Now you are in the subtree of Energetic stability of quasicrystals project.

# Quasicrystals from Hume-Rothery-like rules

Our goal is to explore if it is possible to obtain stable 2D and 3D quasicrystals from a system that contains two components: ions and electrons, by tuning their relative concentrations.

The key observations:

1. The pair-wise interaction between atoms favors a certain distance between them (characteristic ordering wave vector $q_0$). Physically, it comes from balancing the reduction of electron kinetic energy (due to delocalization) and increase of interionic Coulomb repulsion as atom approach each other.

2. Electrons mediate higher-order interactions between ions. These interactions are important for selecting among various (quasi-) crystalline ordered states.

We will take the parwise interaction between ions as an input, but will calculate the 3rd and 4th order in density interactions by integrating out electrons.

We will explore the stability of icosahedral and dodecahedral QC. They will be defined as the 3D QC whose density contains harmonics that are equal in magnitude and have wavevectors corresponding to the vectors connecting the center of the corresponding polyhedron to its vertices (note that in Katz' review, they use a different definition for icosahedral QC -- wavevectors correspond to edges of icosahedron; experimentally though, it seems like our definition is more relevant).

According to Mermin and Troian, the icosahedral QC can be stabilized if the quartic term in the effective ionic density energy

$\rho(q_1)\rho(q_2)\rho(q_3)\rho(q_4)\delta(q_1+q_2+q_3+q_4)$

there is a dominant negative term of the form

$\rho(q_1)\rho(q_2)\rho(-q_2)\rho(-q_1)$,

such that the angle between $q_1$ and $q_2$ is the angle between vectors pointing from the center of icosahedron to the neighboring vertices (Figure icosa.png)

Instead of postulating such unusual term in free energy, we would like to see if it can emerge naturally from interaction between electrons and ions.

Assuming that ions and electrons interact as

$H_{ie} = V \sum_{kq}\rho_q c^\dagger_{k+q} c_k$

the forth order term is a result of a four-tail diagram, shown in Fig. diagram4.png.
A conjecture that we need to test is that the maximum number of vector $q_1$ and $q_2$ pairs have to be on resonance for the 4th order energy to be maximized. The number of pairs is equal to the number of edges of a polyhedron. Both icosahedron and dodecahedron have 30 edges, the largest number our of platonic solids.
Thus we can expect that either QC can be realized if if the "e/a" ratio is right.

For instance, in the case of icosahedron, $q_1$, $q_2$ and the edge to which they connect form almost a perfect triangle. A guess is that the contribution to the 4th order diagram is peaked when this triangle can be inscribed into the equatorial circle of the Fermi sphere.

The formation of quasicrystals can be simulated in momentum space using time-dependent formulation,

$$\partial _t \rho_q = \frac{\delta F(\{\rho\})}{\delta \rho^*_q} + \xi(q,t)$$

where $F$ is the Ginzburg-Landau functional, and $\xi(q,t)$ is thermal noise. For the case of crystallization occurring only on a ring in the momentum space, $|q| = q_0$, the simulation can be done in a reduced dimension (on a 2D sphere for 3D crystals, on a 1D circle for 2D crystals), and can be done very efficiently (see attached note). The simulation takes the angular dependence of the 4th order term as an input. It can indeed be computed, as discussed here.