Crystalline.jl

Tools for crystalline symmetry analysis implemented in the Julia language.

This package provides access to the symmetry operations of crystalline point groups, space groups, Wyckoff positions, their irreducible representations and band representations, as well as tools for their associated manipulation.

The package can be installed via Julia's package manager:

julia> using Pkg; Pkg.add("Crystalline")
julia> using Crystalline

Crystalline.jl provides several functionalities for line groups, plane groups, and space groups, as well as crystallographic point groups.

Example use includes:

# construct a 3D `SymOperation` from its triplet form
julia> S"x,-y,-z"
2₁₀₀ ─────────────────────────── (x,-y,-z)
 ┌ 1  0  0 ╷ 0 ┐
 │ 0 -1  0 ┆ 0 │
 └ 0  0 -1 ╵ 0 ┘

# load the `SymOperation`s of the 3D space group ⋕16 in a conventional setting
julia> sg = spacegroup(16, Val(3))
SpaceGroup{3} ⋕16 (P222) with 4 operations:
 1
 2₀₀₁
 2₀₁₀
 2₁₀₀

# load a dictionary of small irreps and their little groups for space group ⋕16,
# indexed by their k-point labels; then inspect the small irreps at the A point
julia> lgirs = lgirreps(16, Val(3))
julia> lgirs["A"]
2-element Collection{LGIrrep{3}} for ⋕16 (P222) at A = [α, 0, 1/2]:
A₁ ─┬─────────────────────────────────────────────
    ├─ 1: ──────────────────────────────── (x,y,z)
    │     1
    │
    ├─ 2₁₀₀: ─────────────────────────── (x,-y,-z)
    │     1
    └─────────────────────────────────────────────
A₂ ─┬─────────────────────────────────────────────
    ├─ 1: ──────────────────────────────── (x,y,z)
    │     1
    │
    ├─ 2₁₀₀: ─────────────────────────── (x,-y,-z)
    │     -1
    └─────────────────────────────────────────────

# construct the character table for the small irreps at the Γ point
julia> characters(lgirs["Γ"])
CharacterTable{3} for ⋕16 (P222) at Γ = [0, 0, 0]:
──────┬────────────────
      │ Γ₁  Γ₂  Γ₃  Γ₄ 
──────┼────────────────
    1 │  1   1   1   1 
 2₁₀₀ │  1  -1   1  -1
 2₀₁₀ │  1  -1  -1   1
 2₀₀₁ │  1   1  -1  -1
──────┴────────────────

Additional functionality includes e.g. point group operations (pointgroup) and irreps (pgirreps), elementary band representations (bandreps), Wyckoff positions (wyckoffs), conjugacy classes (classes), class-specific characters (classcharacters), group generators (generators), subperiodic groups (subperiodicgroup), 3D magnetic space groups (mspacegroup), and physically real irreps (realify). In addition, Bravais lattice utilities and conventions are accessible via the lightweight stand-alone sub-package Bravais.jl.

For a full description of the public API, see the documentation.

At present, the package's emphasis is on spinless systems (i.e., double groups and spinful irreps are not implemented).

Crystalline.jl is a research package in active development: breaking changes are likely (but will respect semantic versioning).

If you find this package useful in your reseach, please cite our paper:

• T. Christensen, H.C. Po, J.D. Joannopoulos, & M. Soljačić, Location and topology of the fundamental gap in photonic crystals, Phys. Rev. X 12, 021066 (2022).

In addition, please cite any earlier works explicitly referenced in the documentation of individual functions.