Why does the C6z operator have 6 eigenvectors in this paper?

Question

In this paper, I couldn't understand this line in the 3rd section (i.e. III. SYMMETRY ARGUMENTS):

In the absence of SOC, the eigenvalues of $C_{6z}$ rotation operator are $e^{i2\pi n/6}$ where $n = 0$ to $5$ and the corresponding eigenstates are denoted by $\psi_1 , \psi_2 , \psi_3, \psi_4 , \psi_5 , \text{and} \psi_6$, respectively.

But, as we know, the rotation matrix for an angle $\phi$ rotation about z-axis in 3-D cartesian space is given by:

$$ \begin{pmatrix} \cos\phi & -\sin\phi & 0\\ \sin\phi & \cos\phi & 0\\ 0 & 0 & 1 \end{pmatrix}, $$ with eigenvalues $\lambda_1 = 1 , \lambda_2=e^{i\phi} , \lambda_3 =e^{-i\phi} $ and their corresponding eigenvectors $v_1 = (0,0,1) , v_2 = (i,1,0), v_3=(-i,1,0)$. I don't know what I am missing to correctly interpret and find the mathematical logic for the eigenvalues of the $C_{6z}$ rotation operator, and what I am confused about is that I am getting 3 eigenvalues but in the paper, it is 6.

Answer 1

The paper appears to have an error.

When a $C_6$ operator acts on an atom's (x,y,z) coordinates, the operator's matrix representation is the 3x3 matrix that you correctly provided in your question post. The rotation matrix is a 3x3 matrix whether it is representing a $C_{3}$ operator, $C_{6}$ operator, $C_{1402}$ operator, or any other $C_n$ operator with $n$ being a positive integer. In the paper to which you referred, $C_{6z}$ is the specific $C_{6}$ operator for a rotation around the z-axis, meaning that it is the $R_z(\theta)$ operator in which $\theta$ is 360/6 = 60 degrees.

In Fig 2a of the arXiv version of the paper, what they call the "eigenstates" of the $C_{6z}$ operator, are labeled as $\psi_1,\psi_2,\ldots,\psi_6$, even though these just appear to be positions in (x,y,z)-space that are obtained by applying the C6 operator. For example, if a nucleus resides at (x,y,z) = (1,2,3), there are 6 different locations to which the $C_{6z}$ operator can transform the coordinates of the nucleus, one of these being the original coordinates: (1,2,3). This is very different from eigenvectors: what eigenvectors would be in this situation, are (x,y,z) coordinates at which the $C_{6z}$ operator does not change the coordinates. For example, you suggested the vector (x,y,z) = (0,0,1), which after any rotation around the z-axis, remains to be (0,0,1).

If the authors were using the terminology correctly, then in Fig 2a we would expect for $\psi_1,\psi_2,\ldots,\psi_6$ to all be at the same location, because they would represent locations at which the operator does not change the coordinates!

A rotation operator can have 6 eigenvalues and 6 eigenvectors if it is an operator that is acting on two sets of (x,y,z) coordinates (for example if it is acting on the positions of two nuclei, rather than just one), but even then the rotation operator would usually be given by the standard 3x3 matrix that you presented in your post, and it would just be applied to the two sets of (x,y,z) coordinates independently (the 6x6 matrix would be a block diagonal matrix composed of two 3x3 rotation matrices).