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Custom lattices

To define a custom lattice, we need to construct a custom UnitCell. A unit cell consists of 3 elements :

  • vectors : vectors that define how the unite cell translates in space
  • sites : list of site position in the unit cell
  • edges : list of edges between the unit cell and the neighboring unit cells

For example, the unit cell of the square lattice is :

const SQUARE_CELL = UnitCell(
    ((1.0,0.0), (0.0,1.0)),
    [(0.0,0.0)],
    [(1, (0,1), 1),
     (1, (1,0), 1)]
)

where (1, (0,1), 1) means that the first site of the unit cell is connected to the first site of the neighboring unit cell in the direction (0,1). A cell can contain multiple sites, and the edges can connect any site to any site in the neighboring unit cell. For example, the unit cell of the hexagonal lattice is :

const HEXAGONAL_CELL = UnitCell(
    ((1.0,0.0), (0.5,sqrt(3)/2)),
    [(0.0,0.0),
     (0.5, sqrt(3)/6)],
    [
     (1, (0,0), 2),
     (1, (-1,0), 2),
     (1, (0,-1), 2),
    ]
)

(1, (0,0), 2) means that the first site of the unit cell is connected to the second site of the same unit cell, (1, (-1,0), 2) means that the first site of the unit cell is connected to the second site of the neighboring unit cell in the direction (-1,0) and (1, (0,-1), 2) means that the first site of the unit cell is connected to the second site of the neighboring unit cell in the direction (0,-1).

Lets define a custom lattice: a square lattice with an extra site in the middle of the unit cell, connected to one corner of the unit cell.

using CairoMakie
using LatticeMaker
const CUSTOM_CELL = UnitCell(
    ((1.0,0.0), (0.0,1.0)),
    [(0.0,0.0)
     (0.5,0.5)  ], # extra site in the middle of the unit cell
    [(1, (0,1), 1),
     (1, (1,0), 1),
     (2, (0,0), 1)] # extra site connected to the first site of the unit cell]
)
L = Lattice(CUSTOM_CELL, (3, 4); periodic=(true, true))
plot_lattice(L)
Example block output