Directional Dynamics of the Non-Hermitian Skin Effect

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.

💡 Research Summary

This paper addresses the largely unexplored dynamical aspects of the non‑Hermitian skin effect (NHSE) by introducing a directional causal measure—quantum Liang information flow (QLIF)—and applying it to a non‑Hermitian Su‑Schrieffer‑Heeger (SSH) chain with non‑reciprocal intracell hopping t₁ ± γ. The model exhibits the hallmark NHSE: for γ > 0 all eigenstates accumulate at the left boundary, while for γ < 0 they accumulate at the right. The skin length ξ = 1/|ln r| (with r = |(t₁ − γ)/(t₁ + γ)|) quantifies the exponential localization and diverges in the Hermitian limit.

QLIF is defined by comparing the von‑Neumann entropy of a subsystem A under the full Hamiltonian H with that under a “frozen‑B” Hamiltonian H_{/B} where all couplings to subsystem B are removed. The cumulative flow T_{B→A}(t)=S_A(t)−S_A^{/B}(t) is inherently directional (T_{B→A}≠T_{A→B}), making it ideal for probing non‑reciprocal systems where ordinary two‑point correlation functions are symmetric. The authors initialize a single particle at the chain centre and place observation sites A (left) and B (right) at equal distance d, measuring the asymmetry ΔT = T_{R→L}−T_{L→R}.

The first major finding is the “scissors effect”: for γ = 0 the left‑right QLIF curves overlap perfectly, but any non‑zero γ splits them, with the sign of ΔT matching the sign of γ. For small |γ| the dependence is approximately linear (ΔT ≈ k γ). As |γ| increases, ΔT reaches a maximum around |γ| ≈ 0.15–0.3 and then diminishes toward zero as |γ|→t₁, where one hopping direction is completely suppressed and the skin length collapses to a few sites. Consequently, the strongest NHSE paradoxically yields the weakest measurable directional asymmetry—a clear experimental prediction.

Because the SSH chain is bipartite, the choice of sublattice for the initial state and the observation sites matters. Configurations where all three sites belong to the same sublattice (ααα or βββ) respect spatial inversion symmetry at γ = 0, giving ΔT = 0, and obey ΔT_{ααα}(γ)=−ΔT_{βββ}(−γ). Mixed‑sublattice configurations exhibit a finite offset even at γ = 0, reflecting structural left‑right bias unrelated to non‑Hermiticity; the authors therefore focus on same‑sublattice setups for the genuine NHSE analysis.

Temporal evolution reveals three distinct regimes. (I) An early “onset” stage (t ≲ 4) where QLIF is negligible until the information front reaches the observation sites, bounded by a light‑cone reminiscent of the Lieb‑Robinson bound. (II) An intermediate quasi‑steady regime (4 ≲ t ≲ 10) where the γ‑dependent asymmetry is most pronounced; the time‑integrated flow obeys sgn(∫T) ≈ −sgn(γ). (III) A late‑time regime (t ≳ 10) characterized by coherent oscillations with period T_osc ≈ 2π/ΔE set by the intrinsic level spacing; these oscillations persist because skin‑localized eigenstates retain high local probability density.

The authors also extract an effective propagation velocity by defining an onset time t* (the earliest time at which |T_{j0→j0+d}| exceeds a small threshold). Plotting t* versus distance d shows linear scaling for short distances (d ≲ 10) with slopes between the maximal group velocity v_max = 2 min(t₁,t₂)=1 and the Lieb‑Robinson bound v_LR = 2 max(t₁,t₂)=2. However, for larger distances the curve for γ > 0 bends upward, indicating a dramatic NHSE‑induced blocking: information traveling against the skin direction is exponentially suppressed as ∼e^{−d/ξ}. The resulting velocity ordering v_eff(γ < 0) > v_eff(0) > v_eff(γ > 0) directly visualizes the NHSE blocking effect.

Finally, the dependence of ΔT on the skin length ξ is examined. ΔT exhibits a non‑monotonic behavior, peaking at ξ_opt ≈ 3–4 lattice spacings. For ξ ≪ d the eigenstates are confined to the boundary and bulk signals cannot reach the observation sites, suppressing ΔT. For ξ ≫ d the system approaches the Hermitian limit and the asymmetry vanishes. The optimal intermediate ξ balances strong non‑reciprocal symmetry breaking with sufficient bulk transport, providing a practical guideline: moderate non‑Hermiticity maximizes observable directional effects. The two branches (γ > 0 and γ < 0) show slightly different peak positions, reflecting the interplay between skin direction and measurement geometry.

In summary, the paper establishes the first quantitative link between static skin localization and directional information dynamics. By employing QLIF, it captures the “scissors” asymmetry, the non‑monotonic optimality with respect to γ and ξ, the three‑stage temporal evolution, and the NHSE‑induced blocking of information flow. These insights not only deepen our theoretical understanding of non‑Hermitian many‑body dynamics but also offer concrete experimental targets for photonic lattices, topo‑electrical circuits, quantum walks, and ultracold‑atom platforms where the NHSE can be engineered and probed.