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Mechanism for scale-free skin effect in one-dimensional systems

Shu-Xuan Wang

Abstract

Non-Hermitian skin effect (NHSE) is one of the most fascinating phenomena in non-Hermitian systems, which refers to enormous eigenstates localize at the boundary exponentially under open boundary condition (OBC). For typical NHSE, the localization length for a skin mode is independent of the system's size. Recently, some studies have revealed that for specific $1$-dimensional model, the localization length for eigenstates are proportional to the system's length under generalized boundary condition (GBC), and such phenomenon is dubbed as scale-free skin effect (SFSE). Further, SFSE is discovered in $1$-dimensional Hermitian chain with pure imaginary impurity at the end. In this work, we propose a mechanism for SFSE in 1-dimensional systems, which is model-independent. Our work provide a viewpoint for researching SFSE and shed new light on understanding finite size effect in non-Hermitian systems.

Mechanism for scale-free skin effect in one-dimensional systems

Abstract

Non-Hermitian skin effect (NHSE) is one of the most fascinating phenomena in non-Hermitian systems, which refers to enormous eigenstates localize at the boundary exponentially under open boundary condition (OBC). For typical NHSE, the localization length for a skin mode is independent of the system's size. Recently, some studies have revealed that for specific -dimensional model, the localization length for eigenstates are proportional to the system's length under generalized boundary condition (GBC), and such phenomenon is dubbed as scale-free skin effect (SFSE). Further, SFSE is discovered in -dimensional Hermitian chain with pure imaginary impurity at the end. In this work, we propose a mechanism for SFSE in 1-dimensional systems, which is model-independent. Our work provide a viewpoint for researching SFSE and shed new light on understanding finite size effect in non-Hermitian systems.

Paper Structure

This paper contains 10 sections, 47 equations, 2 figures.

Table of Contents

  1. Introduction
  2. framework
  3. examples
  4. Hatano-Nelson model with impurities of boundary couplings
  5. Hatano Nelson model with onsite boundary impurity
  6. Discussion and Conclusion
  7. Acknowledgments
  8. Proof of Eq.\ref{['8']}
  9. Demonstration for Eqs.\ref{['15']}, \ref{['16']}, \ref{['18']} and \ref{['19']}
  10. Theoretical formula of the expection $\langle x \rangle/L$ under thermodynamic limit for skin and scale-free skin mode

Figures (2)

  • Figure 1: Mean positions for eigenstates of HN model with impurities given in Eqs.\ref{['13']} and \ref{['14']}. For (a)-(f) $\mu= 0,-1,-\frac{1}{4},\frac{1}{4},-\frac{1}{2}$ and $\frac{1}{2}$ respectively. The index $n$ about eigenstate is arranged in the ascending order of the eigenvalue real part. The red, orange and green dots correspond to numerical results for $L = 75,100$ and $125$ respectively. The blue curves are the theoretical curves predicted by our theory, and concrete expression for these curves are given in appendix C. Other parameters are $t_r = 2 t_l =2$.
  • Figure 2: Mean positions for eigenstates of HN model with impurities given in Eqs.\ref{['13']} and \ref{['17']}. $V = \frac{-1}{2}$ and $1$ for (a) and (b). The index $n$ about eigenstate is arranged in the ascending order of the eigenvalue real part. The red, orange and green dots correspond to numerical results for $L = 75,100$ and $125$ respectively. The blue curves are the theoretical curves predicted by our theory. Other parameters are $t_r = 2 t_l =2$.