PT phase transition in higher-dimensional quantum systems

We consider a 2d anisotropic SHO with {\bf ixy} interaction and a 3d SHO in an imaginary magnetic field with ${\vec{\mu }}_l$.$\vec{B}$ interaction to study the $PT$ phase transition analytically in higher dimension.Unbroken $PT$ symmetry in the first case is complementary to the rotational symmetry of the original Hermitian system .$PT$ phase transition ceases to occur the moment the 2d oscillator becomes isotropic.Transverse magnetic field in the other system introduces the anisotropy in the system and the system undergoes PT phase transition depending on the strength of the magnetic field and frequency of the oscillator.