Many-body phase transitions in a non-Hermitian Ising chain

We study many-body phase transitions in a one-dimensional ferromagnetic transversed field Ising model with an imaginary field, and show that the system exhibits three phase transitions: one second-order phase transition and two PT phase transitions. The second-order phase transition occurring in the ground state is investigated via biorthogonal and self-normal entanglement entropy, for which we develop an approach to perform finite-size scaling theory to extract the central charge for small systems. Compared with the second-order phase transition, the first PT transition is characterized by the appearance of an exceptional point in the full energy spectrum, while the second PT transition only occurs in specific excited states. Furthermore, we interestingly show that both exceptional points are second-order in terms of scalings of imaginary parts of the energy. This work provides an exact solution for many-body phase transitions in non-Hermitian systems.