Non-Hermitian skin effect in periodic, random, and quasiperiodic systems

Abstract

The non-Hermitian skin effect (NHSE), which drives bulk states toward system boundaries, modifies bulk-boundary correspondence and complicates the identification of topological edge modes. Although breaking translational symmetry is known to influence this behavior, a systematic comparison of different structural classes remains limited. Here we investigate periodic, random, and quasiperiodic (Fibonacci) systems using a one-dimensional non-Hermitian quantum walk model. By matching the local scattering parameters in a topologically nontrivial regime, we isolate the role of spatial structure in the presence of the NHSE. We find that periodic systems exhibit pronounced boundary accumulation of bulk states. Random systems suppress this accumulation through Anderson localization, but the topological gap becomes partially filled with localized in-gap states. In contrast, the Fibonacci quasiperiodic system suppresses large-scale boundary accumulation while maintaining a well-defined topological gap. Analysis of the wave functions suggests that the hierarchical quasiperiodic structure fragments bulk states across multiple length scales, thereby mitigating the NHSE. These results identify deterministic quasiperiodicity as a distinct structural regime for controlling non-Hermitian skin dynamics and isolating topological boundary modes.

Figures (5)

• Figure 1: Schematic one-dimensional site sequences used in the three systems. The system size is fixed at $N=89$. White and gray blocks represent A and B sites, respectively, with coin parameters $\theta_\mathrm{A}=0.1\pi$ and $\theta_\mathrm{B}=0.51\pi$. (a) Periodic, (b) Random, (c) Fibonacci quasiperiodic systems. (d) Enlarged view of the left and right boundaries of the Fibonacci sequence. The parameters $g$ and $\gamma$ represent non-Hermitian pumping toward the left and right directions, respectively.
• Figure 2: Quasienergy spectra of periodic, random, and Fibonacci quasiperiodic systems in the Hermitian limit ($g=0$, $\gamma=0$). (a) Periodic, (b) random, and (c) Fibonacci quasiperiodic spectra. Eigenvalues are plotted as a function of the state index. The isolated zero mode ($E=0$) and $\pi$ mode ($E=\pi$) are indicated by open squares and open circles, respectively.
• Figure 3: (a) Average center of mass (COM) and (b) inverse participation ratio (IPR) of the bulk states as functions of the nonreciprocal hopping parameter $g$. Circles, squares, and triangles represent the periodic, random, and Fibonacci quasiperiodic systems, respectively. The COM is measured in units of the lattice site index $x$. The averages are taken over bulk eigenstates after excluding a few edge-localized states near both boundaries.
• Figure 4: Complex eigenvalue spectra of the three systems in the non-Hermitian regime ($g=0.4$, $\gamma=0.5$). (a) Periodic, (b) random, and (c) Fibonacci quasiperiodic systems. The two points on the real axis correspond to the $\pi$ (open circle) and zero modes (open square). The $\pi$ mode lies outside the unit circle, indicating strong amplification.
• Figure 5: Spatial distribution of the total bulk density $\rho_\mathrm{bulk}(x)$ under non-Hermitian pumping ($\gamma=0.5$). (a) Periodic, (b) random, and (c) Fibonacci quasiperiodic systems. Solid and dashed curves correspond to $g=0.4$ and $g=0.6$, respectively. The gray shaded area indicate the boundary areas excluded from the bulk region. Insets show the logarithmic plots of $\rho_\mathrm{bulk}(x)$ down to $10^{-5}$. $\rho_\mathrm{bulk}(x)$ is obtained by summing over eigenstates after excluding a few edge-localized states near both boundaries, identified by their center of mass within the two sites from each edge.