I am trying to use eq 16 of the following paper to calculate the Chern number of a 4x4 degenerate system: https://arxiv.org/pdf/cond-mat/0503172.pdf [1]. I believe this is the standard scheme used by many.
In my sample system (I am using eq. 1 in https://arxiv.org/pdf/1008.2026.pdf [2]), I chose parameters such that the 2 bands above the gap are degenerate with each other, and the 2 bands below the gap are degenerate with each other.

I correctly used the method in the above paper [1] to make a MATLAB implementation of the Abelian case.

When I solve my 4-band system, I get four $4\times 1$ instantaneous energy eigenstates. When I plot the eigenvalues, I see '2' bands, because each is degenerate to another. So far, so good.
My issue is that I am not sure what the authors mean when they say "multiplet". 

However, I must now calculate the U(1) link variable in eq. 16 of [1] as:
$$
U(k) = \frac{det (\phi^* (k) \phi(k+\mu))}{|det (\phi^* (k) \phi(k+\mu))|},
$$

where I chose $\phi$ to be the eigenstate of one (of four) levels, $^*$ is the complex conjugate transpose, and $\mu$ is a small parameter used to take the numerical derivative.
However, the authors define $\phi$ as a "multiplet" $\{n_1, ...., n_M\}$, where $M$ is a dimension of the $M\times M$ non-Abelian Berry connection matrix $A=\psi^* d \psi$. I believe my problem arises here: What does "multiplet" mean in this context? I assumed it was just an equivalent term for "instantaneous eigenstate". So, I have four $4\times 1$ multiplets in this 2-fold degenerate system. 


I assumed that $\phi^* (k) \phi(k+\mu)$ could be read as $|\phi(k)\rangle\langle\phi(k+\mu)|$, so that for a $4\times 1$ eigenvector $|\phi(k)\rangle$, I would get a $4\times 4$ matrix for $|\phi(k)\rangle\langle\phi(k+\mu)|$. Is this interpretation correct? I do not think it is, because the last two rows of all vectors are zero, making the determinant always zero, and therefore $U(k)$ ill-defined due to division by zero. I am guessing I got something wrong with my interpretation of the word "multiplet". It would make sense numerically if my multiplet was just the first 2 rows of each eigenvector (then, the denominator will be non-zero). However, I cannot justify this. Could someone clarify my confusions please? I have a feeling these "multiplets" should instead be $2\times 1$ vectors... but then, I don't know how to justify getting rid of 2 rows of numerical eigenvectors of a $4\times 4$ system. Thanks.