  For a continuous $\mathbb{N}^d$ or $\mathbb{Z}^d$ action on a compact space,
we introduce the notion of Bohr chaoticity, which is an invariant of
topological conjugacy and which is proved stronger than having positive
entropy. We prove that all principal algebraic $\mathbb{Z}$ actions of positive
entropy are Bohr-chaotic. The same is proved for principal algebraic
$\mathbb{Z}^d$ ($d\ge 2$) actions of positive entropy under the condition of
existence of summable homoclinic points.
