  We study the splitting in the screening mass of pions and the $\eta$-meson
across the chiral crossover. This splitting is determined by the 't Hooft
determinant. We use results for the renormalisation group scale dependence of
the 't Hooft determinant obtained within the functional renomalisation group in
quenched QCD with two flavours. The scale dependence of the 't Hooft
determinant is mapped to its temperature dependence with the help of a
Polyakov-quark-meson model. As a result we obtain the temperature dependence of
the splitting in the screening mass of pions and the $\eta$-meson.
