It was understood that Chem insulators cannot be realized in the presence of PT symmetry.In this paper,we reveal a new class of PT-symmetric Chem insulators,which has intal degrees of freedom forming real representations of a symmetry group with a complex endomorphism field.As a generalization to the conventional 2n-dimensional Ch insulators with integer n ＞ 1,these PT-symmetric Ch insulators have the n-th complex Ch number as their topological invariant,and have a Z classification given by the equivariant orthogonal K theory.Thus,in a fairly different sense,there exist ubiquitously Chem insulators with PT symmetry.By generalizing the Thouless charge pump argument,we find that,for a PT-symmetric Ch insulator with Ch number v,there are equally many v flavors of coexisting left-and right-handed chiral modes.Chiral modes with opposite chirality are complex conjugates to each other as complex representations of the intal symmetry group,but are not isomorphic.For the physical dimensionality d =2,the PT-symmetric Ch insulators may be realized in artificial systems including photonic crystals and periodic mechanical