  Non-Hermitian quantum system recently have attracted a lots of attentions
theoretically and experimentally. However, the results based on the
single-particle picture may not apply to understand the property of
non-Hermitian many-body system. How the property of quantum many-body system
especially the phase transition will be affected by the non-hermiticity remains
unclear. Here we study non-Hermitian quantum contact process (QCP) model, whose
effective Hamiltonian is derived from Lindbladian master equation. We show that
there is a continuous phase transition induced by the non-hermiticity in QCP.
We also determine the critical exponents $\beta$ of order parameter, $\gamma$
of susceptibility and study the correlation and entanglement near phase
transition. We observe that the order parameter and susceptibility display
infinitely singularity even for finite size system, since non-hermiticity endow
many-body system with different singular behaviour from classical phase
transition. Moreover our results show that the phase transition have no
counterpart in Hermitian case and belongs to completely different universality
class.
