Duality between non-reciprocal Ising model and non-reciprocal Anisotropic XY model

Non-reciprocal interactions naturally emerge and are prevalent in out-of-equilibrium systems. In this work, we advance the study of non-reciprocity in many-body systems by formulating a non-reciprocal kinetic Ising model, associating a Hilbert space to it, and deriving its time evolution operator. We demonstrate that this operator corresponds to the Hamiltonian of the anisotropic XY model, modified with additional interactions that introduce non-reciprocity in the quantum domain. Furthermore, through Jordan-Wigner transformations, we map this model to the Hatano-Nelson chain, where the non-reciprocal interactions manifest as the non-Hermitian terms of the Hatano-Nelson model. We diagonalize this Hamiltonian and analyze it in detail, revealing new effects arising from non-reciprocity in this many-body system. While exploring this system, we contrast its quantum and classical versions, highlighting the distinct effects possible in each regime.