netket.graph.space_group.TranslationGroup

class netket.graph.space_group.TranslationGroup

Bases: PermutationGroup

Class to handle translation symmetries of a Lattice.

Corresponds to a representation of the translation group on the given lattice as a permutation group of N_sites variables.

Can be used as a PermutationGroup representing the translations, but the product table is computed much more efficiently than in a generic PermutationGroup.

Inheritance

Attributes

character_table_by_class

Calculates the character table using Burnside’s algorithm.

Each row of the output lists the characters of one irrep in the order the conjugacy classes are listed in conjugacy_classes.

Assumes that Identity() == self[0], if not, the sign of some characters may be flipped. The irreps are sorted by dimension.

conjugacy_classes

The conjugacy classes of the group.

Returns:

A tuple (classes, representatives, inverse) where classes is a boolean array whose rows indicate the elements belonging to each conjugacy class, representatives contains the lowest-indexed member of each conjugacy class, and inverse stores the conjugacy-class index of every group element.

conjugacy_table

The conjugacy table of the group.

Assuming the definitions

g = self[idx_g]
h = self[idx_h]

self[self.conjugacy_table[idx_g,idx_h]] corresponds to \(h^{-1}gh\).

group_shape

Tuple of the number of translations represented by the group along each lattice direction.

self.group_shape[i] is self.lattice.extent[i] // stride_i if both i in self.axes and self.lattice.pbc[i] is True, otherwise 1.

inverse

Indices of the inverse of each element.

Assuming the definitions

g = self[idx_g]
h = self[self.inverse[idx_g]]

gh = product(g, h) is equivalent to Identity()

Computed more efficiently than for a generic PermutationGroup exploiting the Abelian group structure.

product_table

A table of indices corresponding to \(g^{-1} h\) over the group.

Assuming the definitions

g = self[idx_g]
h = self[idx_h]
idx_u = self.product_table[idx_g, idx_h]

self[idx_u] corresponds to \(u = g^{-1} h\) .

Computed more efficiently than for a generic PermutationGroup exploiting the Abelian group structure.

shape

Tuple (<# of group elements>, <degree>).

Equivalent to self.to_array().shape.

lattice: Lattice

The lattice whose translation group is represented.

axes: tuple[int]

Axes translations along which are represented by the group.

strides: tuple[int]

Step size (in unit cells) for translations along each axis in axes.

A stride of 1 (the default) includes all unit-cell translations along that axis. A stride of s includes only every s-th translation, giving a subgroup of size extent // s. Must divide the lattice extent for periodic axes.

degree: int

Number of elements the permutations act on.

elems: list[Element]

List of group elements.

Methods

__call__(initial)

Apply all group elements to all entries of initial along the last axis.

apply_to_id(x)

Returns the image of indices x under all permutations

Parameters:

x (ndarray | Array)

character_table(multiplier=None)

Calculates the character table using Burnside’s algorithm.

Parameters:

multiplier (Union[ndarray, Array, None]) – (optional) Schur multiplier

Return type:

ndarray

Returns:

a matrix of all linear irrep characters (if multiplier is None) or projective irrep characters with the given multiplier, sorted by dimension.

Each row of lists the characters of all group elements for one irrep, i.e. self.character_table()[i,g] gives \(\chi_i(g)\).

It is assumed that Identity() == self[0]. If not, the sign of some characters may be flipped and the sorting by dimension will be wrong.

character_table_readable(multiplier=None, full=False)

Returns a conventional rendering of the character table.

Parameters:

• multiplier (Union[ndarray, Array, None]) – (optional) Schur multiplier

• full (bool) – whether the character table for all group elements (True) or one representative per conjugacy class (False, default)

Return type:

tuple[list[str], Union[ndarray, Array]]

Returns:

A tuple (classes, characters) where classes is a list of string labels for each conjugacy-class representative (or for each group element when full=True), and characters is a matrix whose rows list the characters of one irrep.

check_multiplier(multiplier, rtol=1e-08, atol=0)

Checks the associativity constraint of Schur multipliers.

\[\alpha(x, y) \alpha(xy, z) = \alpha(x, yz) \alpha(y, z).\]

Parameters:

• multiplier (Union[ndarray, Array]) – the array of Schur multipliers \(\alpha(x,y)\)

• rtol – relative tolerance

• atol – absolute tolerance

Return type:

bool

Returns:

whether multiplier is a valid Schur multiplier up to the given tolerance

Raises:

ValueError – if the shape of multiplier does not match the size of the group

irrep_matrices()

Returns matrices that realise all irreps of the group.

Return type:

list[Union[ndarray, Array]]

Returns:

A list of 3D arrays such that self.irrep_matrices()[i][g] contains the representation of self[g] consistent with the characters in self.character_table()[i].

is_subgroup(other, *, proper=False)

Return True if other is a subgroup of self.

For two TranslationGroup objects on the same lattice, uses a cheap stride-divisibility check: other is a subgroup of self iff every axis active in other is also active in self with a stride that divides other’s stride (i.e. ss % fs == 0).

Falls back to the generic element-containment check for non-TranslationGroup arguments.

Parameters:

proper (bool) – if True, also require |other| < |self|.

Return type:

bool

momentum_irrep(*k)

Returns the irrep characters (phase factors) corresponding to crystal momentum \(\vec k\).

\(\vec k\) must be given in Cartesian units. For a lattice with extent \(L\) along an axis, the valid momenta along that axis are \(k = 2\pi m / L\) for \(m = 0, 1, \ldots, L-1\).

Example (1D chain of length 4, momentum \(k = \pi/2\)):

tg.momentum_irrep(2 * np.pi * 1 / 4)   # m=1 -> k = 2π/4 = π/2

Return type:

ndarray

Parameters:

k (ndarray | Array)

projective_characters_by_class(multiplier)

Calculates the character table of projective representations with a given Schur multiplier α using a modified Burnside algorithm.

Parameters:

multiplier (Union[ndarray, Array, None]) – the unitary Schur multiplier. If unspecified, computes linear representation characters.

Return type:

tuple[ndarray, ndarray]

Returns:

If multiplier is None, returns characters_by_class, a 2D array whose rows contain the characters of one linear irrep evaluated on a representative of each conjugacy class. Otherwise it returns a tuple (characters_by_class, class_factors), where class_factors is a 1D array listing the per-element factors needed to recover the character of each element from the character of its class representative.

Note: the algorithm and the definitions above are explained in more detail in https://arxiv.org/abs/2505.14790.

remove_duplicates(*, return_inverse=False)

Returns a new PermutationGroup with duplicate elements (that is, elements which represent identical permutations) removed.

Parameters:

return_inverse – If True, also return indices to reconstruct the original group from the result.

Returns:

The permutation group with duplicate elements removed. If return_inverse==True, it also returns the indices needed to reconstruct the original group from the result.

replace(**updates)

Returns a new object replacing the specified fields with new values.

to_array()

Convert the abstract group operations to an array of permutation indices.

It returns a matrix where the i-th row contains the indices corresponding to the i-th group element. That is, self.to_array()[i, j] is \(g_i^{-1}(j)\). Moreover,

G = # this permutation group...
V = np.arange(G.degree)
assert np.all(G(V) == V[..., G.to_array()])

Return type:

Union[ndarray, Array]

Returns:

A matrix that can be used to index arrays in the computational basis in order to obtain their permutations.