Nearest-neighbour correlation functions for the supersymmetric XYZ spin chain and Painlev\'e VI

We study nearest-neighbour correlation functions for the ground state of the supersymmetric XYZ spin chain with odd length and periodic boundary conditions. Under a technical assumption related to the $Q$-operator of the corresponding eight-vertex model, we show that they can be expressed exactly in terms of the Painlev\'e VI tau functions $s_n$ and $\bar s_n$ introduced by Bazhanov and Mangazeev. Furthermore, we give an interpretation of the correlation functions in terms of the Painlev\'e VI Hamiltonian.