  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. 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 spin correlation function as well as string
parameter, which characterize the emergent phases and critical points with
different patterns of decay and critical exponents. With the help of 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 (PM) phase and the antiferromagnetic (AF) phase. A phase
diagram is then presented to graphically illustrate, based on two "KT-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 ultra-cold atoms, pushing
for progress in exploring novel phases and phase transitions.

Statistical Mechanics;Quantum Gases