# Source code for netket.graph.common_lattices

```# Copyright 2021 The NetKet Authors - All rights reserved.
#
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#
# Unless required by applicable law or agreed to in writing, software
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and

from itertools import permutations
from typing import Union
from collections.abc import Sequence
import numpy as np

from .lattice import Lattice

from netket.utils.group import PointGroup, PGSymmetry, planar, cubic, Identity

def _perm_symm(perm: tuple) -> PGSymmetry:
n = len(perm)
M = np.zeros((n, n))
M[range(n), perm] = 1
return PGSymmetry(M)

def _axis_reflection(axis: int, ndim: int) -> PGSymmetry:
M = np.eye(ndim)
M[axis, axis] = -1
return PGSymmetry(M)

def _grid_point_group(
extent: Sequence[int], pbc: Sequence[bool], color_edges: bool
) -> PointGroup:
"""Point group of `Grid`, made up of axis permutations and flipping each axis."""
ndim = len(extent)
# Cannot exchange two axes if they are colored differently; otherwise,
# can only exchange them if they have the same kind of BC and length.
# Represent open BC by setting kind[i] = -extent[i], so just have to match these
if color_edges:
result = PointGroup([Identity()], ndim=ndim)
else:
axis_perm = []
axes = np.arange(ndim, dtype=int)
extent = np.asarray(extent, dtype=int)
kind = np.where(pbc, extent, -extent)
for perm in permutations(axes):
if np.all(kind == kind[list(perm)]):
if np.all(perm == axes):
axis_perm.append(Identity())
else:
axis_perm.append(_perm_symm(perm))
result = PointGroup(axis_perm, ndim=ndim)

# reflections across axes and setting the origin
# OBC axes are only symmetric w.r.t. their midpoint, (extent[i]-1)/2
origin = []
for i in range(ndim):
result = result @ PointGroup([Identity(), _axis_reflection(i, ndim)], ndim=ndim)
origin.append(0 if pbc[i] else (extent[i] - 1) / 2)

return result.change_origin(origin)

[docs]
def Grid(
extent: Sequence[int],
*,
pbc: Union[bool, Sequence[bool]] = True,
color_edges: bool = False,
**kwargs,
) -> Lattice:
"""
Constructs a hypercubic lattice given its extent in all dimensions.

Args:
extent: Size of the lattice along each dimension. It must be a list with
integer components >= 1.
pbc: If `True`, the grid will have periodic boundary conditions (PBC);
if `False`, the grid will have open boundary conditions (OBC).
This parameter can also be a list of booleans with same length as
the parameter `length`, in which case each dimension will have
PBC/OBC depending on the corresponding entry of `pbc`.
color_edges: generates nearest-neighbour edges colored according to direction
i.e. edges along Cartesian direction #i have color i
cannot be used with `max_neighbor_order` or `custom_edges`
kwargs: Additional keyword arguments are passed on to the constructor of
:ref:`netket.graph.Lattice`.

Examples:
Construct a 5x10 square lattice with periodic boundary conditions:

>>> import netket
>>> g=netket.graph.Grid(extent=[5, 10], pbc=True)
>>> print(g.n_nodes)
50

Construct a 2x2x3 cubic lattice with open boundary conditions:

>>> g=netket.graph.Grid(extent=[2,2,3], pbc=False)
>>> print(g.n_nodes)
12
"""
extent = np.asarray(extent, dtype=int)

ndim = len(extent)
if isinstance(pbc, bool):
pbc = [pbc] * ndim
if color_edges:
kwargs["custom_edges"] = [(0, 0, vec) for vec in np.eye(ndim)]
return Lattice(
basis_vectors=np.eye(ndim),
extent=extent,
pbc=pbc,
point_group=lambda: _grid_point_group(extent, pbc, color_edges),
**kwargs,
)

[docs]
def Hypercube(length: int, n_dim: int = 1, *, pbc: bool = True, **kwargs) -> Lattice:
r"""Constructs a hypercubic lattice with equal side length in all dimensions.
Periodic boundary conditions can also be imposed.

Args:
length: Side length of the hypercube; must always be >=1
n_dim: Dimension of the hypercube; must be at least 1.
pbc: Whether the hypercube should have periodic boundary conditions
(in all directions)
kwargs: Additional keyword arguments are passed on to the constructor of
:ref:`netket.graph.Lattice`.

Examples:
A 10x10x10 cubic lattice with periodic boundary conditions can be
constructed as follows:

>>> import netket
>>> g = netket.graph.Hypercube(10, n_dim=3, pbc=True)
>>> print(g.n_nodes)
1000
"""
if not isinstance(length, int) or length <= 0:
raise TypeError("Argument `length` must be a positive integer")
length_vector = [length] * n_dim
return Grid(length_vector, pbc=pbc, **kwargs)

[docs]
def Cube(length: int, *, pbc: bool = True, **kwargs) -> Lattice:
"""Constructs a cubic lattice of side `length`
Periodic boundary conditions can also be imposed

Args:
length: Side length of the cube; must always be >=1
pbc: Whether the cube should have periodic boundary conditions
(in all directions)
kwargs: Additional keyword arguments are passed on to the constructor of
:ref:`netket.graph.Lattice`.

Examples:
A 10Ã—10Ã—10 cubic lattice with periodic boundary conditions can be
constructed as follows:

>>> import netket
>>> g=netket.graph.Cube(10, pbc=True)
>>> print(g.n_nodes)
1000
"""
return Hypercube(length, pbc=pbc, n_dim=3, **kwargs)

[docs]
def Square(length: int, *, pbc: bool = True, **kwargs) -> Lattice:
"""Constructs a square lattice of side `length`
Periodic boundary conditions can also be imposed

Args:
length: Side length of the square; must always be >=1
pbc: Whether the square should have periodic boundary
conditions (in both directions)
kwargs: Additional keyword arguments are passed on to the constructor of
:ref:`netket.graph.Lattice`.

Examples:
A 10x10 square lattice with periodic boundary conditions can be
constructed as follows:

>>> import netket
>>> g=netket.graph.Square(10, pbc=True)
>>> print(g.n_nodes)
100
"""
return Hypercube(length, pbc=pbc, n_dim=2, **kwargs)

[docs]
def Chain(length: int, *, pbc: bool = True, **kwargs) -> Lattice:
r"""Constructs a chain of `length` sites.
Periodic boundary conditions can also be imposed

Args:
length: Length of the chain. It must always be >=1
pbc: Whether the chain should have periodic boundary conditions
kwargs: Additional keyword arguments are passed on to the constructor of
:ref:`netket.graph.Lattice`.

Examples:
A 10 site chain with periodic boundary conditions can be
constructed as follows:

>>> import netket
>>> g = netket.graph.Chain(10, pbc=True)
>>> print(g.n_nodes)
10
"""
return Hypercube(length, pbc=pbc, n_dim=1, **kwargs)

[docs]
def BCC(
extent: Sequence[int], *, pbc: Union[bool, Sequence[bool]] = True, **kwargs
) -> Lattice:
"""Constructs a BCC lattice of a given spatial extent.
Periodic boundary conditions can also be imposed
Sites are returned at the Bravais lattice points.

Arguments:
extent: Number of primitive unit cells along each direction, needs to be
an array of length 3
pbc: If `True`, the lattice will have periodic boundary conditions (PBC);
if `False`, the lattice will have open boundary conditions (OBC).
This parameter can also be a list of booleans with same length as
the parameter `length`, in which case each dimension will have
PBC/OBC depending on the corresponding entry of `pbc`.
kwargs: Additional keyword arguments are passed on to the constructor of
:ref:`netket.graph.Lattice`.

Example:
Construct a BCC lattice with 3Ã—3Ã—3 primitive unit cells:

>>> from netket.graph import BCC
>>> g = BCC(extent=[3,3,3])
>>> print(g.n_nodes)
27
"""
basis = [[-0.5, 0.5, 0.5], [0.5, -0.5, 0.5], [0.5, 0.5, -0.5]]
# determine if full point group is realised by the simulation box
point_group = cubic.Oh() if np.all(pbc) and len(set(extent)) == 1 else None
return Lattice(
basis_vectors=basis, extent=extent, pbc=pbc, point_group=point_group, **kwargs
)

[docs]
def FCC(
extent: Sequence[int], *, pbc: Union[bool, Sequence[bool]] = True, **kwargs
) -> Lattice:
"""Constructs an FCC lattice of a given spatial extent.
Periodic boundary conditions can also be imposed
Sites are returned at the Bravais lattice points.

Arguments:
extent: Number of primitive unit cells along each direction, needs
to be an array of length 3
pbc: If `True`, the lattice will have periodic boundary conditions (PBC);
if `False`, the lattice will have open boundary conditions (OBC).
This parameter can also be a list of booleans with same length as
the parameter `length`, in which case each dimension will have
PBC/OBC depending on the corresponding entry of `pbc`.
kwargs: Additional keyword arguments are passed on to the constructor of
:ref:`netket.graph.Lattice`.

Example:
Construct an FCC lattice with 3Ã—3Ã—3 primitive unit cells:

>>> from netket.graph import FCC
>>> g = FCC(extent=[3,3,3])
>>> print(g.n_nodes)
27
"""
basis = [[0, 0.5, 0.5], [0.5, 0, 0.5], [0.5, 0.5, 0]]
# determine if full point group is realised by the simulation box
point_group = cubic.Oh() if np.all(pbc) and len(set(extent)) == 1 else None
return Lattice(
basis_vectors=basis, extent=extent, pbc=pbc, point_group=point_group, **kwargs
)

[docs]
def Diamond(
extent: Sequence[int], *, pbc: Union[bool, Sequence[bool]] = True, **kwargs
) -> Lattice:
"""Constructs a diamond lattice of a given spatial extent.
Periodic boundary conditions can also be imposed.

Sites are returned at the 8a Wyckoff positions of the FCC lattice
([000], [1/4,1/4,1/4], and translations thereof).

Arguments:
extent: Number of primitive unit cells along each direction, needs to
be an array of length 3
pbc: If `True`, the lattice will have periodic boundary conditions (PBC);
if `False`, the lattice will have open boundary conditions (OBC).
This parameter can also be a list of booleans with same length as
the parameter `length`, in which case each dimension will have
PBC/OBC depending on the corresponding entry of `pbc`.
kwargs: Additional keyword arguments are passed on to the constructor of
:ref:`netket.graph.Lattice`.

Example:
Construct a diamond lattice with 3Ã—3Ã—3 primitive unit cells:

>>> from netket.graph import Diamond
>>> g = Diamond(extent=[3,3,3])
>>> print(g.n_nodes)
54
"""
basis = [[0, 0.5, 0.5], [0.5, 0, 0.5], [0.5, 0.5, 0]]
sites = [[0, 0, 0], [0.25, 0.25, 0.25]]
# determine if full point group is realised by the simulation box
point_group = cubic.Fd3m() if np.all(pbc) and len(set(extent)) == 1 else None
return Lattice(
basis_vectors=basis,
site_offsets=sites,
extent=extent,
pbc=pbc,
point_group=point_group,
**kwargs,
)

[docs]
def Pyrochlore(
extent: Sequence[int], *, pbc: Union[bool, Sequence[bool]] = True, **kwargs
) -> Lattice:
"""Constructs a pyrochlore lattice of a given spatial extent.
Periodic boundary conditions can also be imposed.

Sites are returned at the 16c Wyckoff positions of the FCC lattice
([111]/8, [1 -1 -1]/8, [-1 1 -1]/8, [-1 -1 1]/8, and translations thereof).

Arguments:
extent: Number of primitive unit cells along each direction, needs to be
an array of length 3
pbc: If `True`, the lattice will have periodic boundary conditions (PBC);
if `False`, the lattice will have open boundary conditions (OBC).
This parameter can also be a list of booleans with same length as
the parameter `length`, in which case each dimension will have
PBC/OBC depending on the corresponding entry of `pbc`.
kwargs: Additional keyword arguments are passed on to the constructor of
:ref:`netket.graph.Lattice`.

Example:
Construct a pyrochlore lattice with 3Ã—3Ã—3 primitive unit cells:

>>> from netket.graph import Pyrochlore
>>> g = Pyrochlore(extent=[3,3,3])
>>> print(g.n_nodes)
108
"""
basis = [[0, 0.5, 0.5], [0.5, 0, 0.5], [0.5, 0.5, 0]]
sites = np.array([[1, 1, 1], [1, 3, 3], [3, 1, 3], [3, 3, 1]]) / 8
# determine if full point group is realised by the simulation box
point_group = cubic.Fd3m() if np.all(pbc) and len(set(extent)) == 1 else None
return Lattice(
basis_vectors=basis,
site_offsets=sites,
extent=extent,
pbc=pbc,
point_group=point_group,
**kwargs,
)

def _hexagonal_general(
extent, *, site_offsets=None, pbc: Union[bool, Sequence[bool]] = True, **kwargs
) -> Lattice:
basis = [[1, 0], [0.5, 0.75**0.5]]
# determine if full point group is realised by the simulation box
point_group = planar.D(6) if np.all(pbc) and extent[0] == extent[1] else None
return Lattice(
basis_vectors=basis,
extent=extent,
site_offsets=site_offsets,
pbc=pbc,
point_group=point_group,
**kwargs,
)

[docs]
def Triangular(extent, *, pbc: Union[bool, Sequence[bool]] = True, **kwargs) -> Lattice:
r"""Constructs a triangular lattice of a given spatial extent.
Periodic boundary conditions can also be imposed
Sites are returned at the Bravais lattice points.

Arguments:
extent: Number of unit cells along each direction, needs to be an array
of length 2
pbc: If `True`, the lattice will have periodic boundary conditions (PBC);
if `False`, the lattice will have open boundary conditions (OBC).
This parameter can also be a list of booleans with same length as
the parameter `length`, in which case each dimension will have
PBC/OBC depending on the corresponding entry of `pbc`.
kwargs: Additional keyword arguments are passed on to the constructor of
:ref:`netket.graph.Lattice`.

Example:
Construct a triangular lattice with 3 Ã— 3 unit cells:

>>> from netket.graph import Triangular
>>> g = Triangular(extent=[3, 3])
>>> print(g.n_nodes)
9
"""
return _hexagonal_general(extent, site_offsets=None, pbc=pbc, **kwargs)

[docs]
def Honeycomb(extent, *, pbc: Union[bool, Sequence[bool]] = True, **kwargs) -> Lattice:
r"""Constructs a honeycomb lattice of a given spatial extent.
Periodic boundary conditions can also be imposed.
Sites are returned at the 2b Wyckoff positions.

Arguments:
extent: Number of unit cells along each direction, needs to be an array
of length 2
pbc: If `True`, the lattice will have periodic boundary conditions (PBC);
if `False`, the lattice will have open boundary conditions (OBC).
This parameter can also be a list of booleans with same length as
the parameter `length`, in which case each dimension will have
PBC/OBC depending on the corresponding entry of `pbc`.
kwargs: Additional keyword arguments are passed on to the constructor of
:ref:`netket.graph.Lattice`.

Example:
Construct a honeycomb lattice with 3 Ã— 3 unit cells:

>>> from netket.graph import Honeycomb
>>> g = Honeycomb(extent=[3, 3])
>>> print(g.n_nodes)
18
"""
return _hexagonal_general(
extent,
site_offsets=[[0.5, 0.5 / 3**0.5], [1, 1 / 3**0.5]],
pbc=pbc,
**kwargs,
)

[docs]
def Kagome(extent, *, pbc: Union[bool, Sequence[bool]] = True, **kwargs) -> Lattice:
r"""Constructs a kagome lattice of a given spatial extent.
Periodic boundary conditions can also be imposed.
Sites are returned at the 3c Wyckoff positions.

Arguments:
extent: Number of unit cells along each direction, needs to be an array
of length 2
pbc: If `True`, the lattice will have periodic boundary conditions (PBC);
if `False`, the lattice will have open boundary conditions (OBC).
This parameter can also be a list of booleans with same length as
the parameter `length`, in which case each dimension will have
PBC/OBC depending on the corresponding entry of `pbc`.
kwargs: Additional keyword arguments are passed on to the constructor of
:ref:`netket.graph.Lattice`.

Example:
Construct a kagome lattice with 3 Ã— 3 unit cells:

>>> from netket.graph import Kagome
>>> g = Kagome(extent=[3, 3])
>>> print(g.n_nodes)
27
"""
return _hexagonal_general(
extent,
site_offsets=[[0.5, 0], [0.25, 0.75**0.5 / 2], [0.75, 0.75**0.5 / 2]],
pbc=pbc,
**kwargs,
)

[docs]
def KitaevHoneycomb(
extent, *, pbc: Union[bool, Sequence[bool]] = True, **kwargs
) -> Lattice:
r"""Constructs a honeycomb lattice of a given spatial extent.

Nearest-neighbour edges are coloured according to direction
(cf. Kitaev, https://doi.org/10.1016/j.aop.2005.10.005).
Periodic boundary conditions can also be imposed.
Sites are returned at the 2b Wyckoff positions.

Arguments:
extent: Number of unit cells along each direction, needs to be an array
of length 2
pbc: If `True`, the lattice will have periodic boundary conditions (PBC);
if `False`, the lattice will have open boundary conditions (OBC).
This parameter can also be a list of booleans with same length as
the parameter `length`, in which case each dimension will have
PBC/OBC depending on the corresponding entry of `pbc`.
kwargs: Additional keyword arguments are passed on to the constructor of
:ref:`netket.graph.Lattice`.

Example:
Construct a Kitaev honeycomb lattice with 3 Ã— 3 unit cells:

>>> from netket.graph import Honeycomb
>>> g = KitaevHoneycomb(extent=[3, 3])
>>> print(g.n_nodes)
18
>>> print(len(g.edges(filter_color=2)))
9
"""
return Lattice(
basis_vectors=[[1, 0], [0.5, 0.75**0.5]],
extent=extent,
site_offsets=[[0.5, 0.5 / 3**0.5], [1, 1 / 3**0.5]],
pbc=pbc,
point_group=planar.C(2) if np.all(pbc) else None,
custom_edges=[
(0, 1, [0.5, 0.5 / 3**0.5]),
(0, 1, [-0.5, 0.5 / 3**0.5]),
(0, 1, [0, -1 / 3**0.5]),
],
**kwargs,
)

```