Universal competitive spectral scaling from the critical non-Hermitian skin effect
Fang Qin, Ye Ma, R. Shen, Ching Hua Lee
Abstract
Recently, it was discovered that certain non-Hermitian systems can exhibit qualitative different properties at different system sizes, such as being gapless at small sizes and having topological edge modes at large sizes $L$. This dramatic system-size sensitivity is known as the critical non-Hermitian skin effect (cNHSE), and occurs due to the competition between two or more non-Hermitian pumping channels. In this work, we rigorously develop the notion of a size-dependent generalized Brillouin zone (GBZ) in a general multicomponent cNHSE model ansatz, and found that the GBZ exhibits a universal $a+{b}^{1/(L+1)}$ scaling behavior. In particular, we provided analytical estimates of the scaling rate $b$ in terms of model parameters, and demonstrated their good empirical fit with two paradigmatic models, the coupled Hatano-Nelson model with offset, and the topologically coupled chain model with offset. We also provided analytic result for the critical size ${L}_{c}$, below which cNHSE scaling is frozen. The cNHSE represents the result of juxtaposing different channels for bulk-boundary correspondence breaking, and can be readily demonstrated in non-Hermitian metamaterials and circuit arrays.