I am reading the paper on, "Entanglement Entropy and Quantum Field Theory"

https://arxiv.org/pdf/hep-th/0405152.pdf

In this paper, the author introduces cut in the path integral which is referred to be the partial trace over space and it gives partial density matrix of a subsystem.

Does the partial density have a branch cut in general ? Is it multivalued ? This question arises because of the fact that while taking Tr$[\rho^{n}]$ ,the density matrix is called as Riemann surface with a cut on it as it is a partial trace. What is the origin of the multivaluedness to term density matrix as Riemann surface ?

https://arxiv.org/pdf/hep-th/0405152.pdf