Non-Hermitian Haldane-Hubbard model: Effective description of an open system with balanced gain and loss

We study the correlated Haldane-Hubbard model with single-particle gain and loss, focusing on its non-Hermitian phase diagram and the ensuing non-unitary dynamic properties. The interplay of interactions and non-hermiticity results in insulating behavior with a phase diagram divided into three distinct regions, exhibiting either topologically gapped or (real) gapless regimes and a trivial phase. The latter is mapped by the emergence of a local order parameter associated with a charge density wave. A ${\cal PT}$-symmetry breaking at the low-lying spectrum occurs when increasing the gain-loss magnitude at a fixed interaction strength, marking the transition from gapped to gapless topological behavior. Further increase leads to the onset of charge ordering in a first-order phase transition in which level crossing takes place in the spectrum's imaginary part. The support that the staggered gain and loss display to robust charge density wave in equilibrium is confirmed in the real-time dynamics in the presence of non-hermiticity, suggesting that engineered gain and loss can be used to tailor an ordered many-body state in experiments.