Pyrochlore Lattice Unit Cell Expansion

This example computes the linked cluster expansion on the pyrochlore lattice up to order 3, this utilizes the unit cell expansion on the lattice.

Setup

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

using Lincege

pyro_exp_uc_basis = [
        [
                [1 / 2, 1 / 2, 1 / 2],
                [1 / 2, -1 / 2, -1 / 2],
                [-1 / 2, 1 / 2, -1 / 2],
                [-1 / 2, -1 / 2, 1 / 2],
        ],
]
pyro_exp_uc_pvecs = [[2.0, 2.0, 0.0], [2.0, 0.0, 2.0], [0.0, 2.0, 2.0]]
pyro_exp_uc_lbonds = [
        # Inside a Unit Cell
        ExpansionBond([1, 1], [1, 2], [0, 0, 0], 1),
        ExpansionBond([1, 1], [1, 3], [0, 0, 0], 1),
        ExpansionBond([1, 1], [1, 4], [0, 0, 0], 1),
        ExpansionBond([1, 2], [1, 3], [0, 0, 0], 1),
        ExpansionBond([1, 2], [1, 4], [0, 0, 0], 1),
        ExpansionBond([1, 3], [1, 4], [0, 0, 0], 1),
        # Between Unit Cells
        ExpansionBond([1, 1], [1, 2], [0, 0, 1], 1),
        ExpansionBond([1, 1], [1, 3], [0, 1, 0], 1),
        ExpansionBond([1, 1], [1, 4], [1, 0, 0], 1),
        ExpansionBond([1, 2], [1, 3], [0, 1, -1], 1),
        ExpansionBond([1, 2], [1, 4], [1, 0, -1], 1),
        ExpansionBond([1, 3], [1, 4], [1, -1, 0], 1),
]
pyro_exp_uc_ebonds = [
        Bond(1, 1, [1, 0, 0], 1),
        Bond(1, 1, [0, 1, 0], 1),
        Bond(1, 1, [0, 0, 1], 1),
        Bond(1, 1, [0, 1, -1], 1),
        Bond(1, 1, [1, 0, -1], 1),
        Bond(1, 1, [1, -1, 0], 1),
]
pyro_exp_uc_uc = ExpansionUnitCell(
        pyro_exp_uc_basis,
        pyro_exp_uc_pvecs,
        pyro_exp_uc_lbonds,
        pyro_exp_uc_ebonds,
        [[1, 1, 1, 1]]
)

ExpansionUnitCell([[0.5 0.5 -0.5 -0.5; 0.5 -0.5 0.5 -0.5; 0.5 -0.5 -0.5 0.5]], [2.0 2.0 0.0; 2.0 0.0 2.0; 0.0 2.0 2.0], [[1, 1, 1, 1]], [[1, 2, 3, 4]], ExpansionBond[ExpansionBond([1, 1], [1, 2], [0, 0, 0], 1), ExpansionBond([1, 1], [1, 3], [0, 0, 0], 1), ExpansionBond([1, 1], [1, 4], [0, 0, 0], 1), ExpansionBond([1, 2], [1, 3], [0, 0, 0], 1), ExpansionBond([1, 2], [1, 4], [0, 0, 0], 1), ExpansionBond([1, 3], [1, 4], [0, 0, 0], 1), ExpansionBond([1, 1], [1, 2], [0, 0, 1], 1), ExpansionBond([1, 1], [1, 3], [0, 1, 0], 1), ExpansionBond([1, 1], [1, 4], [1, 0, 0], 1), ExpansionBond([1, 2], [1, 3], [0, 1, -1], 1), ExpansionBond([1, 2], [1, 4], [1, 0, -1], 1), ExpansionBond([1, 3], [1, 4], [1, -1, 0], 1)], Bond[Bond(1, 1, [1, 0, 0], 1), Bond(1, 1, [0, 1, 0], 1), Bond(1, 1, [0, 0, 1], 1), Bond(1, 1, [0, 1, -1], 1), Bond(1, 1, [1, 0, -1], 1), Bond(1, 1, [1, -1, 0], 1)])

Building the lattice and clusters

Note that we use a StrongClusterExpansionLattice because the expansion does not share sites between different expansion units. If we were to do the tetrahedra expansion, we would use a WeakClusterExpansionLattice instead.

m_order = 3
lattice = StrongClusterExpansionLattice(m_order, pyro_exp_uc_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 5 clusters

Computing the expansion

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

6×4 Matrix{Float64}:
 0.0   0.25  -3.0   16.5
 0.0   0.0    1.5  -27.0
 0.0   0.0    0.0   10.5
 0.0   0.0    0.0    1.0
 0.0   0.0    0.0    1.0
 1.0  -1.0    0.0    0.0

Writing to JSON

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