  A conventional quantum phase transition (QPT) can be accessed by varying a
real parameter at absolute zero temperature. Motivated by the discovery of the
pseudo-Hermiticity of non-Hermitian systems, we explore the QPT in
non-Hermitian PT-symmetric Ising model, which is driven by a staggered complex
transverse field. Exact solution shows that the Laplacian of the groundstate
energy density, with respect to real and imaginary components of the transverse
field, diverges on the boundary in the complex plane. The phase diagram
indicate that the imaginary transverse field has the effect of shrinking the
paramagnet phase. In addition, we also investigate the connection between the
geometric phase and the QPT.
