The global phase diagram of the cluster-XY spin chain with dissipation

We study the ground-state phase diagram of a non-Hermitian cluster-XY spin chain in the language of free fermions. By calculating the second derivative of ground-state energy density and various types of order parameters, we establish the global ground-state phase diagram of the model, exhibiting rich quantum phases and corresponding phase transitions. Specially, the results reveal that the non-Hermitian cluster-XY model contains five different phases and two critical regions, i.e., ferromagnetic (FM), antiferromagnetic (AFM), symmetry-protected topological (SPT), paramagnetic (PM), Luttinger liquid-like phase, as well as critical region I and II. The order parameters and critical behaviors are investigated and the correctness of the theory is confirmed.