Edge-controlled non-Hermitian skin effect in the modified Haldane model

Abstract

The hybrid skin-topological effect (HSTE) arises from the interplay between the non-Hermitian skin modes and topologically protected edge states. Here, we investigate the HSTE associated with antichiral edge states in a modified Haldane nanoribbon with gain and loss applied exclusively at the zigzag edges. We show that in antichiral systems, the HSTE originates from an imbalance of effective gain and loss between edge states and counter-propagating bulk modes, revealing a mechanism distinct from that in conventional chiral systems. Remarkably, in sufficiently narrow ribbons, gain or loss applied to only one edge induces a skin effect in the states localized at the opposite edge, demonstrating a non-Hermitian nonlocal antichiral skin effect. We further show that edge-localized dissipation can induce bulk skin modes only when $\mathcal{PT}$ symmetry is broken, while the bulk non-Hermitian skin effect is strictly forbidden in the $\mathcal{PT}$-symmetric regime. By tuning the gain and loss applied solely at the edges, both the emergence and localization direction of bulk skin modes can be controlled. Our results establish a symmetry-based mechanism for controlling non-Hermitian skin effects via edge dissipation in antichiral systems.

Figures (11)

• Figure 1: (a) Direction of phase accumulation in the NNN hopping of the modified Haldane model. (b) Schematic illustration of antichiral edge states, which propagate in the same direction along opposite edges while counter-propagating modes reside in the bulk.
• Figure 2: Schematic of the modified Haldane nanoribbon with edge-localized gain and loss. Imaginary on-site potentials $i\gamma_1$ and $i\gamma_2$ are applied to the lower and upper zigzag edges, respectively. The dashed arrows indicate the direction of phase $\phi$ in the NNN hopping.
• Figure 3: Illustration of the $\mathcal{PT}$ symmetry of the modified Haldane model. The direction of phase accumulation in the NNN hopping term is unchanged under the combined $\mathcal{PT}$ operation, ensuring the intrinsic $\mathcal{PT}$ symmetry of the Hermitian part of the Hamiltonian. Here, $P$ and $T$ denote the inversion and time-reversal operators, respectively.
• Figure 4: Results for the $\mathcal{PT}$-symmetric case $H(\gamma_1=-0.6,\gamma_2=+0.6)$. (a) Schematic of the system. (b) Band structure of $\mathrm{Re}[E]$ under PBC along the $x$ direction. (c) Complex-energy spectrum under PBC. The red arrows indicate the direction of each loop as $k_x$ increases, and $w$ denotes the point-gap winding number. In (b) and (c), brighter colors indicate stronger localization at the upper and lower edges. The system size is $N_x=16000$. (d) Energy spectrum under OBC along $x$. (e) Spatial distribution of the summed lower-edge states enclosed by the dashed rectangle in (d). (f) Spatial distribution of the summed upper-edge states enclosed by the dashed rectangle in (d). (g) Spatial distribution of the summed bulk states on the real axis in (d). In (e)-(g), brighter colors indicate higher state density. The system size is $N_x=30$.
• Figure 5: Results for $H(\gamma_1=-0.6,\gamma_2=0)$. (a) Schematic of the system. (b) Band structure of $\mathrm{Re}[E]$ under PBC along the $x$ direction. In the main text, we focus on the bands $E_1$ and $E_2$. (c) Complex-energy spectrum under PBC. (d) Complex-energy spectrum of $E_1$ and $E_2$ under PBC. The red arrows indicate the direction of each loop as $k_x$ increases, and $w$ denotes the point-gap winding number. Brighter colors indicate stronger localization at the upper and lower edges. The eigenenergies enclosed by the blue dashed rectangle correspond to those enclosed in (b). The system size is $N_x=1600$. (e) Energy spectrum under OBC along $x$. Red dots indicate eigenenergies originating from $E_1$ and $E_2$ in (c). (f) Spatial distribution of the summed lower-edge states enclosed by the dashed rectangle in (e). (g) Spatial distribution of the summed upper-edge states enclosed by the dashed rectangle in (e). (h) Spatial distribution of the summed bulk states (red dots) enclosed by the dashed rectangle in (e). In (f)-(h), brighter colors indicate higher state density. The system size is $N_x=30$.