We study the dynamics of the transverse field Ising chain after a local quench
in which two independently thermalised chains are joined together and are left
to evolve unitarily. In the emerging non-equilibrium steady state the Rényi
mutual information with different indices are calculated between two adjacent
segments of the chain, and are found to scale logarithmically in the subsystem
size. Surprisingly, for Rényi indices > 2 we find cases where the prefactor of
the logarithmic dependence is negative. The fact that the naively defined Rényi
mutual information might be negative has been pointed out before, however, we
provide the first example for this scenario in a realistic many-body setup. Our
numerical and analytical results indicate that in this setup it can be negative for
any index > 2 while it is always positive for < 2. Interestingly, even for
> 2 the calculated prefactors show some universal features: for example, the
same prefactor is also shown to govern the logarithmic time dependence of the
Rényi mutual information before the system relaxes locally to the steady state.
In particular, it can decrease in the non-equilibrium evolution after the quench
