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We use the exceptional point in Hopfield-Bogoliubov matrix to find the phasetransition points in the bosonic system. In many previous jobs, the excitationenergy vanished at the critical point. It can be stated equivalently thatquantum critical point is obtained when the determinant of Hopfield-Bogoliubovmatrix vanishes. We analytically obtain the Hopfield-Bogoliubov matrixcorresponding to the general quadratic Hamiltonian. For single-mode system theappearance of the exceptional point in Hopfield-Bogoliubov matrix is equivalentto the disappearance of the determinant of Hopfield-Bogoliubov matrix. However,in multi-mode bosonic system, they are not equivalent except in some specialcases. For example, in the case of perfect symmetry, that is, swapping any twosubsystems and keeping the total Hamiltonian invariable, the exceptional pointand the degenerate point coincide all the time when the phase transitionoccurs. When the exceptional point and the degenerate point do not coincide, wefind a significant result. With the increase of two-photon driving intensity,the normal phase changes to the superradiant phase, then the superradiant phasechanges to the normal phase, and finally the normal phase changes to thesuperradiant phase.

Quantum phases transition revealed by the exceptional point in Hopfield-Bogoliubov matrix