Simulating non-Hermitian dynamics of a multi-spin quantum system and an emergent central spin model

It is possible to simulate the dynamics of a single spin-$1/2$ ($\mathsf{PT~}$ symmetric) system by conveniently embedding it into a subspace of a larger Hilbert space with unitary dynamics. Our goal is to formulate a many body generalization of this idea i.e., embedding many body non-Hermitian dynamics. As a first step in this direction, we investigate embedding of "$N$" non-interacting spin-$1/2$ ($\mathsf{PT~}$ symmetric) degrees of freedom, thereby unfolding the complex nature of such an embedding procedure. It turns out that the resulting Hermitian Hamiltonian represents a cluster of $N+1$ spin halves with "all to all", $q$-body interaction terms ($q=1,...,N+1$) in which the additional spin-$1/2$ is a part of the larger embedding space. We can visualize it as a strongly correlated central spin model with the additional spin-$1/2$ playing the role of central spin. We find that due to the orthogonality catastrophe, even a vanishing small exchange field applied along the anisotropy axis of the central spin leads to a strong suppression of its decoherence arising from spin-flipping perturbations.