Alternating quantum-emitter chains: Exceptional-point phase transition, edge state, and quantum walks

We study the long-range hopping limit of a one-dimensional array of $N$ equal-distanced quantum emitters in free space, where the hopping amplitude of emitter excitation is proportional to the inverse of the distance and equals the lattice dimension. For two species of emitters in an alternating arrangement, the single excitation sector exhibits non-Hermitian spectral singularities known as exceptional points. We unveil an unconventional phase transition, dubbed exceptional-point phase transition, from the collective to individual spontaneous emission behaviors. At the transition point, the $N \times N$ Hamiltonian fragments into $N/2-1$ many two-dimensional non-diagonalizable blocks. The remaining diagonalizable block contains a dissipation-induced edge state with algebraically localized profiles, and we provide numerical evidence for its existence in the infinite-array limit. We demonstrate that the edge state can be eliminated via a continuous deformation, consistent with the ill-definedness of bulk topological invariant. We also propose a spatially resolved character to quantify the incoherent flow and loss in the non-unitary quantum walks of single atomic excitations.