Quasiperiodic order

Most of condensed-matter physics takes periodicity for granted. Crystals are invariant under a Bravais lattice of translations, and almost every technique — Bloch theorem, band structure, reciprocal lattices — is a consequence. But in 1982 Shechtman's diffraction measurements of a rapidly cooled Al–Mn alloy showed a pattern with sharp Bragg peaks and 10-fold rotational symmetry, which is forbidden for any 2D or 3D crystallographic space group. The discovery of quasicrystals (Nobel Prize in Chemistry, 2011) forced physics to accept a structural regime that is neither periodic nor disordered: long-range order without translation symmetry.

LatticeCore is designed to host these structures on the same footing as Bravais crystals. This column is the physics story; the construction and Fourier analysis algorithms live in QuasiCrystal.jl and are exposed here through the type skeletons HyperReciprocalLattice, AcceptanceWindow, and BraggPeakSet.

What "quasiperiodic" means

A point set $\mathcal{L} \subset \mathbb{R}^d$ is quasiperiodic if its autocorrelation function — equivalently, its diffraction pattern — is a pure point measure supported on a dense countable set:

\[\mathcal{F}[\rho](\mathbf{k}) = \sum_{\mathbf{k}_n \in M^\ast} I(\mathbf{k}_n)\, \delta(\mathbf{k} - \mathbf{k}_n).\]

Periodic crystals are a special case where the support $M^\ast$ is a discrete Bravais reciprocal lattice. For a quasicrystal, $M^\ast$ is still countable but dense in $\mathbb{R}^d$: you can find a Bragg peak arbitrarily close to any $\mathbf{k}$, yet only countably many peaks carry non-zero intensity. The resulting diffraction pattern is sharp but inexhaustible.

The intensities $I(\mathbf{k}_n)$ are not uniform. They fall off with a form factor, leaving a finite number of bright peaks at any intensity threshold — which is exactly the finite set LatticeCore stores in a BraggPeakSet.

Cut and project

The cleanest way to generate a quasicrystal is cut-and-project. The recipe:

Pick a host periodic lattice in a higher dimension $D_{\text{hyper}}$. For the Fibonacci chain, $D_{\text{hyper}} = 2$; for the Penrose tiling, $D_{\text{hyper}} = 5$; for the Ammann–Beenker tiling, $D_{\text{hyper}} = 4$.
Choose a physical subspace $E_\parallel$ of dimension $D_{\text{phys}}$ inside $\mathbb{R}^{D_{\text{hyper}}}$. The embedding must be irrational — that is, $E_\parallel$ is not spanned by any rational basis of the hyper lattice — so that its parallel projection lands densely in $E_\parallel$ without ever repeating.
Choose a perpendicular subspace $E_\perp$ of dimension $D_{\text{hyper}} - D_{\text{phys}}$, orthogonal to $E_\parallel$.
Choose an acceptance window $W \subset E_\perp$ — a bounded region that acts as a stencil.
Project every hyper lattice point $\mathbf{n} \in \mathbb{Z}^{D_{\text{hyper}}}$ whose perpendicular projection sits inside $W$ onto the physical subspace. The resulting set of physical-space points

\[ \mathcal{L} = \{\, \pi_\parallel(\mathbf{n}) : \mathbf{n} \in \mathbb{Z}^{D_{\text{hyper}}}, \pi_\perp(\mathbf{n}) \in W \,\}\]

is the quasicrystal.

The acceptance window is where all the structure hides. A rectangular window in 2D gives you a Fibonacci chain. Five pentagonal windows arranged with 5-fold symmetry (one per perpendicular-space slab of the Penrose lifting) give you the Penrose P3 tiling. Octagonal windows give Ammann–Beenker.

LatticeCore captures the three ingredients as type skeletons:

abstract type AcceptanceWindow end

struct HyperReciprocalLattice{DPhys, DHyper, DPerp, T, W <: AcceptanceWindow}
    hyper_basis::SMatrix{DHyper, DHyper, T}
    parallel_proj::SMatrix{DPhys, DHyper, T}
    perp_proj::SMatrix{DPerp, DHyper, T}
    window::W
end

Concrete window types (PolygonalWindow, IntervalWindow, ...) and the construction of projection matrices live in QuasiCrystal.jl. The shape lives here because it is the contract every downstream consumer agrees on.

Fibonacci in full

The Fibonacci chain is the simplest non-trivial example, small enough to do by hand:

Hyper lattice: $\mathbb{Z}^2$.
Physical subspace: the line with slope $1/\varphi$ where $\varphi = (1 + \sqrt{5})/2$ is the golden ratio.
Perpendicular subspace: the orthogonal line with slope $-\varphi$.
Acceptance window: an interval of length $\sqrt{1 + \varphi^2}$ on the perpendicular line.

Project every integer point $(m, n) \in \mathbb{Z}^2$ whose perpendicular projection lies in the window, and you get a sequence of "long" ($L$) and "short" ($S$) intervals on the physical line. Asymptotically the ratio $|L|/|S| = \varphi$, and the substitution rule $L \to LS$, $S \to L$ iterated from a single $L$ generates the same sequence.

The substitution matrix

\[M = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}\]

has eigenvalues $\varphi$ and $-1/\varphi$. The leading eigenvalue $\varphi$ is the inflation factor: every substitution step stretches the sequence by $\varphi$. This simple fact — an algebraic integer lurking in a hyperbolic map — is what makes Fibonacci quasicrystals so tractable analytically.

The Fourier module and Bragg peaks

Because a quasicrystal inherits its translation symmetry from the host periodic lattice, its Fourier transform is supported on the projected reciprocal lattice of the host:

\[M^\ast = \pi_\parallel (\mathbb{Z}^{D_{\text{hyper}}})^\ast.\]

This projected set is countable (it is the image of a countable set) and, crucially, dense in $\mathbb{R}^{D_{\text{phys}}}$ — because the projection is irrational. The set $M^\ast$ is the Fourier module of the quasicrystal. Unlike a Bravais reciprocal lattice, $M^\ast$ is not discrete.

The intensity carried by each projected peak is the Fourier transform of the acceptance window evaluated in the perpendicular space:

\[I(\mathbf{k}_n) \propto \bigl|\, \hat{W}(\pi_\perp(\mathbf{g}_n))\, \bigr|^2,\]

where $\mathbf{g}_n \in (\mathbb{Z}^{D_{\text{hyper}}})^\ast$ is the higher-dimensional reciprocal lattice vector that $\mathbf{k}_n = \pi_\parallel(\mathbf{g}_n)$ projects from. For an interval window of width $a$, $\hat{W}(q) \propto \mathrm{sinc}(qa/2)$; for a polygonal window, the window Fourier transform is a sum of sinc factors coming from the polygon's vertices.

The practical consequence is that only a finite number of peaks exceed any given intensity threshold. Pick a cutoff, list those peaks, and you have a BraggPeakSet:

struct BraggPeakSet{DPhys, DHyper, T} <: AbstractMomentumLattice{DPhys, T}
    peaks::Vector{SVector{DPhys, T}}
    intensities::Vector{T}
    hyper_indices::Vector{NTuple{DHyper, Int}}
end

This is a subtype of AbstractMomentumLattice, so the same structure_factor code paths that work on a PeriodicMomentumLattice work on a quasicrystal. In the words of the 05 momentum-space chapter, a Bragg peak set is the quasicrystal's answer to a Monkhorst–Pack mesh.

Why this matters for LatticeCore

You could, in principle, stick a quasicrystal generator on its own hierarchy and never share interfaces with periodic lattices. That is what most of the existing quasicrystal literature does. The argument for the LatticeCore approach — one AbstractLattice hierarchy, one AbstractMomentumLattice hierarchy, one structure factor — is that you want to run the same Monte Carlo on both. A diluted antiferromagnet on a Penrose tiling wants the same observer stack as the same model on a square lattice: the trait machinery picks the right reciprocal-space object through momentum_lattice, the structure factor iterates over it with exactly the same code, and the physics stays comparable.

The cut-and-project recipe is how the hierarchy is populated; the interface is what makes the comparison cheap.