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

TL;DR

This paper investigates quantum phase transitions in bosonic systems by analyzing exceptional points in the Hopfield-Bogoliubov matrix, revealing new insights into phase transition conditions beyond traditional energy vanishing criteria.

Contribution

It analytically links exceptional points in the Hopfield-Bogoliubov matrix to quantum phase transitions in multi-mode bosonic systems, extending understanding beyond single-mode cases.

Findings

Exceptional points indicate phase transition boundaries.

In multi-mode systems, exceptional points and degeneracy points may not coincide.

Two-photon driving induces multiple phase transitions between normal and superradiant phases.

Abstract

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.