Erratic non-Hermitian skin localization

Abstract

A novel localization phenomenon, termed erratic non-Hermitian skin localization, has been identified in disordered globally-reciprocal non-Hermitian lattices. Unlike conventional non-Hermitian skin effect and Anderson localization, it features macroscopic eigenstate localization at irregular, disorder-dependent positions with sub-exponential decay. Using the Hatano-Nelson model with disordered imaginary gauge fields as a case study, this effect is linked to stochastic interfaces governed by the universal order statistics of random walks. Finite-size scaling analysis confirms the localized nature of the eigenstates. This discovery challenges conventional wave localization paradigms, offering new avenues for understanding and controlling localization phenomena in non-Hermitian physics.

Figures (5)

• Figure 1: Schematic of three different types of localization in a one-dimensional NH disordered lattice. The three panels depict typical shapes of four eigenstates $|\psi_n|$ in different color. (a) NH skin localization: all eigenstates are exponentially localized at the lattice edges. (b) Anderson localization: the eigenstates are exponentially localized and their locations are uniformly distributed along the lattice. (c) Erratic skin localization: all eigenstates are localized at around the same position in the lattice, with possible far apart satellite peaks. The localization is lower than exponential and the position of the main and satellite peaks is erratic, i.e. strongly dependent on the realization of disorder.
• Figure 2: (a) Schematic of the Hatano-Nelson model with disordered imaginary gauge field. (b) Energy spectrum of the Hatano-Neslon model under PBC (blue circles) and OBC (red circles) for a Bernoulli distribution $f(h)$ with $\Delta h=0.4$. Lattice size $N=1000$. (c) Behavior of the eigenstate distribution $I_n$ under PBC for four different realizations of the stochastic sequence $\{ h_n\}$, indicated by the four different colors.
• Figure 3: (a) Probability density function of the IPR in a lattice of size $N=2000$ with PBC and a disordered gauge field with a Bernoulli distribution ($\Delta h=0.4$). The probability distribution has been numerically computed by considering $10^5$ realizations of the sequence $\{ h_n \}$. (c) Behavior of the mean value $\overline{\rm{IPR}}$ of the IPR distribution versus lattice size (red circles) on a log scale. The dashed curve is the theoretical prediction based on Eq.(5). The blue circles show, for comparison, the IPR behavior in a disorder-free lattice ($h_n=0$). $\beta$ is the corresponding fractal dimension.
• Figure 4: (a) Schematic of the random walk $X_n=\sum_{k=1}^{n-1} h_l$ defined by the stochastic sequence $\{ h_k \}$ of the gauge field. Note that for the specific walk realization there are two relative maxima of $X_k$, at $k=n_1$ and $k=3$, which define two interfaces around which the gauge field $h$ locally displays opposite signs. (b) Schematic of the eigenfunction $|\psi_k|$, displaying a main peak and a satellite peak at the two interfaces. (c) Ordering procedure used to compute the IPR; $\delta_k$ are the gaps between adjacent ordered values of $X_k$.
• Figure 5: (a) Schematic of the NH Rice-Mele model displaying the reciprocal NHSE. (b) Energy spectrum under PBC (blue circles) and OBC (red points) in the disorder-free lattice for $t_1=1$, $t_2=1.5$, $\Delta=2$, $\gamma=0.5$ and $\lambda=1$. (c) Same as (b), but in the disordered lattice, where each amplitude in the sequence $\{ \lambda_n \}$ can take only the two values $\pm 1$ with equal probability (Bernoulli distribution). (d) Behavior of the mean IPR versus system size $N$ on a log scale (blue circles); a statistical average over 100 different realizations of the sequence $\{ \lambda_n \}$ has been assumed. For comparison, the red circles show the behavior IPR$=1/N$ of a delocalized phase. (e) Behavior of the eigenstate distribution $I_n$ for three different realizations of the stochastic sequence $\{ \lambda_n\}$ (lattice size $N=600$).