<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">PT</mml:mi></mml:math>-symmetry-breaking transition in a chain of trapped interacting ions

Trapped ions are an ideal platform to implement quantum simulation. Previously, the parity-time ($\mathcal{PT}$)-reversal symmetry-breaking transition in the paradigmatic non-Hermitian Hamiltonian ${h}_{\mathcal{PT}}=J{\ensuremath{\sigma}}_{x}\ensuremath{-}i\mathrm{\ensuremath{\Gamma}}{\ensuremath{\sigma}}_{z}$ was observed in a single-ion experiment in a passive way. In this work we propose to study the interaction effects on the $\mathcal{PT}$-symmetry-breaking transition in a chain of $N$ trapped interacting ions. We consider an effective Ising interaction ${H}_{\text{int}}^{\text{Ising}\phantom{\rule{4.pt}{0ex}}x}={\ensuremath{\sum}}_{j&lt;k}{U}_{jk}{\ensuremath{\sigma}}_{x}^{j}{\ensuremath{\sigma}}_{x}^{k}$ between the ions on top of ${h}_{\mathcal{PT}}$. We find that sufficiently strong interaction strength can enhance the $\mathcal{PT}$-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 $\mathrm{\ensuremath{\Gamma}}$, can turn the system into the $\mathcal{PT}$-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.