  We study mutual information for Renyi entropy of arbitrary index n, in
interacting quantum systems at finite-temperature critical points, using
high-temperature expansion, quantum Monte Carlo simulations and scaling theory.
We find that for n>1, the critical behavior is manifest at two temperatures T_c
and n*T_c. For the XXZ model with Ising anisotropy, the coefficient of the
area-law has a t*ln(t) singularity, whereas the subleading correction from
corners has a logarithmic divergence, with a coefficient related to the exact
results of Cardy and Peschel. For T<n*T_c there is a constant term associated
with broken symmetries that jumps at both T_c and n*T_c, which can be
understood in terms of a scaling function analogous to the boundary entropy of
Affleck and Ludwig.
