# Topological phase transitions - breaking of continuous translational invariance [closed]

I'm relatively new to the theoretical side of physics. I have a question about topology, continuous symmetry breaking and phase transitions. Your help is much appreciated!

Ok so I have an infinite array of metallic nanoparticles with a two atom basis, which can equivalently be thought of as two equivalent but independent sub-lattices (think of graphene). Anyway, I start with a configuration such that the lattice is a honeycomb structure (with three nearest neighbours), I then deform the lattice by scaling the position vector of the second basis particle, which can be seen in the figure as 0.9, 0.858, 0.856 and 0.8 times the original second basis particle vector.

I have generated plots which show if the resulting dispersion of the plasmons is gapless (cream) or gapped (orange) for arbitrary dipole orientation (am modelling the LSPs as dipoles).

Ok now I am going to say a paragraph and I would like you to tell me if I'm talking rubbish or not...

"At some value between $\vec{d}_3 = 0.9 \vec{e}_3$ and $\vec{d}_3 = 0.858 \vec{e}_3$ there is a topological change which can be seen in the breaking of the continuity of gapless states around the points $(\frac{\pi}{2},\frac{5\pi}{12})$ and $(\frac{\pi}{2},\frac{8\pi}{12})$. There is another topological change that occurs at a value between $\vec{d}_3 = 0.858 \vec{e}_3$ and $\vec{d}_3 = 0.856 \vec{e}_3$ where initially there is a continuum of gapless states about the lines $\theta = 0$ and $\theta = \pi$, then subsequently a continuum of gapped states about the same lines. At these values of $\vec{d}_3$ where a continuous translational symmetry has been broken, it is reasonable to assume this is accompanied by some form of phase transition."

-

## closed as too localized by Manishearth♦Apr 30 '13 at 20:29

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

thanks @007 for adding the image :) –  Tom Apr 26 '13 at 10:26