netket.nn.blocks.SymmExpSum

netket.nn.blocks.SymmExpSum#

class netket.nn.blocks.SymmExpSum[source]#

Bases: Module

A wrapper module to symmetrise a variational wave function with respect to a permutation group \(G\) by summing over all permuted inputs.

Given the characters \(\chi_g\) of an irreducible representation (irrep) of \(G\), a variational state that transforms according to this irrep is given by the projection formula

\[\psi_\theta(\sigma) = \frac{1}{|G|}\sum_{g\in G} \chi_g \psi_\theta(T_{g}\sigma).\]

Ground states usually transform according to the trivial irrep \(\chi_g=1 \forall g\in G\).

Examples

Symmetrise an netket.models.RBM with respect to the space group of a 2D netket.graph.Square lattice:

>>> import netket as nk
>>> graph = nk.graph.Square(4)
>>> group = graph.space_group()
>>> print("Size of space group:", len(group))
Size of space group: 128
>>> # Construct the bare unsymmetrized machine
>>> machine_no_symm = nk.models.RBM(alpha=2)
>>> # Symmetrize the RBM over the space group
>>> ma = nk.nn.blocks.SymmExpSum(module = machine_no_symm, symm_group=group)

Nontrivial irreps can be specified using momentum and point-group quantum numbers:

>>> from math import pi
>>> print(group.little_group(pi, 0).character_table_readable())
(['1xId()', '1xRefl(0°)', '1xRefl(90°)', '1xRot(180°)'],
array([[ 1.,  1.,  1.,  1.],
       [ 1.,  1., -1., -1.],
       [ 1., -1.,  1., -1.],
       [ 1., -1., -1.,  1.]]))
>>> chi = group.space_group_irreps(pi, 0)[1]
>>> ma = nk.nn.blocks.SymmExpSum(module = machine_no_symm, symm_group=group, characters=chi)

Convolutional networks are already invariant under translations, so they only need to be symmetrised with respect to the point group (e.g., mirrors and rotations).

>>> import netket as nk
>>> graph = nk.graph.Square(4)
>>> print("Size of the point group:", len(graph.point_group()))
Size of the point group: 8
>>> # Construct a translation-invariant RBM
>>> machine_trans = nk.models.RBMSymm(alpha=2, symmetries=graph.translation_group())
>>> # Symmetrize the RBM over the point group
>>> ma = nk.nn.blocks.SymmExpSum(module = machine_trans, symm_group=graph.point_group())
Attributes
character_id: int | None = None#

Index of the character to project onto in the character table of the symmetry group.

The characters are accessed as:

symm_group.character_table()[character_id]

Only one of characters and character_id may be specified. If neither is specified, the character is taken to be all 1, yielding a trivially symmetric state.

characters: HashableArray | None = None#

Characters \(\chi_g\) of the space group to project onto.

Only one of characters and character_id may be specified. If neither is specified, the character is taken to be all 1, yielding a trivially symmetric state.

module: Module#

The unsymmetrised neural-network ansatz.

symm_group: PermutationGroup#

The symmetry group to use. It should be a valid netket.utils.group.PermutationGroup object.

Can be extracted from a netket.graph.Lattice object by calling point_group() or translation_group().

Alternatively, if you have a netket.graph.Graph object you can build it from automorphisms().

graph = nk.graph.Square(3)
symm_group = graph.point_group()
Methods
__call__(x)[source]#

Accepts a single input or arbitrary batch of inputs.

The last dimension of x must match the shape of the permutation group.

Return type:

Array

Parameters:

x (Array)

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