# Energetic stability of quasicrystals

*The goal of this project is to find a physically motivated minimal model that would show formation of quasicrysals.
At the moment, there is a lot of work that has been done within the framework of Ginzburg-Landau, but they typically involve some ad-hoc assumptions about the form of the functional. We want to see if models of this kind can be obtained starting from the simple mixutre or electrons and ions in some proportion*

Quasicrystals are materials that possess unusual ("non-crystallographic") symmetries, which are not consistent with the spatially periodic arrangements of atoms. Despite the very complex arrangement of atoms in the real space, the diffraction patterns in the reciprocal space are highly ordered. They contain resolution-limited Bragg peaks, forming self-similar patterns with non-crystallographic symmetries. The most common are the three-dimensional quasicrystals with the point-group symmetry of icosahedron (iQC). In addition, there are quasi-two dimensional quasicrytals, which are quasiperiodic (with 8, 10, or 12-fold symmetry) in a plane and periodic in orthogonal direction.

Quaiscrystals were first created by rapid cooling of Al-Mn alloys, and were initially believed to be only a metastable phase of matter. However, subsequently they were discovered in a wide variety of alloys, even upon adiabatic cooling, convincingly demonstrating their thermodynamic stability.

Majority of quasicrystal-forming alloys contain metal atoms. Yet, their electrical and thermal conductivities are low. This, and the observation that quasicrystals are stable in a rather narrow window of relative concentrations of the constituent components, lead to the conjecture that their formation is governed by the Hume-Rothery rules ^{[1]}, ^{[2]} that had previously been found to account for particularly stable metallic alloys. Those rules state that the stability of a crystals is governed by the ratio of the density of conduction electrons to the density of atoms (atoms of different type counted all together). Within the standard theory of metals, this ratio corresponds to a geometric matching condition between the nearly-free electron Fermi surface and the boundaries (Bragg planes) of the first Brillouin zone ^{[3]}.

Even though quasicrystals do not posess the Brillouin zone, the existence of sharply defined Bragg spots in their diffraction patterns allows to construct a "quasi-Brillouin" zone, bounded by the Bragg planes attributed to the strongest Bragg diffraction peaks.

Then, the gapping of parts of the Fermi surface due to the umklapp scattering can explain both the suppression of transport as well as the observed reduction of the electronic density of states near the Fermi energy of these materials [Cp, ARPES].

In the past, there have been attempts to construct a microscopic theory that would explain the energetic stability of quasicrystals starting from the Hume-Rothery rules ^{[4]}. They were based on performing comparison of electronic energies of various candidate states, in a way that had been done previously to determine the relative stability of different conventional crystals ^{[3]}. However, an assumption that enables the comparison, that the strength of the Bragg scattering is the same among all crystals, is hard to justify.

On the other hand, many attempts have been made to understand the stability of different phases starting from purely phenomenological description of the phase transition in terms of the Ginzburg-Landau (GL) functional for weak crystallization ^{[5]}. Indeed, several models have been proposed that could lead to QC being the lowest energy states in some range of parameters ^{[6]}, ^{[7]}, ^{[8]}, ^{[9]}. However, they had not provided any microscopic justification of their assumptions, and thus cannot acount for the fact that many QC's follow the Hume-Rothery rules.

Here we provide the microscopic derivation of the effective GL functional starting from the basic microscopic model of interacting electrons and ions. Most notably, we find that electrons provide an interaction between different Fourier components of ionic densities that favors certain angle between the modulation directions. We find based on variational calculation and direct simulations, that when this angle equals approximately to the angle between icosahedron vertices , iQC takes the best advantage of the interaction. The other states that appear nearby in the phase diagram are the FCC (not BCC!), romboherdal, and smectic. In addition, from the simulations and based on analytical arguments we predict the possibility of an *imperfect* quasicrystalline state, which corresponds to a distorted arrangement icosahedral of Bragg peaks and unequal magnitudes of the corresponding density modulations.

The condition for stability of iQC that we find, very closely corresponds to the HR rules known empirically for iQC; namely, the magnitude of the primary Bragg peaks is approximately equal to the diameter of the Fermi surface.

The GL functional that we derive is related to the one discussed by Mermin and Troyan ^{[7]}, with similar results. The difference is that the fictitious component introduced by MT is replaced by conduction electrons.