  We use the exceptional point in Hopfield-Bogoliubov matrix to find the phase
transition points in the bosonic system. In many previous jobs, the excitation
energy vanished at the critical point. It can be stated equivalently that
quantum critical point is obtained when the determinant of Hopfield-Bogoliubov
matrix vanishes. We analytically obtain the Hopfield-Bogoliubov matrix
corresponding to the general quadratic Hamiltonian. For single-mode system the
appearance of the exceptional point in Hopfield-Bogoliubov matrix is equivalent
to the disappearance of the determinant of Hopfield-Bogoliubov matrix. However,
in multi-mode bosonic system, they are not equivalent except in some special
cases. For example, in the case of perfect symmetry, that is, swapping any two
subsystems and keeping the total Hamiltonian invariable, the exceptional point
and the degenerate point coincide all the time when the phase transition
occurs. When the exceptional point and the degenerate point do not coincide, we
find a significant result. With the increase of two-photon driving intensity,
the normal phase changes to the superradiant phase, then the superradiant phase
changes to the normal phase, and finally the normal phase changes to the
superradiant phase.
