (Union[ndarray, Array]) – Batch of states to be converted into the corresponding integers.

Returns:

Array of integers corresponding to states.

Return type:

numpy.darray