Cut-and-Project Density Functional Theory for Quasicrystals

Abstract

Cut-and-project from a symmetric structure in a higher-dimensional space is a standard method for describing the structure of a large class of quasicrystals. By means of a novel localization procedure, we now show how local physical interactions within these quasicrystals are also accurately described by cut-and-project, from corresponding physical interactions in the higher-dimensional space. A density functional theory (DFT++) formulation allows the cut-and-project method to handle the Schroedinger equation for interactions in quasicrystals. The theory is both rigorous and computationally tractable. The resulting ab initio approach specifies quasicrystalline quantum states, in contrast to previous approaches which only worked with crystalline approximants of the quasi-periodic structures.

Figures (2)

• Figure 1: The Fibonacci QC in both physical space (PS) and higher space (HS). (a) Construction of the Fibonacci QC in the PS with atoms spaced by either $R$ or $S$, generated by a string substitution rule. The starting seed string is $S$, and the substitutions are $S\rightarrow SR$ and $R\rightarrow S$. In this example $S$ and $R$ may denote arbitrary spatial or typological information about the material. In the lower figures, (b) illustrates the toric unit cell (as a unit square, or as the union of $a^2$ and $b^2$), while (c) illustrates the translationally invariant HS associated with this description. The red squares have side $a=1/\sqrt{1+\tau^2}$ for the $R$ step, while the blue squares have side $b = \tau / \sqrt{1+\tau^2}$ for the $S$ step. In (b), the cut line $\mathbb L$ of slope $1/\tau=a/b$ wraps infinitely often around the torus without meeting itself. If atoms are placed at distances $a$ and $b$ on $\mathbb L$ according to (a), they will not fill up the square densely, but do densely fill the atomic torus arcs pictured as the two line segments in the square. By taking the closure of these atomic positions, the C+P construction is completed atoms-as-linesatoms-as-lines-new. The unit cell is tiled out to the whole HS plane in (c). Here, the extended line $\mathbb L$ now copies the PS in the plane. The color-coded atoms appear in (a), (b), and (c).
• Figure 2: Potentials for the HS of the Fibonacci QC and its approximants. (a) The potential corresponding to the full Fibonacci QC. The abrupt discontinuities at each end of the atomic arcs radiate out as finer and finer discontinuities along the dashed line $\mathbb L$ in either direction. (b) The potential corresponding to the approximant with slope $13/21$ possessing $13+21=34$ atoms in the crystalline unit cell. (c) The potential corresponding to the approximant with slope $3/5$ containing $3+5=8$ atoms in the crystalline unit cell.