Correlation functions of integrable $O(n)$ spin chains

We study the correlation functions of the integrable $O(n)$ spin chain in the thermodynamic limit. We addressed the problem of solving functional equations of the quantum Knizhnik Zamolodchikov type for density matrix related to the $O(n)$ spin chain. We give the explicit solution for two-sites density matrix elements for the $O(n)$ which are then evaluated for the $n=3,4,\dots ,8$ cases at zero temperature.