Impurity-induced loss bursts from anomalous scale-free localization in a non-Hermitian dissipative lattice

Abstract

We identify anomalous scale-free localization and associated impurity-induced loss bursts in a non-Hermitian dissipative cross-stitch lattice. By a local basis rotation, the model is mapped onto an effective non-Hermitian Su-Schrieffer-Heeger lattice, where local impurities act as tunable effective boundaries. For the parameter choice considered here, the impurity-free periodic-boundary-condition point is reached at $η=1$. Tuning $η$ away from this point drives the system through generalized-boundary-condition regimes toward two effective open-boundary-condition-like limits at $η\to0$ and $η\to\infty$. For finite $η\notin\{0,1\}$, the loop-sector spectra remain separated from the real-energy axis, while the corresponding eigenstates exhibit scale-free localization pinned by the impurity. Unlike conventional impurity-induced scale-free localization, the Lyapunov exponent depends explicitly on the eigenenergy, making the localization strength eigenstate dependent. We further show that this anomalous eigenmode structure produces an impurity-induced loss burst: The long-time integrated dissipation probability develops a pronounced local enhancement relative to the surrounding background near an impurity-generated effective boundary, even when the initial wave packet is prepared far away. In the single-impurity case, the burst region consists of the impurity site and its adjacent effective-boundary site, and the effect occurs without imaginary-gap closing. For multiple impurities, local burst regions emerge around all impurities, while the dominant burst boundary is selected by the initial wave-packet position and the nonreciprocal drift direction. These results connect anomalous scale-free localization with controllable dissipation dynamics in non-Hermitian lattices.