Anderson-skin dualism: A boundary-dependent effect in non-Hermitian disordered coupled systems

TL;DR

We address a boundary-dependent localization phenomenon in non-Hermitian disordered coupled systems, termed Anderson-Skin dualism, where states inside a non-Hermitian point-gap loop are Anderson-localized under PBC but transform into skin modes under OBC. Using a minimal 1D Hatano-Nelson coupled to Aubry-André model, we derive Lyapunov exponents γ_L(E) and γ_R(E) via an asymmetric transfer-matrix method and corroborate with IPR and winding number ω to reveal boundary-controlled localization. The study extends to nonzero AA hopping, reciprocal systems, and higher dimensions (2D and 3D), showing AS dualism persists in localized regimes and can occur in reciprocal settings, with richer phenomena such as hybrid and corner skin states. These results reveal a universal boundary-controlled localization mechanism linked to point-gap topology, with potential realizations in photonic lattices, ultracold atoms, and topological circuits.

Abstract

We report a novel localization phenomenon that emerges in non-Hermitian and quasiperiodic coupled systems, which we dub ``Anderson-Skin (AS) dualism". The emergence of AS dualism is due to the fact that non-Hermitian topological systems provide non-trivial topological transport channels for disordered systems, causing the originally localized Anderson modes to transform into skin modes, i.e., the localized states within the point gap regions have dual characteristics of localization under periodic boundary condition (PBC) and skin effects under open boundary conditions (OBC). As an example, we analytically prove the 1D AS dualism through the transfer matrix method. Moreover, by discussing many-body interacting systems, we confirm that AS dualism is universal.

Figures (9)

• Figure 1: Schematic diagram of AS dualiam.
• Figure 2: (a) The IPR $\xi$ of all eigenstates in the complex energy plane. Right axis: The corresponding winding number $\omega$ versus $E=Re(E)$. (b) Eigenvalues under different boundary conditions. (c) The positions of center-of-mass for all eigenstates under different boundary conditions. (d) The left and right LEs versus $E$. Probability distributions of eigenstates inside (e) and outside (f) the point gap. Throughout, parameters $t=0.5$, $g=0.5$, $\lambda=2$, and $N=144$.
• Figure 3: Many-body IPRs of all eigenstates with PBCs (a) and OBCs (c), where the gray dots represent the eigenvalues under PBCs. The inset illustrates the sublattice correlation $C_{m}$ versus $Re(E)$. Particle distributions for $E=-0.7522$ (b1), $-2.9020+4.5682i$ (b2), $E=-2.8849+6.5858i$(b3), and $E=96.7366$ (b4). I-$\mu$ means that $\mu$ particles are occupied on the AA chain. The inset of (c) shows the OBC particle distribution for the eigenstates in cluster I and II, respectively. (d) The mean positions $\tilde{n}_c$ versus Re($E$). Thourghout, $g=2$, $\lambda=2$, $t=0.5$, $U=100$. $N=34$.
• Figure S1: Under the condition of $J_{\mathrm{rand}}=0$ [$J_{\mathrm{rand}}=1$], the IPR (a1) [(b1)] corresponding to the eigenvalues under PBC, as well as the comparison of eigenstates (a2) [(b2)] and the positions of the center-of-mass (a3) [(b3)] with different boundary conditions. Throughout, $W=4$, $t=0.5$, $J=1$, $g=0.5$, and $N=100$.
• Figure S2: Under the condition of $\lambda=0.2$ [$\lambda=2$], the IPR (a) [(d)] corresponding to the eigenvalues under PBC, as well as the comparison of eigenstates (b) [(e)] and the positions of the center-of-mass (c) [(f)] with different boundary conditions. Throughout, $t=0.5$, $J=1$, $g=0.5$, and $N=144$.