“…In the case of a complex Hamiltonian, the situation is more severe on account of the fact that the Rayleigh-Schrödinger perturbation theory breaks down altogether in the vicinities of exceptional points. Nevertheless, for a given Hamiltonian, one can expand the eigenstates and eigenvalues in the form of a Newton-Puiseux series in order to identify the metric geometry close to exceptional points (see, for example, [54,55] for effective use of the Newton-Puiseux expansion for the investigation of properties of the eigenstates of complex Hamiltonians in the vicinities of exceptional points; see, also, [56,57] for a more general discussion on related mathematical ideas). This line of investigation, therefore, leads to a new application of information geometry in the sensitivity analysis of physical systems characterised by Hermitian or more generally complex Hamiltonians (we remark that properties of exceptional points of higher order, where more than two eigenstates coalesce, can be quite intricate; see, e.g., [58,59]).…”