Observation of Erratic Non-Hermitian Skin Localization and Transport

TL;DR

The work identifies and characterizes erratic non-Hermitian skin localization (ENHSL), a regime where macroscopic, disorder-dependent bulk localization coexists with ballistic transport on average in globally reciprocal non-Hermitian lattices. It introduces a disordered Hatano–Nelson chain with zero net imaginary bias and establishes a link between localization peaks and the maximal excursion of a symmetric random-walk landscape $X_n$, yielding a universal Lévy-arcsine law for the dominant propagation site. The authors realize ENHSL in an acoustic lattice, reconstruct the full complex spectrum with Green's-function measurements, and perform time-resolved propagation experiments to connect spectral features to dynamics. The resultant transport theory demonstrates that extremal statistics of $X_n$ control transport, decoupling spectral localization from ensemble transport, with potential implications for designing stochastic yet predictable wave confinement in a broad range of wave systems.

Abstract

Localization is a pervasive phenomenon across physics, shaping transport from electrons in solids to light and sound in engineered media. In traditional settings, disorder strongly impedes transport, resulting in dynamical localization or, at best, sub-ballistic or diffusive dynamics. A distinct and previously unobserved regime, erratic non-Hermitian skin localization (ENHSL), can arise in globally reciprocal non-Hermitian lattices with disorder. It features macroscopic, disorder-dependent localization at irregular bulk positions with subexponential decay, linked to stochastic interfaces governed by the universal order statistics of random walks. We realize this regime experimentally in an acoustic lattice implementing a disordered Hatano-Nelson chain with imaginary gauge fields. Using Green's-function-based spectroscopy together with time-resolved measurements on the same platform, we reconstruct the full complex spectrum and eigenstates, and directly observe wave-packet dynamics. Remarkably, we observe ballistic transport despite strong spectral localization. We develop a transport theory that connects the dominant propagation site to the maximal random-walk excursion within an expanding light cone and predicts a universal Levy-arcsine statistics, in quantitative agreement with experiment. Our results decouple eigenstate localization from transport and establish ENHSL as a new paradigm for wave dynamics.

Figures (5)

• Figure 1: Conceptual comparison between ENHSL and Hermitian Anderson localization.a, Tight-binding schematic of a globally reciprocal non-Hermitian chain with disordered imaginary gauge fields (hopping disorder). Sites share a uniform onsite potential $(\omega_0)$, while the link colors indicate spatially fluctuating non-reciprocal hoppings ($J_n^{\mathrm{L/R}}$). b, Hermitian Anderson chain with onsite disorder (site colors, $\omega_n$) and Hermitian uniform hoppings ($J$). c, Spectral signatures of ENHSL. The inset sketches the complex spectrum, which remains (approximately) real despite non-reciprocal hoppings under global reciprocity. The main panel illustrates a typical ENHSL eigenstate profile featuring a dominant bulk peak accompanied by two weaker satellite peaks; the dashed curve indicates the summed eigenstate intensity $\Psi_n=\sum_m|\psi_{m,n}|^2$, which is dominated by the same peak structure. d, Spectral signatures of Anderson localization: many exponentially localized eigenstates centered at different sites, yielding a relatively featureless mode-summed profile $\Psi_n$ (dashed line). e, Dynamical response in ENHSL following a bulk-centered excitation. The 3D traces show the spatiotemporal evolution of the wave-packet envelope $\psi_n(t)$ for a single disorder realization. Dashed guidelines show the symmetric ensemble-averaged spreading envelope, $n=\pm \langle d(t)\rangle$ (gray shading: realization-to-realization fluctuations), highlighting the ballistic transport. f, Dynamical response in the Anderson case, where the wave packet remains dynamically localized and the ensemble-averaged spreading saturates at long times (dashed guideline).
• Figure 2: Unified spectral-and-dynamical measurement protocol on the same acoustic lattice.a, Photograph of the experimental platform: a 1D chain of acoustic resonators (interior view shown at left, containing a microphone and two loudspeakers) coupled by electroacoustic feedback implemented with external amplifiers and phase shifters integrated in a multi-channel controller (right). Here, a representative hopping from site $n$ to site $n+1$ is illustrated. b,c, Green's-function-based spectroscopy for full spectral reconstruction. b, Site-by-site excitation (pump $j$) together with full spatial readout (probe $i$) yields the frequency-resolved response (Green's-function) matrix $G_{ij}(\omega)$. Shown is an example lattice of size $N=20$ under OBC ($\omega_0 = 1040\,\mathrm{Hz} - 6i\,\mathrm{Hz}$, $J=1.5\,\mathrm{Hz}, \Delta h = 2\,\mathrm{Hz}$), with $G_{ij}(\omega)$ visualized at several representative frequencies. The inset color wheel indicates the visualization scheme: hue encodes the phase $\angle G_{ij}$, while brightness encodes the magnitude $|G_{ij}|$ (log-scaled). c, Extraction of complex eigenenergies from the measured Green’s function. Dots show the measured magnitudes of the eigenvalues $\lambda_m(\omega)$ of $G(\omega)$ as functions of frequency, and solid curves show fits to a single-pole form $\lambda_m(\omega)\simeq A_m/(\omega-E_m)$, from which the complex eigenenergies $E_m$ are obtained. d--f, Time-resolved wave-packet measurements using the same lattice. d, Gaussian-modulated tone burst $s(t)$ centered at $\Re (\omega_0)=1040\,\mathrm{Hz}$, illustrating the carrier and the Gaussian envelope (width $\sigma$) with temporal center $t_0 = 5\sigma$. e,f, Excitation waveforms $s(t)$ (e) and their spectra $|s(\omega)|$ (f) for four representative envelope widths $\sigma=5,\,10,\,20,$ and $40\,\mathrm{ms}$.
• Figure 3: Spectral signatures of ENHSL, comparing experiments and tight-binding simulations for a chain of size $N=50$ with $\omega_0 = 1040\,\mathrm{Hz} - 6i\,\mathrm{Hz}$, $J=1.5\,\mathrm{Hz}$, and $\Delta h = 2\,\mathrm{Hz}$.a,b, Complex spectra for a representative realization under both PBC and OBC. Circles (squares) denote PBC (OBC) energies, and colors encode $\mathrm{IPR}_m$. The dashed horizontal line marks the nominal uniform background-loss level ($-6\,\mathrm{Hz}$) used as a reference. c,d, Spatial distributions of all eigenstates under PBC (top) and OBC (bottom) for experiment (c) and simulation (d). Heat maps show the eigenstate amplitudes $|\psi_{m,n}|$ as functions of eigenstate index $m$ and site coordinate $n$. The overlaid purple curve (right axis) shows the random-walk landscape $X_n$ associated with the imaginary-gauge-field disorder, highlighting that prominent localization peaks occur near large excursions of $X_n$. e, Scaling of localization with system size under OBC. Symbols show the ensemble-averaged $\langle{\mathrm{IPR}}\rangle$ for $N=5,10,15,...,70$ (each $N$ averaged over $R=8$ disorder realizations), comparing experiment and simulation; shaded bands indicate the realization-to-realization spread (standard deviation). The dashed guideline ($\beta=1$) indicates the extended-state scaling $\mathrm{IPR}\propto N^{-1}$. A fit to $\langle{\mathrm{IPR}}\rangle \propto N^{-\beta}$ yields $\beta\simeq 0$, consistent with size-independent localization in ENHSL.
• Figure 4: Experimental observation of wave-packet propagation following excitation at the central site ($n=0$) in a chain of length $N=61$ under OBC. The parameters set in experiments are $\omega_0 = 1040\,\mathrm{Hz} - 1.75i\,\mathrm{Hz}$, $J=4\,\mathrm{Hz}$, and $\Delta h = 1.6\,\mathrm{Hz}$. a,b,c, Three representative disorder realizations of the ENHSL. d, Disorder-free Hermitian chain (clean reference). e, Hermitian chain with onsite disorder showing Anderson localization (reference). For each panel, rows from top to bottom show: (i) the spatial profile of the eigenstate intensity summed over all states, $\Psi_n=\sum_{m}\lvert\psi_{m,n}\rvert^{2}$; (ii) the random-walk landscape $X_n$; (iii) the experimentally measured spatiotemporal evolution of the acoustic pressure envelope $\abs{\psi_n(t)}$; (iv) the corresponding tight-binding simulation; and (v) the spreading distance $d(t)$ extracted from experiment (blue solid) and simulation (red solid), with the ballistic law shown for comparison (orange dashed).
• Figure 5: Ensemble statistics of wave-packet spreading for the ENHSL (OBC, $N=61$), obtained from $R=100$ disorder realizations. The parameters set in experiments are $\omega_0 = 1040\,\mathrm{Hz} - 1.75i\,\mathrm{Hz}$, $J=4\,\mathrm{Hz}$, and $\Delta h = 1.6\,\mathrm{Hz}$. a, Ensemble-averaged spreading distance $\langle d(t)\rangle$ from experiments and tight-binding simulations, compared with the ballistic scaling of a clean Hermitian chain and with a Hermitian Anderson-localized reference (shown by simulations, exhibiting saturation). b--e, Distributions of the spreading distance at fixed times $t=0.1,\,0.2,\,0.3,$ and $0.4~\mathrm{s}$, respectively. In each panel, purple markers indicate the values $d^{(r)} (t)$ from individual realizations $r$ (realization index shown on the right axis). The shaded curves show the probability density function (PDF) $F(d,t)$ (left axis) estimated from the finite ensemble using a bounded KDE method on the interval $(0,30)$. The dashed lines denote the Lieb-Robinson (LB) bound, $n_\mathrm{LB}(t) = 2Jt$. f, PDF of the dominant propagation site, $n_0(t)=\arg\max_n \abs{\psi_n(t)}$, at $t=0.5~\mathrm{s}$, obtained from the same $R=100$ realizations, and estimated using a bounded KDE method on the interval $(-30,30)$. Experimental and simulated distributions are compared. The dashed vertical lines indicate the corresponding LB bounds, $\pm n_{\mathrm{LB}}$, and the black curve shows the Lévy arcsine-law prediction restricted to the interval $(-n_{\mathrm{LB}},\,n_{\mathrm{LB}})$.