  We study the finite-temperature behavior of the Lipkin-Meshkov-Glick model,
with a focus on correlation properties as measured by the mutual information.
The latter, which quantifies the amount of both classical and quantum
correlations, is computed exactly in the two limiting cases of vanishing
magnetic field and vanishing temperature. For all other situations, numerical
results provide evidence of a finite mutual information at all temperatures
except at criticality. There, it diverges as the logarithm of the system size,
with a prefactor that can take only two values, depending on whether the
critical temperature vanishes or not. Our work provides a simple example in
which the mutual information appears as a powerful tool to detect
finite-temperature phase transitions, contrary to entanglement measures such as
the concurrence.
