The $\mathcal{P}\mathcal{T}$-symmetry-breaking transition in a chain of trapped interacting ions

Trapped ions are an ideal platform to implement quantum simulation. Previously the parity-time reversal ($\mathcal{P}\mathcal{T}$) symmetry-breaking transition in the paradigmatic non-Hermitian Hamiltonian $h_{PT}=J{\sigma }_x-i\mathrm{\Gamma }{\sigma }_z$ has been observed in a single ion experiment in a passive way. In this work, we propose to study the interaction effects on the $\mathcal{P}\mathcal{T}$-symmetry-breaking transition in a chain of $N$ trapped interacting ions. We consider an effective Ising interaction $H_{\text{int}}^{\text{Ising-x}}={\sum }_{j<k}{U}_{jk}{\sigma }_x^j{\sigma }_x^k$ between the ions on top of $h_{PT}$. We find that sufficiently strong interaction strength can enhance the $\mathcal{P}\mathcal{T}$-symmetric phase for even $N$ while the phase is suppressed in all the other cases. In particular, the suppression can be so strong that even infinitesimal dissipation, quantified by $\mathrm{\Gamma }$, can turn the system into the $\mathcal{P}\mathcal{T}$-symmetry-breaking phase. In addition, we assess the convolved effects due to the coupling and the spin phase shifts. Our findings can be readily tested in ion chain experiments.