Instability of Icosahedral quasicrystals wrt distortions

The reason for high stability of iQC is the very special feature that the angles between all vectors pointing from the center of icosahedron to its vertices are either equal to $\alpha_0$ or $\pi - \alpha_0$. Therefore, if the interaction in the GL theory favors $\alpha_0$ (and hence also $\pi - \alpha_0$), then we can expect that the energy of iQC state is going to be low.

However, it is natural to ask: What if the interaction $u(\alpha)$ reaches the minimum at a different angle? Will iQC remain stable, or will it immediately distort? For comparison, the rombohedral state always can follow the angle of the potential minimum.

To answer this question, let us expand the interaction energy in the vicinity of the iQC.

$E_{int} = \sum_{i\lt j} u(\alpha _{ij}) |\rho_i|^2 |\rho_j|^2$.

For the sake of argument will neglect the fact that the amplitudes of the order parameter can also react to distortions -- this will only further lower the energy of distorted state. Then, defining $\delta_{ij} = \alpha_{ij} -\alpha_0$,

$E_{int} = E_0 + u'(\alpha _0)\sum_{i\lt j}\delta_{ij} + 0.5 u''(\alpha _0)\sum_{i\lt j}\delta^2_{ij}+ ... $.

To make further progress, we can define convenient coordinates for the Bragg peaks on the sphere, and explore whether the energy can be lowered by a distortion. Both the first and second derivate terms define quadratic forms with non-negative eigenvalues (due to the nonlinear dependence of $\delta_{ij}$ on local coordinates, even the first order term produces quadratic form upon expansion). Out of 12 total eigenvalues the quadrartic form of $\sum_{i\lt j}\delta_{ij}$ has only 4 non-zeros; in contrast $\sum_{i\lt j}\delta^2_{ij}$ has only 3 zero modes that correspond to rigid global rotations. When put together, for $u'(\alpha _0) \lt - (2/3) u''(\alpha _0)$ negative stiffness modes emerge, signifying distortive instability of icosahedron. The strongest instability occurs at the largest possible quasimomenta $\pm 4\pi/5$ (see attached Mathematica code). At the critical point $u'(\alpha _0) = - (2/3) u''(\alpha _0)$ , four zero modes simultaneously appear, forming a flat zero-frequency band as a function of quasimomentum on icosahedron.

Hence, the conclusion is that even if $u(\alpha)$ reaches the minimum at non-icosahedral point, the iQC remains (at least) locally stable for $u' \gt 0$ (tensile strain between Bragg peaks), and even for compressive strain it remain stable until a critical value of negative $u'$ is reached.

This is quite relevant to our numerical studies, which reveal that the first order derivative of the potential can be quite large relative to the second derivative. Indeed, distorted iQC appear regularly in numerics.