A non-Hermitian $PT-$symmetric Bose-Hubbard model: eigenvalue rings from unfolding higher-order exceptional points

We study a non-Hermitian $PT-$symmetric generalization of an $N$-particle, two-mode Bose-Hubbard system, modeling for example a Bose-Einstein condensate in a double well potential coupled to a continuum via a sink in one of the wells and a source in the other. The effect of the interplay between the particle interaction and the non-Hermiticity on characteristic features of the spectrum is analyzed drawing special attention to the occurrence and unfolding of exceptional points (EPs). We find that for vanishing particle interaction there are only two EPs of order $N+1$ which under perturbation unfold either into $[(N+1)/2]$ eigenvalue pairs (and in case of $N+1$ odd, into an additional zero-eigenvalue) or into eigenvalue triplets (third-order eigenvalue rings) and $(N+1)\mod 3$ single eigenvalues, depending on the direction of the perturbation in parameter space. This behavior is described analytically using perturbational techniques. More general EP unfoldings into eigenvalue rings up to $(N+1)$th order are indicated.