Square Lattice Cluster Expansion

This example computes the linked cluster expansion on the square lattice up to order 3, this utilizes the corner-sharing square expansion

Setup

Define the unit cell with basis positions, primitive vectors, and nearest-neighbour bonds:

using Lincege

square_cluster_basis = [[[-1 / 2, -1 / 2], [-1 / 2, 1 / 2], [1 / 2, -1 / 2], [1 / 2, 1 / 2]]]
square_cluster_pvecs = [[1.0, 1.0], [1.0, -1.0]]
square_cluster_lbonds = [
        ExpansionBond([1, 1], [1, 2], [0, 0], 1),
        ExpansionBond([1, 1], [1, 3], [0, 0], 1),
        ExpansionBond([1, 2], [1, 4], [0, 0], 1),
        ExpansionBond([1, 3], [1, 4], [0, 0], 1),
]
square_cluster_ebonds = [
        Bond(1, 1, [1, 0], 1),
        Bond(1, 1, [0, 1], 1),
]
square_cluster_uc = ExpansionUnitCell(
        square_cluster_basis,
        square_cluster_pvecs,
        square_cluster_lbonds,
        square_cluster_ebonds,
        [[1, 1, 1, 1]]
)

ExpansionUnitCell([[-0.5 -0.5 0.5 0.5; -0.5 0.5 -0.5 0.5]], [1.0 1.0; 1.0 -1.0], [[1, 1, 1, 1]], [[1, 2, 2, 1]], ExpansionBond[ExpansionBond([1, 1], [1, 2], [0, 0], 1), ExpansionBond([1, 1], [1, 3], [0, 0], 1), ExpansionBond([1, 2], [1, 4], [0, 0], 1), ExpansionBond([1, 3], [1, 4], [0, 0], 1)], Bond[Bond(1, 1, [1, 0], 1), Bond(1, 1, [0, 1], 1)])

Building the lattice and clusters

Note that we use a WeakClusterExpansionLattice because the expansion shares sites between different expansion units (the corners of the squares).

m_order = 3
lattice = WeakClusterExpansionLattice(m_order, square_cluster_uc)

trans_clusters = TranslationClusterSet(lattice)
clusters_from_lattice!(trans_clusters, lattice)

iso_clusters = IsomorphicClusterSet(lattice)
clusters_from_clusters!(iso_clusters, trans_clusters)

ClusterSet{IsomorphicHasher} with 4 clusters

Computing the expansion

expansion = Expansion(iso_clusters, lattice, m_order)
summation!(expansion, m_order)
expansion.weights

5×4 Matrix{Float64}:
 0.0   0.5  -2.0   3.0
 0.0   0.0   1.0  -6.0
 0.0   0.0   0.0   2.0
 0.0   0.0   0.0   1.0
 1.0  -2.0   1.0   0.0

Writing to JSON

write_to_json(expansion, lattice, iso_clusters, "pyrochlore_lattice_uc_exp.json")