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Braiding of dynamical eigenvalues of Hermitian bosonic Kitaev chains

In quantum mechanics, observables correspond to Hermitian operators, and the spectra are restricted to be real. However, the dynamics of the underlying fields may allow complex eigenvalues and therefore create the possibility of braiding structures. Here we study the braiding of dynamical eigenvalues in quantum systems by considering Hermitian bosonic Kitaev chains with multiple bands. The dynamics of the quantum fields in these systems are described by their dynamic matrices, which have complex eigenvalues. We show that there are symmetry constraints imposed on these dynamic eigenvalues. Despite these constraints, braiding is possible for frequencies within the effective gain and loss regions of the complex plane. We explicitly construct two- and three-strand braidings using the exceptional points found in the system and discuss possible implementations.