Spontaneous Breakdown of a PT-Symmetry in the Liouvillian Dynamics at a Non-Hermitian Degeneracy Point

Abstract

We consider the prevalent phenomenon that a pair of eigenvalues of the Liouville-von Neumann operator (Liouvillian) changes from pure imaginary to complex values with a common imaginary part for resonance states in an extended function space outside the Hilbert space. Such a transition point is an exceptional point, where non-Hermitian degeneracy occurs and both the pairs of eigenvalues and of eigenvectors coalesce. The transition can be attributed to a spontaneous breakdown of a parity and time-reversal (PT)-symmetry. This PT-symmetry in the Liouvillian dynamics results from the microscopic dynamics based on the fundamental physical laws. The kinetic equation of the Boltzmann type for a particle weakly coupled with a bath consists of the collision term, which is similar to a Hermitian operator and has even parity, and the flow term, which is anti-Hermitian and has odd parity. As a result of the competition between the two terms, a pair of PT-symmetric eigenstates of the effective Liouvillian converts to a PT-symmetry related pair as the flow term becomes more dominant than the collision term beyond an exceptional point.