Stability mapping of bipartite tight-binding graphs with losses and gain: PTPT-symmetry and beyond

We consider bipartite tight-binding graphs composed by N nodes split into two sets of equal size: one set containing nodes with on-site loss, the other set having nodes with on-site gain. The nodes are connected randomly with probability p. Specifically, we measure the connectivity between the two sets with the parameter α, which is the ratio of current adjacent pairs over the total number of possible adjacent pairs between the sets. For general undirected-graph setups, the non-Hermitian Hamiltonian H(γ,α,N) of this model presents pseudo-Hermiticity, where γ is the loss/gain strength. However, we show that for a given graph setup H(γ,α,N) becomes PT-symmetric. In both scenarios (pseudo-Hermiticity and PT-symmetric), depending on the parameter combination, the spectra of H(γ,α,N) can be real even when it is non-Hermitian. Then we demonstrate, for both setups, that there is a well-defined sector of the γα-plane (which grows with N) where the spectrum of H(γ,α,N) is predominantly real.