A physical system is said to satisfy a thermal area law if the mutual
information between two adjacent regions in the Gibbs state is controlled by
the area of their boundary. Thermal area laws have been derived for systems
with bounded local interactions such as quantum spin systems. However, for
lattice bosons these arguments break down because the interactions are
unbounded. We rigorously derive a thermal area law for a class of bosonic
Hamiltonians in any dimension which includes the paradigmatic Bose-Hubbard
model. The main idea to go beyond bounded interactions is to introduce a
quasi-free reference state with artificially decreased chemical potential by
means of a double Peierls-Bogoliubov estimate.
