When one thinks of analyzing a quantum system, the first step is to define the concerned Hilbert Space \(\mathcal{H}\). On this, there is defined the Hamiltonian of the system. This hamiltonian describes the symmetries of the theory, as well as evolution of the wave vectors.

Usually, we deal with hermitian hamiltonians which guarantee two things, viz.

Unitarity of time evolution (probability conservation)
Reality of eigenvalues (experimental observations)

Bender (paper) showed that hermiticity is not necessary for ensuring these conditions. This can be replaced by the more physical notion of \(\mathcal{PT}\)-Symmetry.

In parallel, there have been tremendous developments in the theory of topological insulators and topological superconductors. In this project, we survey what happens to the well established hermitian topological theory when hermiticity is relinquished. We observe surprising effects, and review recent developments.

One can find the review paper and slides. Further, the paper is rendered below.