Minimum of η/s and the phase transition of the linear sigma model in the large-N limit

We reexamine the possibility of employing the viscosity over entropy density ratio as a diagnostic tool to identify a phase transition in hadron physics to the strongly coupled quark-gluon plasma and other circumstances where direct measurement of the order parameter or the free energy may be difficult. It has been conjectured that the minimum of eta/s does indeed occur at the phase transition. We now make a careful assessment in a controlled theoretical framework, the linear sigma model at large N, and indeed find that the minimum of eta/s occurs near the second-order phase transition of the model due to the rapid variation of the order parameter (here the sigma vacuum expectation value) at a temperature slightly smaller than the critical one.