Anti- PT symmetry transition and scale-free localization in a non-Hermitian Su-Schrieffer-Heeger model with defects

We investigate the influence of on-site defects on topological properties, anti-$\mathcal{PT}$ symmetry, and scale-free localization in the non-Hermitian Su-Schrieffer-Heeger model. In the topologically nontrivial phase region, defects give rise to conventional localized states at the defect sites. Remarkably, in the topologically trivial phase region, defects generate a number of scale-free localized states, starkly contrasting with the absence of such states typically associated with the topologically nontrivial phase. Furthermore, we establish a numerical indicator to predict the emergence of scale-free localized states. This indicator relies on the convergence behavior of the eigenvalues. In addition, we prove the feasibility of constructing a non-Hermitian Su-Schrieffer-Heeger model utilizing cavity fields weakly coupled to engineered resonators. Our work deepens the understanding of anti-$\mathcal{PT}$ symmetry and provides insights into scale-free localization in dimerized non-Hermitian systems.