I am trying to replicate a calculation from the linked paper but I am unsure if I understand their math. A locally invariant function defined as the characteristic polynomial is as follows: $$F_U(t) = det[\Re[M^{\dagger}UM] + t \cdot \Im[M^{\dagger}UM]]$$ Then it is said for $U$ with interaction coefficients $(x,y,z)$, we have $$F_U(t) = (t^2+1)(Ct^2 + Bt + A) -t^2$$ I'm having trouble understanding how $(x,y,z)$ are related to $t$, and because neither $M$ nor $U$ are defined as functions of $t$, then it seems like $F_U(t)$ is not a degree-4 polynomial.
In this paper, equation (9), they define $U$ with parameters $(\alpha, \beta, \gamma_1, \gamma_2, \delta_1, \delta_2)$ and find the corresponding coefficient $C=\frac{1}{16}(\cos{2\alpha}-\cos{2\beta})^2$. When I plug the same values in, since $U$ is not a function of $t$, then $F_U$ is not degree-4 and hence $C=0$. Any guidance would be appreciated, thank you.
Source: https://arxiv.org/abs/2105.06074
