Directional dynamics of the non-Hermitian skin effect

Abstract

The dynamical consequences of the non-Hermitian skin effect (NHSE), while increasingly studied, have so far received less systematic attention than its now well-characterized static properties. Here we contribute to this developing front by applying quantum Liang information flow (QLIF) — an inherently directional measure of causal influence — to the non-Hermitian Su−Schrieffer−Heeger model with non-reciprocal hopping. Unlike symmetric correlation functions, QLIF directly captures the directional asymmetry {\mathbb{T}}_{R\to L}\ne {\mathbb{T}}_{L\to R} characteristic of non-reciprocal systems. We demonstrate a “scissors effect” where the asymmetry {\mathrm{\Delta }}_{\mathbb{T}} varies approximately linearly with the non-reciprocity parameter \gamma for small |\gamma |, and exhibits non-monotonic dependence on the skin length \xi, 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.

Graphical abstract

Keywords

quantum information / non-Hermitian system / quantum causal relation

1 Introduction

The non-Hermitian skin effect (NHSE) — the dramatic accumulation of all eigenstates at system boundaries under open boundary conditions — has emerged as a defining phenomenon in non-Hermitian physics [1–11]. Experimental realizations now span photonic lattices [12, 13], topolectrical circuits [14, 15], quantum walks [16, 17], and ultracold atoms [18]. While the relationship between non-reciprocal couplings and skin accumulation direction is well established in one-dimensional models [6, 10, 11] and has been extended to multilayer settings [19], recent works have shown that this correspondence can be nontrivially modified or even reversed in more general settings [20–22]. Much of this body of work has emphasized static properties — spectra, topological invariants, and eigenstate distributions [2, 23–30] — while dynamical aspects have only more recently received systematic attention [31–41]. A central open question is how the non-reciprocity parameter \gamma affects information propagation.

Traditional dynamical probes are inadequate for this task. Time-dependent correlation functions ⟨{\sigma }_i(t){\sigma }_j(0)⟩ are inherently symmetric and cannot capture directional bias [7, 32]. The Lieb−Robinson bound [42, 43], which constrains information spreading in Hermitian systems, breaks down in the non-Hermitian case [44] — as demonstrated by supersonic modes observed on trapped-ion quantum computers [45]. The quasiparticle picture [46] also faces challenges with complex spectra and biorthogonal eigenstates [8, 47]. What is needed is a measure that directly quantifies directional asymmetry in information flow.

Quantum Liang information flow (QLIF) [48–50] provides precisely such a measure. QLIF quantifies the causal influence of one subsystem on another by comparing the entropy evolution with and without the coupling to a frozen subsystem [51, 52]. Unlike correlations, QLIF is inherently directional ({\mathbb{T}}_{B\to A}\ne {\mathbb{T}}_{A\to B}), making it ideally suited for non-reciprocal systems. Here we apply QLIF to the non-Hermitian SSH model [53] with non-reciprocal hopping t_1\pm \gamma. We discover: (i) a “scissors effect” where {\mathrm{\Delta }}_{\mathbb{T}}={\mathbb{T}}_{R\to L}-{\mathbb{T}}_{L\to R} varies linearly with \gamma for small \gamma; (ii) non-monotonic dependence on skin length \xi; (iii) velocity ordering revealing NHSE-induced blocking; and (iv) three distinct temporal regimes. These results establish the first quantitative link between skin localization and directional information dynamics.

2 Model

We consider a non-Hermitian Su−Schrieffer−Heeger (SSH) chain with N unit cells under open boundary conditions [6]:

(1)H={\displaystyle \sum _{j=1}^N{\mathbf{c}}_j^{†}{\mathcal{T}}_0{\mathbf{c}}_j+{\displaystyle \sum _{j=1}^{N-1}\left({\mathbf{c}}_{j+1}^{†}{\mathcal{T}}_{+}{\mathbf{c}}_j+H.c.\right),}}

where {\mathbf{c}}_j=(c_{\alpha ,j},c_{\beta ,j}{)}^{\mathrm{T}} and

(2){\mathcal{T}}_0=\left(\begin{array}{cc}0& t_1+\gamma \\ t_1-\gamma & 0\end{array}\right),{\mathcal{T}}_{+}=\left(\begin{array}{cc}0& 0\\ t_2& 0\end{array}\right).

Here t_1 (t_2) is the intracell (intercell) hopping, and \gamma parametrizes non-reciprocity [1]. The non-Hermitian SSH model is the canonical platform for studying the NHSE: Yao and Wang [6] established the modern theory of skin localization in this model, and it has since been realized experimentally in photonic lattices [12], topolectrical circuits [14, 15], and ultracold atoms [18]. Its bipartite structure additionally enables sublattice-resolved QLIF measurements that reveal measurement-geometry effects absent in single-band models (see Fig. 4).

For \gamma \ne 0, the non-Hermitian skin effect (NHSE) causes all eigenstates to localize exponentially at one boundary [6, 8]. The eigenstate envelope |{\psi }_n|\sim r^n is governed by the skin parameter

(3)r=\sqrt{\left|{\displaystyle \frac{t_1-\gamma }{t_1+\gamma }}\right|},

and the associated skin length

(4)\xi ={\displaystyle \frac{1}{|\ln r|}},

which measures the localization length in units of lattice sites. \gamma >0 () gives left-boundary localization; (r>1) gives right-boundary localization. In the Hermitian limit \gamma \to 0, r\to 1 and \xi \to \mathrm{\infty }.

We set t_1=1, t_2=0.5, and L=42 sites, scanning \gamma up to |\gamma |=0.4 (the physically allowed range is ). Details on the bulk spectrum and light-cone velocity are in the Electronic Supplementary Materials.

3 Quantum Liang information flow

The quantum Liang information flow (QLIF) [48–50, 52] quantifies the causal influence of subsystem B on A through a freezing operation. Let S_A(t)=-\mathrm{t}\mathrm{r}[{\rho }_A\ln {\rho }_A] be the von Neumann entropy of A, and H_{\text{⧸}B} a modified Hamiltonian with all couplings to B removed. The instantaneous QLIF is defined as [48, 49]

(5)T_{B\to A}(t)={\displaystyle \frac{\mathrm{d}{S}_A}{\mathrm{d}t}}-{\displaystyle \frac{\mathrm{d}{S}_{A\text{⧸}B}}{\mathrm{d}t}},

where S_{A\text{⧸}B} is the entropy of A under frozen-B evolution (Fig. 1). Since both evolutions share the same initial state, S_A(0)=S_{A\text{⧸}B}(0), integrating Eq. (5) over time yields the equivalent cumulative form [48, 52]:

(6){\mathbb{T}}_{B\to A}(t)={\displaystyle {\int }_0^t{T}_{B\to A}(t^{\prime })\mathrm{d}{t}^{\prime }=S_A(t)-S_{A\text{⧸}B}(t),}

which directly measures the total causal influence accumulated up to time t. We adopt this cumulative form throughout, as it avoids numerical differentiation and provides a more robust measure for finite-time dynamics. Unlike correlation functions, QLIF is inherently directional: {\mathbb{T}}_{B\to A}\ne {\mathbb{T}}_{A\to B} in general.

Since the non-Hermitian time evolution |\psi (t)⟩={\mathrm{e}}^{-\mathrm{i}Ht}|{\psi }_0⟩ does not preserve the norm, we employ real-time normalization throughout

(7)\rho (t)={\displaystyle \frac{|\psi (t)⟩⟨\psi (t)|}{⟨\psi (t)|\psi (t)⟩}},

which ensures \mathrm{t}\mathrm{r}[\rho ]=1 and positive semidefiniteness at all times. The reduced density matrix {\rho }_A and all entropy quantities are computed from this normalized state. We note that an alternative biorthogonal density matrix {\rho }_{\mathrm{b}\mathrm{i}}=|{\psi }_R⟩⟨{\psi }_L|/⟨{\psi }_L|{\psi }_R⟩, constructed from both right- and left-evolving states, has been considered in the literature [55]. However, {\rho }_{\mathrm{b}\mathrm{i}} is not Hermitian in general and therefore the von Neumann entropy is not well-defined for it. Our choice of the right-state normalized density matrix in Eq. (7) is the standard framework for computing entanglement entropy in non-Hermitian systems [51].

The initial excitation is placed at the central unit cell j_0=\lceil N/2\rceil (N=L/2), with observation sites A (left) and B (right) at equal distance d on opposite sides. For the scissors, sublattice, and skin-length analyses (Figs. 2–5), all three sites lie on the \alpha sublattice (\alpha \alpha \alpha configuration), ensuring {\mathrm{\Delta }}_{\mathbb{T}}=0 at \gamma =0 by inversion symmetry. For the light-cone analysis (Fig. 6), the initial excitation is on the \beta sublattice; this does not affect the bulk dynamics, as confirmed in Fig. 4. Specific parameters are stated in each figure caption; implementation details are in the Supplemental Material.

4 Results

4.1 Scissors effect

In the Hermitian limit (\gamma =0), spatial inversion symmetry ensures {\mathbb{T}}_{R\to L}(t)={\mathbb{T}}_{L\to R}(t); Fig. 2(a) confirms this with deviations below machine precision (). For \gamma \ne 0, this symmetry is broken: Figs. 2(b) and (c) display the “scissors” pattern where the directional asymmetry {\mathrm{\Delta }}_{\mathbb{T}}={\mathbb{T}}_{\mathrm{R}\to \mathrm{L}}-{\mathbb{T}}_{\mathrm{L}\to \mathrm{R}} is positive for \gamma >0 (left-boundary skin localization enhances right-to-left flow) and negative for . At intermediate times (t\approx 10), the cumulative QLIF can be either positive or negative, depending on the interplay between wavepacket spreading and interference effects.

To systematically characterize this dependence, we scan |\gamma |\le 0.4 and extract {\mathrm{\Delta }}_{\mathbb{T}} at fixed times. Figure 3 reveals a key result: the sign rule

(8)\mathrm{s}\mathrm{g}\mathrm{n}({\mathrm{\Delta }}_{\mathbb{T}})=\mathrm{s}\mathrm{g}\mathrm{n}(\gamma )

holds throughout the entire range, directly addressing the open question of “how the behavior of NHSE is influenced by the variation of nonreciprocity” [40]. Crucially, the asymmetry exhibits non-monotonic \gamma-dependence: |{\mathrm{\Delta }}_{\mathbb{T}}| increases approximately linearly for small |\gamma |, peaks at moderate values (|\gamma |\approx 0.15–0.3), then decreases toward zero as |\gamma |\to t_1. The suppression at large |\gamma | has a clear microscopic origin: the intracell hopping amplitudes are t_1+\gamma (\alpha \to \beta) and t_1-\gamma (\beta \to \alpha). As \gamma \to t_1, one direction becomes completely suppressed (t_1-\gamma \to 0) while the other doubles (t_1+\gamma \to 2t_1), making hopping perfectly unidirectional. The skin length \xi \to 0 [Eq. (4)] in this limit, so that the eigenstate amplitude ratio between the two observation sites, r^{2d}={\mathrm{e}}^{-2d/\xi }, diverges — one QLIF channel is exponentially suppressed relative to the other. However, because both channels vanish when the reverse hopping t_1-\gamma \to 0 shuts down, their difference {\mathrm{\Delta }}_{\mathbb{T}} also vanishes. Paradoxically, the strongest NHSE produces the weakest measurable directional asymmetry — a key prediction for experiments seeking optimal operating regimes. The numerical convergence of {\mathrm{\Delta }}_{\mathbb{T}} with respect to time step, system size, and freezing-operation precision is documented in the Electronic Supplementary Materials B.6, where the residual uncertainty on the displayed values is bounded by \sim 2\times {10}^{-9}.

4.2 Sublattice configuration

Figure 4 compares five sublattice configurations. Same-sublattice configurations (\alpha \alpha \alpha, \beta \beta \beta; solid lines) pass through the origin (\gamma ,{\mathrm{\Delta }}_{\mathbb{T}})=(0,0) as required by inversion symmetry, and satisfy {\mathrm{\Delta }}_{\mathbb{T}}^{\beta \beta \beta }(\gamma )={\mathrm{\Delta }}_{\mathbb{T}}^{\alpha \alpha \alpha }(-\gamma ), reflecting the sublattice exchange symmetry \alpha \leftrightarrow \beta combined with \gamma \to -\gamma. Mixed-sublattice configurations (dashed lines) exhibit nonzero {\mathrm{\Delta }}_{\mathbb{T}} at \gamma =0 — a structural asymmetry arising from the inequivalent intracell and intercell hopping pathways. Despite these offsets, all curves show \mathrm{d}{\mathrm{\Delta }}_{\mathbb{T}}/\mathrm{d}\gamma >0 for small |\gamma |: a universal trend reflecting non-reciprocal hopping. All subsequent results use the \alpha \alpha \alpha configuration to isolate the genuine NHSE signature.

4.3 Three temporal regimes

The time evolution exhibits three distinct regimes (Fig. 2): (i) At early times (t\lesssim 4), QLIF remains negligible until information propagates to observation sites; for \gamma \ne 0, the NHSE creates asymmetric onset times. (ii) After the onset transient (4\lesssim t\lesssim 10), QLIF enters a quasi-stationary regime where \gamma-dependent asymmetry is most clearly resolved. (iii) At late times (t\gtrsim 10), persistent oscillations appear with period T_{\text{osc}}\approx 2\pi /\mathrm{\Delta }E determined by intrinsic energy-level spacing, independent of system size (see the Electronic Supplementary Materials [54]). The NHSE protects oscillation amplitude because skin-localized eigenstates maintain high local probability density. All three regimes and their quantitative boundaries are L-independent for L=20–80, confirming these as bulk properties [54].

4.4 Skin length dependence

While \gamma directly enters the Hamiltonian, the physically relevant quantity controlling eigenstate localization is the skin length \xi [Eq. (4)]. Recasting the \gamma-scan of Fig. 3 in terms of \xi (Fig. 5) identifies a characteristic scale: the QLIF asymmetry |{\mathrm{\Delta }}_{\mathbb{T}}| peaks at {\xi }_{\text{opt}}\approx d, comparable to the observation distance, delineating three regimes. In the strong localization regime (\xi \ll d), the controlling quantity is the relative probability ratio r^{4d}={\mathrm{e}}^{-4d/\xi } between the two observation sites; the state becomes so directionally polarized that |{\mathrm{\Delta }}_{\mathbb{T}}| vanishes. In the optimal regime (\xi \sim d), symmetry breaking is strong while bulk transport remains viable, producing maximal asymmetry. In the weak localization regime (\xi \gg d), the system approaches the Hermitian limit. The \gamma >0 and branches peak at different \xi values, reflecting the interplay between skin direction and measurement geometry; this branch asymmetry and the system-size independence of {\xi }_{\text{opt}} are confirmed in the Electronic Supplementary Materials.

4.5 Mechanism summary

The behavior reported above can be traced through a chain linking the microscopic non-reciprocity \gamma to the QLIF asymmetry {\mathrm{\Delta }}_{\mathbb{T}}. The first three steps are exact algebraic relations, verified to machine precision (see the Electronic Supplementary Materials A.6): (i) non-reciprocal intracell hopping t_1\pm \gamma fixes the amplitude ratio r [Eq. (3)]; (ii) r sets the skin length \xi =1/|\ln r| [Eq. (4)] and the eigenstate envelope |{\psi }_n|\sim r^n; (iii) the similarity transformation translates this envelope into a propagator amplitude bias r^{\pm d} between opposite directions [Eq. (9) below]. The final step (iv) is dynamical: the bias enters the two opposite-direction QLIF channels asymmetrically, and the resulting {\mathrm{\Delta }}_{\mathbb{T}} is a protocol response — a functional of the freezing geometry, observation time and initial state on the same footing as the system parameters. This protocol-level character is what makes {\mathrm{\Delta }}_{\mathbb{T}} a dynamical witness of the NHSE, complementary to static localisation measures, and it underlies the qualitative features of Fig. 5 (the peak at \xi \sim d, the sign rule, and the multi-d behaviour).

4.6 NHSE blocking and velocity ordering

To probe the causal structure of information propagation, we measure the onset time t^{\ast } — the earliest time at which |{\mathbb{T}}_{j_0\to j_0\pm d}|>ϵ (ϵ={10}^{-6}) — as a function of distance d for both rightward (j_0+d) and leftward (j_0-d) propagation. The frozen site is placed at the initial excitation j_0, so that QLIF directly measures causal influence from the source.

The light-cone structure is governed by the similarity transformation H_{\text{NH}}=S^{-1}{H}_{\text{eff}}S (see the Electronic Supplementary Materials), which maps the non-Hermitian Hamiltonian to a Hermitian SSH chain with renormalized intracell hopping {\tilde{t}}_1=\sqrt{t_1^2-{\gamma }^2} and unchanged t_2. This has two consequences. First, the wavefront speed v_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\text{GBZ}}=2min({\tilde{t}}_1,t_2)=2t_2=1.0 is direction-independent and \gamma-independent for our parameters. Second, the signal amplitude is direction-dependent: the propagators satisfy

(9)G_{\text{NH}}(j_0\pm d,j_0;t)=r^{\pm d}{G}_{\text{eff}}(j_0\pm d,j_0;t),

where d is the unit-cell displacement and for \gamma >0. The non-Hermitian amplitudes in opposite directions therefore differ by a factor r^{-2d}={\mathrm{e}}^{2d/\xi } (leftward amplified, rightward suppressed for \gamma >0). The wavefront arrives simultaneously, but the skin-amplified direction crosses the detection threshold ϵ earlier, producing a larger apparent velocity.

This mechanism is confirmed by our two-directional analysis (Fig. 6): the velocity ordering reverses between directions,

(10)

with v_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\text{right}}(\gamma )\approx v_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\text{left}}(-\gamma ) as required by the \gamma \to -\gamma plus spatial inversion symmetry. At large distances (d\gtrsim 10), NHSE blocking becomes dramatic: propagation against the skin direction shows a sharp upturn as the exponential suppression r^d pushes the signal below the detection threshold. This provides a dynamical manifestation of the static eigenstate localization, demonstrating that the NHSE fundamentally alters not the speed but the detectability of information transport. A direct numerical verification of the propagator identity Eq. (9) and a side-by-side comparison of {\mathrm{\Delta }}_{\mathbb{T}} with the wavepacket center-of-mass drift \mathrm{\Delta }x(t) are provided in the Electronic Supplementary Materials A.6: the two quantities share a common imaginary-gauge origin (same sign, same antisymmetry under \gamma \to -\gamma) but differ in shape, with the \xi-peak of {\mathrm{\Delta }}_{\mathbb{T}} being absent in the monotone \mathrm{\Delta }x(\xi ).

5 Discussion

A central experimental implication of our results is that moderate non-Hermiticity (|\gamma |\approx 0.15–0.3) maximizes the observable directional asymmetry, as confirmed by both the \gamma-scan (Fig. 3) and \xi-scan (Fig. 5). This provides concrete guidance for experimental realizations.

As a methodological advance, QLIF operates directly at the level of entropy production [51], bypassing the limitations of symmetric correlation functions and the breakdown of Lieb−Robinson bounds [42–44] in non-Hermitian systems. Our results extend recent Hermitian applications [52] to the non-Hermitian regime.

We note that our analysis relies on the standard correspondence between the non-reciprocity parameter \gamma and the skin accumulation direction in the one-dimensional SSH model [6]. However, in more general settings — such as higher-dimensional lattices, photon-mediated interactions, or waveguide-coupled arrays — the direction of eigenstate accumulation does not always follow that of the asymmetric couplings, and skin effect reversal can occur [20–22]. Investigating how QLIF signatures — particularly the sign rule Eq. (8) and the velocity ordering Eq. (10) — are modified in systems exhibiting such reversed NHSE constitutes an intriguing direction for future work.

We fix t_2/t_1=0.5 throughout, placing the system in the topological phase. While QLIF has been shown to capture Hermitian quantum phase transitions [52] — including both spectrum-wide localization transitions and ground-state critical points — the interplay between non-Hermitian topology and QLIF involves additional complexities such as the generalized Brillouin zone and non-Hermitian topological invariants [6, 28], which are beyond the scope of the present work. The systematic study of QLIF signatures across the non-Hermitian topological phase boundary (t_2=t_1) will be addressed in a forthcoming publication. As a partial check that the phenomena reported here are not artifacts of the particular topological choice t_2/t_1=0.5, we repeat the scissors-effect analysis in the trivial phase (t_2/t_1=1.5) in Appendix D; the qualitative features — vanishing asymmetry in the Hermitian limit, the sign rule, and the approximate linearity of {\mathrm{\Delta }}_{\mathbb{T}}(\gamma ) near \gamma =0 — are preserved across the phase boundary. More fundamentally, the directional QLIF signatures reported here do not rely on the SSH bipartite structure at all: we confirm in Appendix E that the scissors asymmetry, the sign rule \mathrm{s}\mathrm{g}\mathrm{n}({\mathrm{\Delta }}_{\mathbb{T}})=-\mathrm{s}\mathrm{g}\mathrm{n}(\text{skin direction}), and the non-monotonic optimal-regime peak in {\mathrm{\Delta }}_{\mathbb{T}}(\gamma ) are all reproduced on the Hatano–Nelson chain [7] — a single-band non-reciprocal model with no sublattice and no line-gap topology — establishing these features as intrinsic to non-Hermitian directionality rather than to SSH topology.

A directly implementable testbed is the class of photonic single-photon non-unitary quantum walks pioneered by Xiao et al. [17, 34], which already realise chiral-symmetric non-Hermitian topological dynamics with tunable gain–loss contrast. There, an internal two-level degree of freedom (the photon polarisation) plays the role of the model’s two-band structure, polarisation-dependent loss provides the non-Hermitian knob, and walk steps furnish a discretised time axis. The site-resolved coincidence counts and chiral-basis wavefunction tomography demonstrated on this platform already supply, in principle, the single-particle reduced density matrix needed to evaluate the entropy S({\rho }_A) on a chosen subsystem. The QLIF protocol additionally requires a reference evolution in which the coupling to the chosen subsystem is removed at the freezing time; in a Floquet quantum walk this corresponds to inserting, at the relevant step, an operation that effectively decouples that subsystem from the rest of the network, after which the two entropy time-series are subtracted. Because the platform realises a Floquet rather than continuous-time model, the correspondence is qualitative and the predicted signatures should be regarded as universal consequences of the chirality of the NHSE rather than as quantitative parameter matches; in particular, the \gamma-sign scissors asymmetry (Fig. 3) and the skin-direction–dependent enhancement (Fig. 5) depend only on this chirality and should be accessible within the walk depths already demonstrated. Classical analogues in topolectrical circuits [15] and photonic mesh lattices [12] remain accessible as complementary platforms, though the von Neumann entropy underlying QLIF requires the genuinely single-particle setting that the photonic walk provides. Several directions also remain open theoretically: extensions to interacting systems [41] and higher-order skin effects [40].

6 Conclusion

We have applied quantum Liang information flow to the non-Hermitian SSH model, establishing the first quantitative connection between skin localization and directional information dynamics. Our principal findings are: (i) a scissors effect with {\mathrm{\Delta }}_{\mathbb{T}}\propto \gamma for small |\gamma |; (ii) non-monotonic \xi-dependence with optimal asymmetry at moderate localization; (iii) velocity ordering demonstrating NHSE blocking; and (iv) three distinct temporal regimes. These results establish QLIF as a powerful probe for non-Hermitian dynamics, opening new avenues for understanding directional information propagation in non-reciprocal quantum systems.

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