Equations of motion governing the dynamics of the exceptional points of parameterically dependent nonhermitian Hamiltonians

Abstract

We study exceptional points (EPs) of a nonhermitian Hamiltonian $\hat{H}(\lambda ,\delta )$ whose parameters $\lambda \in \mathbb{C}$ and $\delta \in \mathbb{R}$. As the real control parameter $\delta$ is varied, the $k$-th EP (or $k$-th cluster of simultaneously existing EPs) of $\hat{H}(\lambda ,\delta )$ moves in the complex plane of $\lambda$ along a continuous trajectory, ${\lambda }_k(\delta )$. We derive a self contained set of equations of motion (EOM) for the trajectory ${\lambda }_k(\delta )$, while interpreting $\delta$ as the propagation time. Such EOM become of interest whenever one wishes to study the response of EPs to external perturbations or continuous parametric changes of the pertinent Hamiltonian. This is e.g.~the case of EPs emanating from hermitian curve crossings/degeneracies (which turn into avoided crossings/near-degeneracies when the Hamiltonian parameters are continuously varied). The presented EOM for EPs have not only their theoretical merits, they possess also a substantial practical relevance. Namely, the just presented approach can be regarded even as an efficient numerical method, useful for generating EPs for a broad class of complex quantum systems encountered in atomic, nuclear and condensed matter physics. Performance of such a method is tested here numerically on a simple yet nontrivial toy model.

28 pages, 9 figures

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