Higher-order time-symmetry-breaking phase transition due to meeting of an exceptional point and a Fano resonance

Abstract

We have theoretically investigated the time-symmetry breaking phase transition process for two discrete states coupled with a one-dimensional continuum by solving the nonlinear eigenvalue problem for the effective Hamiltonian associated with the discrete spectrum. We obtain the effective Hamiltonian with use of the Feshbach-Brillouin-Wigner projection method. Strong energy dependence of the self-energy appearing in the effective Hamiltonian plays a key role in the time-symmetry breaking phase transition: as a result of competition in the decay process between the Van Hove singularity and the Fano resonance, the phase transition becomes a higher-order transition when both the two discrete states are located near the continuum threshold.

Characteristic dynamics near two coalescing eigenvalues incorporating continuum threshold effects

“…In many such familiar circumstances, we can generally think of these processes as following a simple exponential decay law. We note from the outset that the exponential decay is associated with the resonance in quantum mechanics [1][2][3][4][5][6][7][8][9][10][11][12][13][14], which can be thought of as a generalized eigenstate that resides outside the ordinary Hilbert space [2-4, 8, 12, 15-19].…”

Fano absorption spectrum with the complex spectral analysis

“…Note that Eq. (18) is nonlinear in the sense that the operator itself depends on its eigenvalue due to the energy dependance of the self-energy as pointed out in the Introduction [21,23,28]. It is only when taking into account this nonlinearlity that the spectrum of the effective HamiltonianĤ eff coincides with that of the total HamiltonianĤ, and the effective problem is dynamically justified.…”

“…(23). On the other hand, the other two trajectories numbered by (ii) and (iii) are separated from the trajectory (i).…”

“…It should be emphasized that the complex eigenvalue problem of the effective Hamiltonian becomes nonlinear in this approach in the sense that the effective Hamiltonian depends on its eigenvalue. It has been revealed recently that this nonlinearity of the eigenvalue problem of the effective Hamiltonian causes interesting phenomena, such as the dynamical phase transition [23], bound states in continuum (BIC) [24,25], as well as non-analytic spectral features [26][27][28], and modified time evolution near excep-tional points [29].…”

“…In this section we briefly summarize the complex eigenvalue problem with use of the BWF projection method. One could refer to the literatures for details [13,21,23].…”