Emergent phase transitions in a cluster Ising model with dissipation

We study a cluster Ising model with non-Hermitian external field which can be exactly solved in the language of free fermions. By investigating the second derivative of energy density and fidelity, the possible new critical points are tentatively located. The string order parameter and staggered magnetization are then detected to reveal emergent phases of brand new characteristics. To categorize the exotic phases and phase transitions induced by non-Hermiticity, we calculate the variation mode of the spin-correlation function as well as of the string correlation function, which characterize the emergent phases and critical points with different patterns of decay and critical exponents. With the help of the string order parameter and staggered magnetization, we find that there are four phases after introducing the non-Hermiticity???the cluster phase, the gapless phase, the paramagnetic phase, and the antiferromagnetic phase. A phase diagram is then presented to graphically illustrate, based on two ???Kosterlitz-Thouless-like??? phase transitions and an Ising phase transition, respectively, the generation of three critical lines as non-Hermitian strength increases. Our theoretical work is expected to be realized in the experiment of ultracold atoms, pushing for progress in exploring novel phases and phase transitions.