$\mathcal{P}\mathcal{T}$-symmetry phase transition in a bipartite lattice with long-range interactions

We investigate the parity–time ($\mathcal{P}\mathcal{T}$) symmetry-breaking quantum phase transition in a one-dimensional (1D) bosonic lattice featuring cavity-mediated long-range interactions and spatially staggered dissipation. By mapping the system to an effective spin chain under the constraints of hard-core bosons and integrating the mean-field decoupling approach with biorthogonal basis formalism, we derive a self-consistency equation. Numerical simulation results validate that the derived equation quantitatively captures the $\mathcal{P}\mathcal{T}$-symmetry order parameter’s phase diagram. Our findings reveal that coherent hopping maintains $\mathcal{P}\mathcal{T}$ symmetry through quantum fluctuations. Conversely, cavity-engineered long-range interactions, in synergy with staggered dissipation, act in opposition to drive symmetry breaking. This competitive interplay can inspire further exploration of tunable quantum phase transitions in non-Hermitian systems.