Lattice2D.jl

AbstractBoundaryCondition

格子の境界条件の抽象型。

• PBC: 周期境界条件
• OBC: 開境界条件
• SSD: Sine-Square Deformation 境界条件

AbstractLattice{D}

abstract type for lattices in D dimensions.

AbstractLatticeConnection

abstract type for lattice connections (edges, bonds).

AbstractTopology

Abstract type for lattice topologies.

Bond

struct for a bond (edge) in the lattice.

• src::Int: start site index
• dst::Int: destination site index
• type::Int: type of the bond for categorization
• vector::Vector{Float64}: expresses the bond vector from src to dst

Connection

Connection rules within or between unit cells.

• src_sub: sublattice index of the start point (1, 2, ...)
• dst_sub: sublattice index of the end point
• dx, dy: relative cell position of the end point (0,0 means within the same unit cell)
• type: type of the connection

Dice <: AbstractTopology{2}

struct which represents Dice lattice (T3 lattice) Bipartite lattice with coordination numbers 6 (Hub) and 3 (Rim).

Honeycomb <: AbstractTopology{2}

struct which represents Honeycomb lattice

Kagome <: AbstractTopology{2}

struct which represents Kagome lattice

Lattice{Topology<:AbstractTopology, T, B<:AbstractBoundaryCondition, I<:AbstractIndexing} It mainly represents 2-dimiensional lattice, but it can be used as 1-dimensional lattice as well.

• Lx::Int: x direction lattice size
• Ly::Int: y direction lattice size
• N::Int: total number of sites
• positions::Vector{Vector{T}}: position vectors of each site
• nearest_neighbors::Vector{Vector{Int}}: nearest neighbor indices for each site
• bonds::Vector{Bond}: list of bonds (edges) in the lattice
• basis_vectors::Vector{Vector{T}}: lattice basis vectors
• reciprocal_vectors::Union{Vector{Vector{T}}, Nothing}: reciprocal lattice vectors
• sublattice_ids::Vector{Int}: sublattice IDs of each site
• is_bipartite::Bool: whether the lattice is bipartite
• site_map::Union{Matrix{Int}, Nothing}: mapping of site indices on the lattice
• translation_x::Vector{Int}: x direction translation vector
• translation_y::Vector{Int}: y direction translation vector
• boundary::B: boundary condition
• index_method::I: indexing method

Lieb <: AbstractTopology{2}

struct which represents Lieb lattice

ShastrySutherland <: AbstractTopology{2}

struct which represents Shastry-Sutherland lattice

Square <: AbstractTopology{2}

struct which represents Square lattice

Triangular <: AbstractTopology{2}

struct which represents Triangular lattice

UnionJack <: AbstractTopology{2}

struct which represents Union Jack (Centered Square) lattice.

UnitCell{D, T}

Geometric definition data of the lattice. Basically, the lattice is constructed based on this information. The get_unit_cell(::Type{T}) function retrieves the unit cell data corresponding to each topology.

_coord_to_index(lat::Lattice, x::Int, y::Int, s::Int)

this function converts lattice coordinates to a linear index based on the lattice's indexing method.

• ColMajorIndexing
• RowMajorIndexing
• SnakeIndexing

are available indexing methods.

build_lattice(Topology::Type{<:AbstractTopology}, Lx::Int, Ly::Int; boundary::AbstractBoundaryCondition=PBC())

Construct a lattice with the specified topology, size, and boundary conditions. this function is available if unitcell information is defined.

calc_reciprocal_vectors(basis)

Calculate reciprocal lattice vectors from the given basis vectors.

check_bipartite_bfs(N, neighbors)

check whether the given lattice is bipartite using BFS.

get_coordinates(lat::Lattice, idx::Int)

Get the lattice coordinates (x, y, sublattice) from a linear site index.

Arguments

• lat::Lattice: The lattice structure
• idx::Int: The linear site index

Returns

• Tuple{Int, Int, Int}: A tuple (x, y, s) where x, y are unit cell coordinates and s is sublattice index

Examples

lat = build_lattice(Honeycomb, 4, 4)
x, y, s = get_coordinates(lat, 10)

get_position(lat::Lattice, idx::Int)

Get the spatial position (coordinates) of a site given its linear index.

Arguments

• lat::Lattice: The lattice structure
• idx::Int: The linear site index

Returns

• Vector{Float64}: The position vector [x, y] in real space

Examples

lat = build_lattice(Square, 4, 4)
pos = get_position(lat, 5)  # Get position of site 5

get_site_index(lat::Lattice, x::Int, y::Int, s::Int=1)

Get the linear site index from lattice coordinates (x, y) and sublattice index s. For single-sublattice lattices, s defaults to 1.

Arguments

• lat::Lattice: The lattice structure
• x::Int: x-coordinate (1 to Lx)
• y::Int: y-coordinate (1 to Ly)
• s::Int: sublattice index (1 to number of sublattices), defaults to 1

Returns

• Int: The linear site index

Examples

lat = build_lattice(Square, 4, 4)
idx = get_site_index(lat, 2, 3)  # Get index at (2,3)

lat = build_lattice(Honeycomb, 4, 4)
idx_A = get_site_index(lat, 1, 1, 1)  # Get sublattice A at (1,1)
idx_B = get_site_index(lat, 1, 1, 2)  # Get sublattice B at (1,1)

get_unit_cell(::Type{T}) where T <: AbstractTopology

this function returns the UnitCell associated with the given Topology type. If the Topology type is not recognized, it throws an error.