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arXiv:2604.27585

Observation of Universal Spectral Moments and the Dynamic Dispersive-to-Proliferative Transition

Jia-Xin Zhong, Chang Shu, Nan Cheng, Jee Woo Kim, Kai Zhang, Kai Sun, Yun Jing

quant-phcond-mat.mes-hallSubmitted Apr 30, 2026• Announced May 1, 2026
arXivHTMLPDF

Abstract

In non-Hermitian systems, spectra can be maximally boundary-sensitive, yet bulk physics need not be. Here we experimentally show that spectral moments provide boundary-robust bulk observables in finite non-Hermitian lattices, even when the spectra undergo dramatic geometry-dependent reshaping due to the skin effect. Using a unified acoustic platform with full spectral reconstruction and time-domain access, we probe one-, two- and three-dimensional lattices and demonstrate that spectral moments remain nearly invariant across distinct boundary geometries while the corresponding complex spectra differ strongly. To connect the thermodynamic theorem to realistic finite systems, we develop a loop-counting theory that identifies the physical origin of finite-size deviations in terms of missing boundary loops, quantitatively captures the corrections, and predicts a scaling law, which we verify experimentally. Beyond acoustic spectroscopy, we reveal a counterintuitive dynamical consequence of moment invariance: a dispersive-to-proliferative bulk transition governed by bulk moment structure rather than spectral boundary sensitivity. As a result, local bulk dynamics can remain stable (dispersive) even in a $\mathcal{PT}$-broken spectral regime, challenging the conventional expectation that $\mathcal{PT}$ breaking necessarily implies feedback-induced dynamical instability (proliferation) through exponentially amplifying spectral components. These results establish spectral moments as practical bulk descriptors for finite non-Hermitian matter and open a route to extracting and controlling intrinsic bulk behavior in realistic wave-based non-Hermitian devices.

Comments
20 pages, 4 figures

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In non-Hermitian systems, spectra can be maximally boundary-sensitive, yet bulk physics need not be. Here we experimentally show that spectral moments provide boundary-robust bulk observables in finite non-Hermitian lattices, even when the spectra undergo dramatic geometry-dependent reshaping due to the skin effect. Using a unified acoustic platform with full spectral reconstruction and time-domain access, we probe one-, two- and three-dimensional lattices and demonstrate that spectral moments remain nearly invariant across distinct boundary geometries while the corresponding complex spectra differ strongly. To connect the thermodynamic theorem to realistic finite systems, we develop a loop-counting theory that identifies the physical origin of finite-size deviations in terms of missing boundary loops, quantitatively captures the corrections, and predicts a scaling law, which we verify experimentally. Beyond acoustic spectroscopy, we reveal a counterintuitive dynamical consequence of moment invariance: a dispersive-to-proliferative bulk transition governed by bulk moment structure rather than spectral boundary sensitivity. As a result, local bulk dynamics can remain stable (dispersive) even in a PT\mathcal{PT}PT-broken spectral regime, challenging the conventional expectation that PT\mathcal{PT}PT breaking necessarily implies feedback-induced dynamical instability (proliferation) through exponentially amplifying spectral components. These results establish spectral moments as practical bulk descriptors for finite non-Hermitian matter and open a route to extracting and controlling intrinsic bulk behavior in realistic wave-based non-Hermitian devices.
Non-Hermitian lattices can host the non-Hermitian skin effect, a boundary-induced collapse of all bulk eigenstates into exponentially localized edge modes. This effect underlies anomalous bulk-boundary correspondence and remarkable enhancements in non-Hermitian sensing, yet direct energy-resolved access to the eigenmodes of non-Hermitian lattices has remained limited. Here we report band- and energy-resolved eigenmode spectroscopy of skin modes in a frequency synthetic dimension. By introducing strong frequency-domain boundaries in an electro-optically modulated ring resonator, we realize finite non-Hermitian lattices and use laser detuning as a spectroscopic axis for the eigenenergies of the effective Hamiltonian. Site-resolved heterodyne measurements then reconstruct the spatial profile of each mode, revealing boundary-localized skin states throughout the spectrum and their eigenenergy-dependent displacement from the edge. Beyond 1D, the same frequency-boundary architecture, upon incorporating long-range couplings between finite lattices, produces genuine 2D frequency lattices rather than the hitherto-realized folded 1D systems on twisted tubes. In these lattices we observe tunable directional transport and edge localization in two synthetic dimensions. Our results introduce eigenmode spectroscopy as a direct probe of non-Hermitian physics and establish strongly bounded frequency lattices as a flexible platform for Hamiltonian engineering.
Boundary perturbations are generally irrelevant for bulk properties in the thermodynamic limit, as they are edge-confined and subextensive. We show that this expectation breaks down in boundary-driven systems exhibiting the non-Hermitian skin effect, where arbitrarily weak boundary Floquet driving reconstructs bulk quasienergy spectra and dynamics. We develop a Floquet non-Bloch band theory that extends generalized Brillouin-zone methods to boundary-driven systems at arbitrary driving frequencies, overcoming the lack of a general framework beyond high-frequency approximations. With representative single- and two-band models, we demonstrate that the boundary driving frequency tunes non-Bloch parity-time symmetry breaking, while its amplitude acts as a finite-size control parameter. Our work establishes boundary Floquet control as a general route for manipulating bulk properties, opening a new avenue for dynamical engineering in driven open systems.
Non-Hermitian skin effect (NHSE) is a distinctive phenomenon in non-Hermitian systems, marked by the accumulation of eigenstates at system boundaries. While well understood in one dimension, unraveling the NHSE in higher dimensions is challenging due to the diversity of lattice geometries. Here, we present a geometry-adaptive non-Bloch band theory in arbitrary dimensions, through the lens of spectral potential. Our formulation precisely determines the energy spectra and generalized Brillouin zone in the thermodynamic limit, unveiling their geometric dependency. We establish exact spectral relations that elucidate the geometric dependence of non-Bloch spectra and their connection to amoeba spectra. Moreover, we show that critical skin modes in higher dimensions exhibit scale-free localization along the boundary, thereby making the spectrum highly sensitive to the system size and boundary ratios. We demonstrate that these critical skin modes lead to spectral non-convergence and instability. Our findings pave the way toward a unified understanding of NHSE and non-Bloch bands in arbitrary dimensions. In non-Hermitian quantum systems, the skin effect causes states to cluster at the edges, which is straightforward in one dimension but gets complicated in higher dimensions because of different lattice shapes. Here, the authors report a flexible theory that adapts to any dimension and shape, revealing how these factors influence energy patterns and lead to unstable edge states sensitive to system size.
Non-Hermitian lattice Hamiltonians generally exhibit strong boundary sensitivity, with periodic and open boundary conditions producing distinct density of states (DOS) in the complex-energy plane. This has led to the view that extended non-Hermitian systems lack a unique bulk DOS, with different prescriptions representing inequivalent bulk physics. Here, we show that this apparent ambiguity is largely illusory. For any finite-range tight-binding Hamiltonian, we establish a universal bulk structure: all DOS definitions arising as thermodynamic limits of finite systems share identical multipole moments and generate identical bulk dynamics at finite times and for observables measured far from boundaries. This universality is intimately tied to the thermodynamic Green's functions, which we show to be independent of the boundary condition for large enough complex frequencies. Among all equivalent descriptions, we identify the Brown measure - obtained via Hermitization and resolvent analysis - as a canonical and convenient representative of the bulk DOS, defined directly from the infinite-volume Hamiltonian. We further show that point-gap topology imposes additional universal constraints: boundary-dependent Green's functions are forced to coincide throughout topologically trivial point gaps. This, in particular, provides a systematic criterion, valid in arbitrary dimension, for determining where and how eigenvalues of different boundary truncations can accumulate in the complex plane, and precisely delineates the regime in which the DOS ambiguity retains physical significance.
Non-Hermitian systems exhibit unique spectral properties, including the non-Hermitian skin effect and exceptional points, often influenced by boundary conditions. The modulation of these phenomena by generalized boundary conditions remains unexplored and not understood. Here, we analyze the Hatano-Nelson model with generalized boundary conditions induced by complex hopping amplitudes at the boundary. Using similarity transformations, we determine the conditions yielding real energy spectra and skin effect, and identify the emergence of exceptional points where spectra transition from real to complex. We demonstrate that tuning the boundary hopping amplitudes precisely controls the non-Hermitian skin effect, i.e., the localization of eigenmodes at the lattice edges. These findings reveal the sensitivity of spectral and localization properties to boundary conditions, providing a framework for engineering quantum lattice models with tailored spectral and localization features, with potential applications in quantum devices.
The non-Hermitian skin effect (NHSE), which drives bulk states toward system boundaries, modifies bulk-boundary correspondence and complicates the identification of topological edge modes. Although breaking translational symmetry is known to influence this behavior, a systematic comparison of different structural classes remains limited. Here we investigate periodic, random, and quasiperiodic (Fibonacci) systems using a one-dimensional non-Hermitian quantum walk model. By matching the local scattering parameters in a topologically nontrivial regime, we isolate the role of spatial structure in the presence of the NHSE. We find that periodic systems exhibit pronounced boundary accumulation of bulk states. Random systems suppress this accumulation through Anderson localization, but the topological gap becomes partially filled with localized in-gap states. In contrast, the Fibonacci quasiperiodic system suppresses large-scale boundary accumulation while maintaining a well-defined topological gap. Analysis of the wave functions suggests that the hierarchical quasiperiodic structure fragments bulk states across multiple length scales, thereby mitigating the NHSE. These results identify deterministic quasiperiodicity as a distinct structural regime for controlling non-Hermitian skin dynamics and isolating topological boundary modes.