  The Heisenberg group is one of the simplest sub-Riemannian settings in which
we can define non-elliptic H\"ormander type generators. We can then consider
coercive inequalities associated to such generators. We prove that a certain
class of nontrivial Gibbs measures with quadratic interaction potential on an
infinite product of Heisenberg groups satisfy logarithmic Sobolev inequalities.
