Directional Dynamics of the Non-Hermitian Skin Effect

TL;DR

This work addresses how information propagates in systems with the non-Hermitian skin effect (NHSE) by employing Quantum Liang information flow (QLIF), a directional measure of causal influence. Using a non-Hermitian SSH chain with non-reciprocal hopping, the authors quantify the directional asymmetry of information transfer and reveal a scissors-like pattern where the asymmetry grows linearly with small nonreciprocity, peaks at moderate localization, and then diminishes as skin localization becomes extreme. They identify a velocity ordering $v_{\text{eff}}(\gamma<0) > v_{\text{eff}}(0) > v_{\text{eff}}(\gamma>0)$ signaling NHSE-induced blocking, and show three temporal regimes: onset, stabilization, and coherent oscillations, with oscillation period set by intrinsic energy spacing. The results establish a quantitative link between static skin localization and dynamic information transfer, and suggest experimental tests in photonic lattices and topolectrical circuits, while opening paths to many-body and higher-order skin phenomena.

Abstract

The dynamical consequences of the non-Hermitian skin effect (NHSE) remain largely unexplored despite extensive studies of its static properties. Here we address this gap by applying quantum Liang information flow (QLIF) an inherently directional measure of causal influence to the nonHermitian Su Schrieffer Heeger model with non reciprocal hopping. Unlike symmetric correlation functions, QLIF directly captures the directional asymmetry characteristic of non reciprocal systems. We demonstrate a scissors effect where the asymmetry varies approximately linearly with the non-reciprocity parameter gamma for small gamma, and exhibits non-monotonic dependence on the skin length, with optimal asymmetry at moderate skin localization. The velocity ordering reveals NHSE-induced blocking of information flow against the skin direction. Three distinct temporal regimes emerge: light-cone-bounded spreading, gamma-dependent stabilization, and coherent oscillations. These results establish the first quantitative connection between static skin localization and directional information dynamics, offering new insights into information propagation in non-reciprocal quantum systems.

Figures (6)

• Figure 1: Freezing operation for QLIF. (a) Normal evolution: all couplings active. (b) Frozen-$B$ evolution: couplings to $B$ removed. The QLIF is the entropy difference between (a) and (b).
• Figure 2: Scissors effect in cumulative QLIF. Time evolution of $\mathbb{T}_{R \to L}$ (solid) and $\mathbb{T}_{L \to R}$ (dashed) for (a) $\gamma = 0$ (Hermitian, perfect overlap), (b) $\gamma = +0.3$, and (c) $\gamma = -0.3$. The shaded region highlights the directional asymmetry. Three temporal regimes are visible: (I) onset, (II) stabilization, (III) oscillations. Parameters: $L = 42$, $d = 6$, $t_1 = 1$, $t_2 = 0.5$.
• Figure 3: Directional asymmetry $\Delta_{\mathbb{T}}$ versus $\gamma$ at $t = 10$ (circles) and $t = 15$ (squares) across the full range $|\gamma| < t_1$. The sign rule $\mathrm{sgn}(\Delta_{\mathbb{T}}) = \mathrm{sgn}(\gamma)$ holds throughout. The non-monotonic behavior---peaking at moderate $|\gamma|$ and vanishing as $|\gamma| \to t_1$---reflects competition between symmetry breaking and extreme skin localization that confines dynamics to boundaries.
• Figure 4: Sublattice configuration effect on QLIF asymmetry. Solid lines: same-sublattice ($\alpha\alpha\alpha$, $\beta\beta\beta$) pass through the origin. Dashed lines: mixed-sublattice configurations show nonzero offset at $\gamma = 0$ due to structural asymmetry from bipartite hopping.
• Figure 5: QLIF asymmetry $\Delta_{\mathbb{T}}$ versus skin length $\xi$ for $\gamma > 0$ (red circles) and $\gamma < 0$ (blue squares). Black diamond: Hermitian limit. The non-monotonic dependence reflects competition between symmetry breaking (favoring small $\xi$) and signal suppression (when eigenstates are too boundary-localized). Three regimes are visible: strong localization (left), optimal asymmetry (center), and weak localization approaching the Hermitian limit (right).