Eigenfunction Delocalization Transition in Amir-Hatano-Nelson Model of Random Asymmetric Ring Neural Networks: A Topological Field Theory Approach

Random matrix theory serves as a fundamental tool in the analysis of random dynamical systems, particularly those involving non-symmetric interactions leading to complex spectra. In some random non-Hermitian systems, there can be the interplay between eigenfunction delocalization due to strong asymmetric interactions and the standard Anderson localization due to on-site disorder. In this work, we investigate such infterplay in Amir-Hatano-Nelson (AHN) model which describes the linearized dynamics of asymmetric ring recurrent neural networks obeying Dale's principles, and show that the effective action of this system is described by a topological field theory (TFT). As the system progresses from a delocalized phase to a localized phase, the inverse participation ratio abruptly increases from the minimum value. We show that the localization-delocalization transition of neural activities in the AHN model is determined by the change in the Chern-Simons coupling of the effective TFT description, suggesting the topological origin of such transition.