Carlo, thanks for the answer now! I skimmed over it and seems that I follow your reasoning. I am going to work out the details later, but I am already accepting your answer. Related: do you think $U$ can be written in an exponential form as well? I will try to figure it out later too.

Carlo, thanks for the comments. I am trying to figure this out for a long time now and I ran out of ideas. I simply cannot understand what they are doing (even in the linked article, which I also consulted a couple times but it did not help me).

I would be happy to have an explicit formula for $U$ and $U^{-1}$ which would justify (\ref{1}) and (\ref{2}).

Mikael, thanks for the answer. As I said, the problem as it is is still a bit imprecise in its formulation. I chose to index the operators by $x \in \mathbb{Z}$ because I thought the boundary conditions would not play any role; the authors themselves say that the transformations (\ref{1}) and (\ref{2}) assume implicitly a thermodynamic limit. Maybe the correct approach is in fact to consider $\mathbb{Z}/N\mathbb{Z}$ instead. In this case, can a explicit realization for $U$ and $U^{-1}$ be obtained?