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 circumflex expressionccent H(lambda, delta) whose parameters lambda is an element of C and (delta is an element of R. As the real control parameter (delta is varied, the kth EP (or kth cluster of simultaneously existing EPs) of circumflex expressionccent H(lambda, delta) moves in the complex plane of lambda along a continuous trajectory, lambda(k)(delta).
Using an appropriate non-hermitian formalism (based upon the c-product and not upon the conventional Dirac product), 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/neardegeneracies 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.