Equivariant PT-symmetric real Chern insulators

It was understood that Chern insulators cannot be realized in the presence of PT symmetry. In this paper, we reveal a new class of PT-symmetric Chern insulators, which has internal degrees of freedom forming real representations of a symmetry group with a complex endomorphism field. As a generalization to the conventional 2n-dimensional Chern insulators with integer n >= 1, these PT-symmetric Chern insulators have the n-th complex Chern 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 Chern insulators with PT symmetry. By generalizing the Thouless charge pump argument, we find that, for a PT-symmetric Chern insulator with Chern number nu, there are equally many nu 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 internal symmetry group, but are not isomorphic. For the physical dimensionality d = 2, the PT-symmetric Chern insulators may be realized in artificial systems including photonic crystals and periodic mechanical systems.