# How to describe charge density waves in k-space?

I am reading this article about variational ground state wavefunction. In equation 10 they said that charge density wave (CDW) for spinless fermionic system at half-filling can be written as following:

$$|CDW\rangle=\prod_{k\epsilon RBZ} \frac{1}{\sqrt2} (c_k^\dagger+c_{k+\pi}^\dagger) |0\rangle$$

According to article charge density wave is alternating occupied and empty sites.

I can't understand how this equation is representing CDW. Can someone please give me little explanation of this equation? What is meaning of $c_{k+\pi}^\dagger$ ? Or any reference regarding this CDW equation?

I believe this argument is due to Peierls...

A breaking of a lattice symmetry $\mathbb{Z}$ down to a subgroup $n \mathbb{Z}$ is equivalent to changing the periodicity of $k$ from $2\pi$ to $2\pi/n$. This means that each of the original bands wraps the new Brillouin zone $n$ times, causing some bands to intersect each other. Then interactions among the new bands will generically separate these crossings yielding a band structure which cannot be unwound back to a $2\pi$ periodic one.

In particular an $n=2$ CDW comes from an interaction between $k$-points separated by $\pi$ momentum in the original band structure, but which sit right on top of each other in the $\pi$-periodic Brillouin zone. A ground state like the one you wrote is exactly the sort of wavefunction that will appear in this case. Your product is over the reduced Brillouin zone ($\pi$-periodic) and says that particle modes at $k$ and $k+\pi$ are phase-locked. It's easy to see that the unit translation does not preserve the state you wrote down because it messes up the phases:

$$c_k^\dagger + c_{k+\pi}^\dagger \mapsto e^{i k} c_k^\dagger - e^{ik} c_{k+\pi}^\dagger$$ but if we do it twice we get

$$c_k^\dagger + c_{k+\pi}^\dagger \mapsto e^{2 i k} c_k^\dagger + e^{2 ik} c_{k+\pi}^\dagger$$

• Thank you for your reply. I am afraid that I am too new to this field to understand your argument completely. I request you to please recommend me some introductory material for this problem and Peierls. – Luqman Saleem Apr 20 '18 at 17:26
• It's very simple! You just need to understand how the Fourier transform on the lattice behaves, and a bit of second quantization. Any introduction to band theory will explain these things. – Ryan Thorngren Apr 20 '18 at 17:40
• This seems like a nice article: en.wikipedia.org/wiki/Peierls_transition – Ryan Thorngren Apr 20 '18 at 18:29