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Competing Localizations on Disordered Non-Hermitian Random Graph Lattice

S Rahul, A Harshitha

Abstract

Phase transitions in one-dimensional lattice systems are well established and have been extensively studied within both Hermitian and non-Hermitian frameworks. In this work, we extend this understanding to a more general setting by investigating localization and delocalization transitions and the behavior of the non-Hermitian skin effect (NHSE) using a tight-binding model on a generalized random graph lattice. Our model incorporates three key parameters, asymmetric hopping $Δ$, on-site disorder $W$, and a random long-range coupling $p$ that together define the underlying random graph structure. By varying $p$, $Δ$, and the disorder strength, we explore the interplay between topology, randomness, and non-Hermiticity in determining localization properties. Our results show a strong competition between skin effect driven and Anderson driven localizations across parameter regimes. Notably, even in the presence of strong disorder, skin effect driven localization coexists with Anderson-driven localization. We further discuss the relevance of these results to machine-learning architectures and information propagation in complex networks and other real-world problems.

Competing Localizations on Disordered Non-Hermitian Random Graph Lattice

Abstract

Phase transitions in one-dimensional lattice systems are well established and have been extensively studied within both Hermitian and non-Hermitian frameworks. In this work, we extend this understanding to a more general setting by investigating localization and delocalization transitions and the behavior of the non-Hermitian skin effect (NHSE) using a tight-binding model on a generalized random graph lattice. Our model incorporates three key parameters, asymmetric hopping , on-site disorder , and a random long-range coupling that together define the underlying random graph structure. By varying , , and the disorder strength, we explore the interplay between topology, randomness, and non-Hermiticity in determining localization properties. Our results show a strong competition between skin effect driven and Anderson driven localizations across parameter regimes. Notably, even in the presence of strong disorder, skin effect driven localization coexists with Anderson-driven localization. We further discuss the relevance of these results to machine-learning architectures and information propagation in complex networks and other real-world problems.

Paper Structure

This paper contains 6 sections, 6 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Model Hamiltonian
  3. Interplay of Skin Effect and Anderson Driven Localization Enhancement
  4. Relevance to the real world problems
  5. Conclusion
  6. Appendix

Figures (13)

  • Figure 1: Schematic representation of random graph lattice for the range of values of directed long-range connections $p.$
  • Figure 2: Phase diagram in the $(\Delta,W)$ plane represented in terms of the average IPR for $p=0$. The computation of average IPR is performed for the system size $N=300.$
  • Figure 3: Spatial distribution of eigenstates with respect to site index in the absence of directed long-range connections $p$. Plots (a)-(c) are for clean limit $W=0$. (d)-(f) are plotted for $W=1.0$ and (g)-(i) are plotted for $W=3.0$ respectively.
  • Figure 4: Spatial distribution of eigenstates with respect to site index in the absence of on-site disorder $W$. Plots (a)-(c) are for $p=0.1$. (d)-(f) are plotted for $p=0.5$ and (g)-(i) are plotted for $p=1.0$ respectively.
  • Figure 5: Average IPR is computed with respected to $p$ at $\Delta=1$ for different disorder strengths. The system size under consideration is $N=300.$
  • ...and 8 more figures