Topological triple phase transition in non-Hermitian quasicrystals with complex asymmetric hopping

The triple phase transitions or simultaneous of three different phases, namely, topological, parity-time $(\mathcal{P}\mathcal{T})$ symmetry breaking, and metal-insulator transitions, are observed in an extension the $\mathcal{P}\mathcal{T}$ symmetric non-Hermitian Aubry-Andr\'e-Harper model. In this model, besides complex quasiperiodic on-site potential, non-Hermiticity is also included nearest-neighbor hopping terms. Moreover, terms quasiperiodic. presence two parameters, one from potential another part, ensures transition system. addition, tuning these we identify a parameters regime, where observe transition. Following some recent studies, electrical circuit based experimental realization model discussed.