Simulating many-body non-Hermitian PT -symmetric spin dynamics

It is possible to simulate the dynamics of a single spin-$1/2$ ($\mathcal{PT}$-symmetric) system by conveniently embedding it into a subspace of a larger Hilbert space, undergoing unitary dynamics governed by a Hermitian Hamiltonian. Our goal is to analyze 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 ``$N$'' noninteracting spin-$1/2$ ($\mathcal{PT}$-symmetric) degrees of freedom, thereby unfolding the complex nature of the embedding Hamiltonian. It turns out that the resulting Hermitian Hamiltonian of $N+1$ spin halves comprises ``all to all'', $q$-body interaction terms ($q=1,...,N+1$) where the additional spin-$1/2$ is a part of the larger embedding space. We show that the presence of finite entanglement in the eigenstates of the resulting cluster of $N+1$ spin halves ensures the nonvanishing probability of post-selection of the additional spin-1/2, which is essential for the embedding to be practicable. Finally, we also note that our study can be identified with a central spin model where orthogonality catastrophe owing to the finite entanglement plays a central role in protecting the additional spin-1/2 degree of freedom from decoherence.