Harmonic non-Hermitian skin effect

Domain: neuroscience
Published: 2026-01-30
Editorial outcome: minor_revision

Abstract
The non-Hermitian skin effect (NHSE), an exotic phenomenon of localization at the boundary, has recently drawn extensive attention and Floquet engineering offers a powerful approach to expanding its scope. The fundamental role of harmonic response in Floquet systems has nonetheless yet to be elucidated. Here, we introduce the concept of harmonic NHSE and demonstrate it experimentally in an acoustic platform. Within our theoretical framework, we demonstrate that incorporating dynamic couplings into the paradigmatic Hatano-Nelson model causes a single-frequency input to generate multiple harmonics with diverse skin morphologies, each dictated by the underlying spectral winding topology at its respective frequency. Experimentally, we implement the requisite dynamic couplings in an acoustic lattice via programmable unidirectional couplers. By simply reconfiguring the excitation and modulation frequencies, we demonstrate controllable switching of unipolar and bipolar harmonic NHSEs, which exhibit uniform and opposite sound gathering toward the sample boundaries, respectively. By introducing the class of harmonic NHSE, our work sets the stage for further investigating non-Hermitian physics with harmonic effects.

Introduction
Non-Hermitian systems, typically featuring asymmetric couplings and gain/loss, have garnered extensive attention owing to their unique phenomena and broad applications beyond Hermitian regimes1–3. In particular, the non-Hermitian skin effect (NHSE), referring to the accumulation of bulk states at the lattice boundary, has emerged as a cornerstone of non-Hermitian physics4–6. Over the past few years, a number of theoretical frameworks, such as the generalized Brillouin zone (GBZ), Bloch spectral winding and dynamical degeneracy splitting, have been developed to characterize the NHSE and its related topological properties7–14. These advances have propelled the experimental observation of diverse NHSE morphologies across photonic15, circuit16,17, acoustic18–21, mechanical22,23, quantum walk24, and ultracold atom25,26 platforms, paving the way for potential applications in optical funneling15, robust directional transport27, and non-Hermitian lasing28.

Floquet engineering, an established approach for tailoring material properties via time-periodic modulation29–31, has recently been extended to non-Hermitian systems. This development has led to richer phenomena, including non-Hermitian Floquet topological states32–34, multiple phase transitions35,36, and Floquet exceptional points37,38. In particular, a periodically driven system can exhibit Floquet NHSE even when the instantaneous Hamiltonian at each step lacks NHSE38–43. Along with breaking the bulk-boundary correspondence, Floquet NHSE enables the relocation of topological edge states to a single boundary or the separation of distinct edge states at opposite boundaries, giving rise to the Floquet skin-topological effect40,41. In these studies, the key roles played by Floquet engineering are breaking time-reversal symmetry and introducing artificial gauge fields (or complex long-range couplings). Despite these advances, however, the implications of harmonic generation—a fundamental feature of Floquet systems—has yet to be explored in the context of NHSE. The primary reasons are twofold. Theoretically, most studies focus on the stroboscopic dynamics governed by the effective Hamiltonian38–40; the higher-order Floquet states (harmonics), which share the same eigen-properties with the zeroth-order one, are typically disregarded. Experimentally, Floquet NHSE has so far been implemented in waveguide and ring resonator systems, where spatial modulation emulates temporal periodicity41–43; such setups inherently exclude harmonic generation due to their time-invariant nature.

Here, we introduce and experimentally observe a new type of NHSE, dubbed harmonic NHSE, in Floquet non-Hermitian lattices. This phenomenon results from a direct synergy between Floquet engineering and non-Hermitian topology—time-periodic driving generates a spectrum of harmonics in open-boundary lattices, while the skin localization of each harmonic is dictated by the spectral winding topology at its own frequency. The concept is demonstrated through a simple Hatano-Nelson (HN) model44 incorporating time-periodic couplings. In addition to the unipolar harmonic NHSE induced by uniform spectral winding, more intriguingly, as sketched in Fig. 1, the harmonics can exhibit bipolar NHSE in a long-range lattice with twisted winding spectrum, despite the extended nature of the fundamental wave. Experimentally, we design two nonreciprocal acoustic lattices to observe these harmonic NHSEs, with both dynamic and nonreciprocal couplings reconfigurable via active unidirectional couplers. By tuning the excitation and modulation frequencies, we clearly demonstrate the ability of selectively localizing different harmonics, with their weights flexibly controllable by simply adjusting the duty ratio of dynamic couplings. These results not only extend the existing NHSE framework but also open new avenues for harmonic wave manipulation.

Results
In contrast to the above eigenvalue problem, in practice, one needs to consider an excitation problem to identify the harmonic NHSE. Given the real-space Hamiltonian H r (t) for an N-site finite lattice with open boundaries, the response to a monochromatic excitation at frequency f 0 is described by (see Methods):

(Δ 0 +mΩ) Q m + Δ 1 Q m-1 + Δ -1 Q m+1 = P m (1)

where Δ m = T 1 T 0 H k (t)ei2mπΩtdt is the mth-order Fourier component of H k (t), with Ω = 1/T denoting the modulation frequency. In particular, Δ 0 = T 1 T 0 H k (t)dt = H(cid:2) k denotes the time-averaged Hamiltonian. The spectrum of H k under periodic boundary conditions (PBCs) comprises infinitely many branches, each shifted by Ω, as a result of the unitary transformation UyH k U = diag(H(cid:2) k+mΩ, ..., H(cid:2) k+Ω, H(cid:2) k, H(cid:2) k-Ω, ..., H(cid:2) k-mΩ)45. Due to the single-band nature of our model, these spectral branches remain non-interacting even when they overlap, unlike in two- or multi-band cases46. Specifically, the mth-order Floquet spectrum of H k corresponds to that of H(cid:2) k - mΩ, and its winding topology governs the NHSE under open boundary conditions (OBCs) within the associated Floquet replica.

Moreover, as demonstrated above, the NHSE within the mth-order Floquet replica is governed by the mth-order Floquet spectrum of H k, or equivalently by the spectrum of H(cid:2) k - mΩ. Consequently, the harmonic NHSE under realistic excitation can be fully characterized by examining solely the zeroth-order Floquet spectrum of H k, or equivalently, the spectrum of H(cid:2) k for single-band models. (Remarkably, for two- or multi-band models, the zeroth-order Floquet spectrum of H k can differ fundamentally from the spectrum of H(cid:2) k, due to the interaction between different Floquet branches upon closing46). The above theoretical framework provides a general route to account for harmonic effects in Floquet non-Hermitian systems.

First, we introduce time-periodic dynamic couplings into a prototypical static HN lattice44, forming a Floquet HN (FHN) lattice (see Fig. 2a). The Bloch Hamiltonian of the FHN lattice is H k (t) = κ L + κ D (t) eik + κ R + κ D (t) e-ik. Physically, the nonreciprocal static couplings κ L and κ R establish the fundamental NHSE, while the reciprocal dynamic couplings κ D (t) contribute to the harmonic generation. For simplicity, we assume κ D (t) = κ 0 cos(2πΩt), as it couples only Floquet harmonics of adjacent orders without introducing any additional static component. As shown in Fig. 2b, both the spectrum of H(cid:2) k and the zeroth-order Floquet spectrum of H k form a counterclockwise loop in the complex frequency plane. A reference frequency f b enclosed by the spectrum f(k) carries a nontrivial spectral winding number and their associated NHSEs. Specifically, when choosing the excitation frequency f 0 = 0 and modulation frequency Ω = 1.5, the fundamental frequency f 0 lies at the Bloch-like point of the spectrum, while the +1st- and -1st-order harmonic frequencies f +1 and f -1 fall within the counterclockwise and clockwise spectral loops, respectively (see Fig. 2e). Protected by the twisted winding topology, the incident fundamental wave maintains an extended spatial distribution, whereas the +1st- and -1st-order harmonics undergo unidirectional accumulation toward the left and right boundaries, respectively, as shown in Fig. 2f. Notably, the extended nature at the Bloch-like point can be interpreted in terms of both the vanishing spectral winding number and the specific GBZ with purely real Bloch wave vectors4,8. Furthermore, different harmonics can be selectively engineered to exhibit distinct skin and evanescent behaviors by tuning f 0 and Ω, as detailed in Supplementary Note 2.

We further achieve control over the harmonic intensity distribution by tuning the duty ratio R of dynamic couplings. Since the Fourier components of the dynamic couplings determine the conversion between harmonics, varying R effectively modulates their magnitudes and thereby tailors the intensity weights of individual harmonics within the lattice. This controllability is readily implemented by leveraging the flexible tunability of the real-time unidirectional couplers in our experiments. To ensure that the PBC spectra of time-averaged Hamiltonians (and hence the harmonic NHSE) remain quantitatively unchanged during variation of R, we keep the zeroth-order Fourier component of κ D constant, namely, Rκ 0 = κ(cid:2). Under this condition, the coefficient of the leading order (±1st) Fourier component of κ D is given by c ±1 = ±κ(cid:2)sinc(πR), whose absolute value decreases monotonically with increasing R in the interval [0, 1]. Specifically, when R approaches zero, c ±1 reaches its maximum value κ(cid:2); when R approaches one, c ±1 decreases to its minimum value zero.

Methods
Eigenvalue problem of the Floquet non-Hermitian lattice

The Bloch Hamiltonian H k (t) of the Floquet non-Hermitian lattice is governed by the Schrödinger equation:

-i d/dt |ψ(t) > = 2πH k (t) |ψ(t) > (5)

The Floquet state solutions to this equation take the Floquet form:

|ψ(t) > = e-i2πϵt |u(t) > (6)

where |u(t + T) > = |u(t) > is the time-periodic Floquet mode, and ϵ is the quasi-energy. Substituting Equation (6) into Equation (5) yields:

2πϵ |u(t) > = 2πH k (t) + i d/dt |u(t) > (7)

Specifically, |u(t) > can be expanded as a Fourier series:

|u(t) > = Σ m |u m > e-i2πmΩt (8)

where |u m > are the Fourier components. Similarly, the Hamiltonian H k (t) can be expanded as:

H k (t) = Σ m Δ m e-i2πmΩt (9)

where Δ m = T 1 T 0 H k (t)ei2mπΩtdt is the mth-order Fourier component of H k (t). Substituting Equations (8) and (9) into Equation (7) and regrouping terms, we obtain a time-independent eigenvalue problem in the frequency domain:

ϵ |u j > = H k |u j > (10)

where the time-independent Floquet mode is |u j > = [..., |a -m >, ..., |a -1 >, |a 0 >, |a +1 >, ..., |a +m >]T, with the subscripts denoting the Floquet order. The time-independent Hamiltonian H k takes the form:

(See Equation 11 in original text)

Note that, in practical calculations, we truncate the infinite-dimensional H k at a sufficiently high order of harmonics (m = ±5), ensuring the eigenvalue problem can be numerically solved.

Excitation problem of the Floquet non-Hermitian lattice. For an N-site Floquet non-Hermitian lattice described by the real-space Hamiltonian H r (t), the system’s dynamics under excitation is governed by the nonhomogeneous Schrödinger equation:

-i d/dt Q(t) = 2πH r (t) Q(t) + P(t) (12)

where

P(t) = 2πe-i2πf 0 t [δ 1s, δ 2s, ..., δ ns, ..., δ Ns]T (13)

represents a time-domain excitation at a specific site n = s with monochromatic frequency f 0, and Q(t) represents the time-domain response across all sites. H r (t) and Q(t) can be written as Fourier series:

H r (t) = Σ m Δ m e-i2πmΩt (14)

Q(t) = Σ m Q m e-i2π(f 0+mΩ)t (15)

where Δ m = T 1 T 0 H r (t)ei2mπΩtdt is the mth-order Fourier component of H r (t). Inserting Equations (14) and (15) into Equation (12), and using the orthogonality properties of Fourier harmonics: e-i2πmΩt, e-i2πm0Ωt = δ mm0, we get

(See Equation 16 in original text)

where ζ m = f 0 + mΩ denotes the mth-order harmonic frequency generated by the monochromatic excitation, Q m = [a 1 m, a 2 m, ..., a n m, ..., a N m]T represents the frequency-domain response of the mth-order harmonic across the lattice, P m = 2π [δ m0 δ 1s, δ m0 δ 2s, ..., δ m0 δ ns, ..., δ m0 δ Ns]T describes the excitation applied at sites. Notably, only the component with m = 0 and n = s satisfies δ m0 δ ns = 1, while all other components vanish. This reflects the fact that, in realistic settings, only the zeroth-order harmonic component at site s can be directly excited.

Experimental setup and measurements. Our experimental cavity-tube sample (shown in Figs. 3a and 4a) is fabricated using 3D printing with acoustically rigid photosensitive resin, featuring a wall thickness of 2.0 mm and a fabrication tolerance of 0.1 mm. Each air-filled cavity, having a height of h = 43.0 mm, a width of w = 28.0 mm, and a length of l = 28.0 mm, supports a dipole mode with resonant frequency ω 0 ≈ 4000 Hz and intrinsic loss γ 0 ≈ 38 Hz. Adjacent cavities are connected by two narrow tubes, each having a length of y = 19.8 mm and a cross-sectional area of A = 6.4 × 6.4 mm2. These tubes generate a reciprocal acoustic coupling of κ N ≈ 75 Hz. At both the top and bottom surfaces of each cavity, five holes are drilled for inserting the loudspeakers and microphones. The holes are sealed when not in use to prevent acoustic leakage. The unidirectional dynamic (or static) couplings between cavities are implemented using real-time unidirectional couplers, each consisting of a microphone, a preamplifier (Type LM386), a voltage-controlled amplifier (Type VCA810), a phase shifter (Type MCP41010), and a loudspeaker. The gain factor of the voltage-controlled amplifier is modulated in real time by a signal generator (Aigtek Type ATA-214). When the signal generator outputs a square-wave signal, the real-time unidirectional couplers realize square-form dynamic unidirectional couplings κ D, alternating between a high level of κ 0 and a low level of 0. For the experiments presented in Figs. 3 and 4, κ 0 is tuned to approximately 120 Hz and 56 Hz, respectively. In addition, when the signal generator outputs a direct current signal, the real-time unidirectional couplers realize static unidirectional couplings κ R ≈ -28 Hz for the experiment in Fig. 4. The values of the above acoustic parameters are calibrated from independent measurements of a binary-cavity unit (or a ternary-cavity unit) with a single real-time unidirectional coupler (see details in Supplementary Note 4). In each measurement, the sound signal is launched from a loudspeaker driven by the output generator module of an audio analyzer (B&K Type 3160-A-042). The sound pressure response is measured with a detector microphone (B&K Type 4958) and processed with the signal acquisition module of the audio analyzer.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript reports a novel class of non-Hermitian skin effects (NHSEs), termed the harmonic NHSE, which emerges from the intricate interplay between the characteristic harmonic generation in Floquet systems and the unique NHSE in non- Hermitian systems. The authors demonstrate that the NHSEs of different frequency harmonics are governed by the zeroth- order Floquet spectral topology, leading to distinct localizations across these harmonics. They illustrate this discovery using two modified Hatano-Nelson models with dynamic modulations, wherein unidirectional spectral winding produces unipolar harmonic NHSE, while twisted winding gives rise to bipolar harmonic NHSE. Experimentally, the harmonic NHSE is observed in an array of acoustic cavities with genuine time modulation. By tuning the excitation and modulation frequencies, the authors clearly demonstrate the selective localization and controllable weights of different harmonics.\nIn my view, the NHSE associated with harmonic generation in Floquet non-Hermitian systems represents a previously unexplored regime. While several studies have proposed the concept of Floquet NHSE, the role of harmonics has never been addressed, either theoretically or experimentally. This work identifies the critical gap and uncovers novel physics. From the perspective of innovation, this work is timely and groundbreaking, tackling an untapped topic. Regarding the results, the authors provide extensive details in both theoretical analysis and experimental approaches. Overall, the manuscript is clearly organized and well written, I would like to recommend its publication if the following questions are well addressed.\n1. In the theoretical model in Fig. 2, both nearest-neighbor and next-nearest-neighbor non-reciprocal couplings are considered. Would longer-range non-reciprocal couplings lead to richer behaviors in harmonic NHSE?\n2. What is the specific definition of the spectral winding number used in this work? Could the authors provide it explicitly? How to distinguish harmonic NHSE from the Floquet-induced NHSE?\n3. Could the presence of loss in experiments affect the observation of harmonic NHSE? What are the main sources of loss?\n4. In the experiments shown in Fig. 5, why is the lower limit of the duty ratio R set to 0.4? Is there a specific consideration for this choice?\n5. How universal is this control mechanism? Does it persist for sinusoidal (rather than square-wave) modulation, or in systems with stronger dissipation?\n6. Compared to conventional Floquet NHSE, does harmonic NHSE will show unique dynamic characteristics?\n7. For multi-band models, does the coupling between different Floquet branches lead to phenomena distinct from those in single-band models, or affect the formation of the Harmonic NHSE? Can the topological phases still be distinguished by the low-order Floquet spectrum of the Hamiltonian Hk? Does it introduce new topological phenomena? Please provide a brief discussion.\n8. In figure3c, 4b, only s21 and s31 are considered to extract the coupling strength. If s12/s21,s13/s31 are all considered, can the coupling still fit well with the experiment data?\n9. How about the energy distribution at the other frequencies? Can you provide the spectra response at single stie (at edge / middle of the model)?", "reviewer_2": "This submitted manuscript by Zhang et al. presented a novel and compelling study on the topic of harmonic non-Hermitian skin effect (NHSE). The authors successfully bridged the fields of time-dependent Floquet engineering and non-Hermitian topology by introducing the harmonic waves, which were experimentally generated by time-periodic driving. The distinct and controllable skin effect morphologies were then exhibited. Overall, I think the concept is innovative, the theoretical framework is solid, and the experimental realization in an acoustic platform with real-time modulations is impressive and provides clear, convincing evidence. This work may open a new direction in the study of dynamic non-Hermitian systems and is likely to be of significant interest to the broader physics community. I believe this paper is well-suited for publication in Nature Communications after the following suggested revisions are addressed.\n1. The main text presented simulated results. It would be beneficial for the authors to describe the simulation methods, particularly explaining how the active unidirectional couplers were realized in the simulations.\n2. I am interested in several experimental details. Specifically, more information on the feed-back microphones and loudspeakers (such as types or models) used for the real-time unidirectional couplers. Additionally, how to ensure the consistent performance characteristics of these compact acoustic components?\n3. In the Supplementary Information Note 3, the authors employed the transmission amplitude spectra |(cid:1)_21| and |(cid:1)_31| to fit the relevant parameters. Since it is challenging to directly measure the acoustic pressure at the source, I am interested in the measurement methodology for the individual transmission coefficients. Additionally, the fitting equation in Eq. (S6) differs from the one used in the authors' prior work (10.1103/PhysRevLett.130.017201). Could the authors elaborate on the consideration behind this modification in the methodology please?\n4. Other minor suggestions:\n4.1 Some figure captions (e.g., Figs. 2, 4) are very dense with parameter lists. Consider moving some of these details to the main text or methods for improved readability.\n4.2 On Page 7, lines 128-130, the text \"...the +1st- and −1st-order harmonics undergo unidirectional accumulation toward the right and left boundaries, respectively...\" seems to contradict the caption of Fig. 2f, which states the +1st harmonic goes left and the -1st goes right. Please verify and correct this for consistency.", "reviewer_3": "In this manuscript, Zhang et al. introduce and experimentally demonstrate the concept of a \"harmonic non-Hermitian skin effect (NHSE)\" in a Floquet-engineered acoustic lattice. The core idea is that when a monochromatic wave excites a time-periodically modulated non-Hermitian lattice, it can generate multiple harmonics, each of which can exhibit a distinct skin localization phenotype—unipolar or bipolar—dictated by the spectral winding topology at its respective frequency. The authors provide a theoretical framework based on a frequency-domain analysis and present compelling experimental evidence using a reconfigurable acoustic platform with active, real-time modulations. The ability to selectively control and spatially separate different harmonic components via a single-frequency excitation is a key experimental achievement.\nI find this work to be of high quality, timely, and highly suitable for publication in Nature Communications. The concept of \"harmonic NHSE\" is novel and represents a significant extension of the non-Hermitian skin effect into the realm of Floquet systems with genuine time modulation. The study successfully bridges a gap in the field by focusing on the harmonic generation aspect, which is a fundamental feature of Floquet systems but had been overlooked in previous studies of Floquet NHSE, particularly those relying on synthetic dimensions (e.g., waveguide arrays). The combination of a clear theoretical proposal with a sophisticated and convincing experimental demonstration in acoustics is a major strength. The observation of both unipolar and, more impressively, bipolar harmonic NHSE, where different harmonics localize at opposite ends of the system, is a striking result. The findings are likely to stimulate immediate interest across various platforms in photonics, circuit QED, and cold atoms.\nI am therefore strongly in favor of publication, provided that the authors can address the following points to further enhance the clarity and theoretical rigor of their otherwise excellent manuscript.\n1. GBZ Calculation for Floquet Model: The topological origin of the skin effect for each harmonic is central to the present approach. To make this connection more transparent and rigorous, I encourage the authors to calculate and present the Generalized Brillouin Zone (GBZ) for their Floquet model, following the established non-Bloch band theory (e.g., Yao and Wang, PRL, 2018). Demonstrating how the GBZ and the associated complex wavevectors dictate the skin localization direction for the fundamental wave and its harmonics would significantly strengthen the topological foundation of the \"harmonic NHSE\" concept.\n2. Clarification on the Extended Nature of the Zeroth Harmonic (f₀): In Figure 1 and the corresponding discussion of the bipolar NHSE, the zeroth-order harmonic (f₀) is depicted as extended. This appears counterintuitive because the underlying static lattice, with its nonreciprocal static hoppings (κ_L, κ_R), should itself exhibit clear features of NHSE, localizing all eigenstates. The authors should provide a clearer explanation for this observation. Is this extended behavior a result of a specific parameter choice (e.g., f₀ being exactly at a Bloch-like point in the spectrum)? A detailed discussion on why f₀ resists skin localization in this specific scenario, while the system remains non-Hermitian, is necessary to resolve this apparent contradiction for the reader.\n3. Justification for the Excitation Problem Framework: The authors state that \"the harmonic NHSE can be adequately characterized by focusing solely on the zeroth-order Floquet spectrum of mathcal{H}_k, or equivalently, the spectrum of Ḥ_k\", justifying this by the orthogonality of Fourier harmonics in the excitation vector P_m. While this simplification is crucial for the analysis, the reasoning provided is somewhat obscure. I find it difficult to fully grasp why the topology of the time-averaged Hamiltonian \\bar{H}_k alone determines the skin effect for all higher harmonics generated in the excitation problem. The authors should elaborate on this point, perhaps with a more detailed derivation or a physical argument in the Methods or Supplementary Information, to clarify the validity of this approach."}, "author_response": "Response Letter to Reviewers\nWe sincerely appreciate the efforts of the three reviewers in thoroughly evaluating our manuscript.\nWe are pleased that all reviewers recognize the novelty, significance, and broad appeal of our work.\nThe reviewers raised several insightful questions and constructive suggestions, which have allowed\nus to significantly clarify and improve our manuscript. Below, we respond to each point raised by\nthe reviewers and highlight (in yellow) the corresponding changes made in the revised version.\nReviewer #1\nThis manuscript reports a novel class of non-Hermitian skin effects (NHSEs), termed the harmonic\nNHSE, which emerges from the intricate interplay between the characteristic harmonic generation\nin Floquet systems and the unique NHSE in non-Hermitian systems. The authors demonstrate that\nthe NHSEs of different frequency harmonics are governed by the zeroth-order Floquet spectral\ntopology, leading to distinct localizations across these harmonics. They illustrate this discovery\nusing two modified Hatano-Nelson models with dynamic modulations, wherein unidirectional\nspectral winding produces unipolar harmonic NHSE, while twisted winding gives rise to bipolar\nharmonic NHSE. Experimentally, the harmonic NHSE is observed in an array of acoustic cavities with\ngenuine time modulation. By tuning the excitation and modulation frequencies, the authors clearly\ndemonstrate the selective localization and controllable weights of different harmonics.\nIn my view, the NHSE associated with harmonic generation in Floquet non-Hermitian systems\nrepresents a previously unexplored regime. While several studies have proposed the concept of\nFloquet NHSE, the role of harmonics has never been addressed, either theoretically or\nexperimentally. This work identifies the critical gap and uncovers novel physics. From the\nperspective of innovation, this work is timely and groundbreaking, tackling an untapped topic.\nRegarding the results, the authors provide extensive details in both theoretical analysis and\nexperimental approaches. Overall, the manuscript is clearly organized and well written, I would like\nto recommend its publication if the following questions are well addressed.\nReply: We sincerely thank you for the high evaluation of our work. We also greatly appreciate your\nconstructive suggestions and insightful questions, all of which have been carefully considered and\naddressed below.\n1. In the theoretical model in Fig. 2, both nearest-neighbor and next-nearest-neighbor non-\nreciprocal couplings are considered. Would longer-range non-reciprocal couplings lead to richer\nbehaviors in harmonic NHSE?\nReply: This is a good question. In fact, in static lattices, longer-range non-reciprocal couplings have\nbeen demonstrated to generate complex spectral winding topology, leading to much richer forms\nof the NHSE [e.g., Science 371, 1240 (2021); Phys. Rev. B 107, 094111 (2023)]. Accordingly, when\nFloquet modulation is introduced into such static systems and a fundamental excitation is applied,\ndifferent harmonic frequencies can fall into spectral loops with distinct winding numbers. As a\nresult, different harmonics are expected to exhibit richer NHSE behaviors.\n2. What is the specific definition of the spectral winding number used in this work? Could the\nauthors provide it explicitly? How to distinguish harmonic NHSE from the Floquet-induced NHSE?\n(cid:3029)\nas\n1 (cid:3095) 𝜕𝑓(cid:4666)𝑘(cid:4667)⁄𝜕𝑘\n𝑣 (cid:3404) (cid:3505) 𝑑𝑘,\n2𝜋𝑖 𝑓(cid:4666)𝑘(cid:4667)(cid:3398)𝑓\n(cid:2879)(cid:3095) (cid:3029)\nwhere 𝑓(cid:4666)𝑘(cid:4667) is the eigenfrequency band.\nWhile Both the harmonic NHSE and Floquet-induced NHSE can be characterized by the nontrivial\nspectral winding numbers, they can be distinguished by the following two points. First, the Floquet\nNHSE, primarily involving two- or multi-band models, emphasizes the nontrivial point-gap topology\n(and hence the NHSE) induced by Floquet modulation, even when the constituent instantaneous\nHamiltonians lack such topology. In contrast, our harmonic NHSE, which arises even in single-band\nmodels, highlights the harmonics generated by Floquet modulation, whose NHSE can be flexibly\ntailored according to the underlying spectral winding topology. Second, the Floquet NHSE focuses\non the eigenvalue problem of effective Hamiltonian, where higher-order Floquet states share the\neigen-properties of the zeroth-order one and are generally not considered. By contrast, the\nharmonic NHSE emerges in the excitation problem of a finite real-space lattice. In realistic settings,\nsince only the zeroth-order Floquet component is directly excited, the generated higher-order ones\ncan exhibit distinct and independently observable responses.\nInspired by your questions, we have added the explicit definition of the spectral winding number\nin the revised manuscript (Page 5), i.e., “As shown in Fig. 2b, both the spectrum of 𝐻(cid:3365) and the\n(cid:3038)\nzeroth-order Floquet spectrum of ℋ form a counterclockwise loop in the complex frequency\n(cid:3038)\nplane. A reference frequency 𝑓 enclosed by the spectrum 𝑓(cid:4666)𝑘(cid:4667) carries a nontrivial spectral\n(cid:3029)\nwinding number\n1 (cid:3095) 𝜕𝑓(cid:4666)𝑘(cid:4667)⁄𝜕𝑘\n𝑣 (cid:3404) (cid:3505) 𝑑𝑘 (cid:3404)(cid:3397)1, (cid:4666)3(cid:4667)\n2𝜋𝑖 𝑓(cid:4666)𝑘(cid:4667)(cid:3398)𝑓\n(cid:2879)(cid:3095) (cid:3029)\nindicating the presence of a unipolar harmonic NHSE10,11.”\n3. Could the presence of loss in experiments affect the observation of harmonic NHSE? What are\nthe main sources of loss?\nReply: Yes. The presence of loss can obscure the observation of harmonic NHSE, since the\nattenuation caused by dissipation competes with the amplification induced by NHSE. This issue has\nbeen analyzed in more detail in our previous study [Phys. Rev. B 110, L140305 (2024)]. In our cavity-\ntube setup, the dominant losses mainly stem from viscous and frictional effects associated with air\nand the cavity walls.\n4. In the experiments shown in Fig. 5, why is the lower limit of the duty ratio R set to 0.4? Is there a\nspecific consideration for this choice?\nReply: In the experiments shown in Fig. 5, to ensure that the PBC spectra of the time-averaged\nHamiltonians (and thus the harmonic NHSE) remain quantitatively unchanged when varying the\nduty ratio 𝑅, we keep the zeroth-order Fourier component of 𝜅 constant, i.e., 𝑅𝜅 (cid:3404)𝜅̅.\n(cid:3005) (cid:2868)\nUnder\nthis condition, a smaller 𝑅 requires a larger magnitude 𝜅 for the square-wave dynamic coupling,\n(cid:2868)\nwhich may lead to energy saturation and eventual breakdown of the experimental system. In our\npractical experiments, we found that 𝑅 (cid:3404)0.4 is the lowest value that can be achieved without\n2\ncausing system breakdown.\nTo clarify this point, we have modified the related sentences in the revised manuscript (Page 12),\ni.e., “It is worth noting that a small value of 𝑅 requires a large 𝜅 of square-wave dynamic coupling,\n(cid:2868)\nwhich may lead to energy saturation and eventual breakdown of the experimental system.\nTherefore, we limit our experimental demonstrations to 0.4(cid:3409)𝑅 (cid:3409)0.8.”\n5. How universal is this control mechanism? Does it persist for sinusoidal (rather than square-wave)\nmodulation, or in systems with stronger dissipation?\nReply: The control mechanism of Floquet modulation is universal and can be, in principle,\nreconfigured to realize arbitrary forms of dynamic coupling, including the sinusoidal one. This\nreconfigurability originates from the working principle of our real-time unidirectional coupler: the\nsound signal in a cavity is collected by a microphone, then modulated through a preamplifier, a\nvoltage-controlled amplifier, and a phase shifter, before being output into another cavity via a\nloudspeaker. In particular, the gain factor of the voltage-controlled amplifier is controlled in real\ntime by a waveform generator. Therefore, to achieve sinusoidal Floquet modulation, one simply\nneeds to replace the square-wave signal with a sinusoidal one. Remarkably, we employ square-\nwave modulation in our work because of its simplicity and ease of experimental implementation.\nMoreover, the Floquet modulation mechanism is independent of dissipation. As long as a steady\nstate can be established in the system, the Floquet modulation mechanism remains valid.\n6. Compared to conventional Floquet NHSE, does harmonic NHSE will show unique dynamic\ncharacteristics?\nReply: This is an excellent question. We first clarify that conventional Floquet NHSE typically\nfocuses on eigenstates rather than excitation states, and therefore largely neglects harmonic\neffects induced by time-periodic modulation. In contrast, the harmonic NHSE addressed in our\nwork explicitly emphasizes the role of harmonic responses under a fundamental excitation, which\nleads to distinct dynamical characteristics. As shown in Figs. R1 and R2, we compute the temporal\nwave dynamics of both the Floquet Hatano–Nelson (FHN) lattice and the long-range FHN lattice\nunder a continuous cosinoidal excitation. The lattice parameters are identical to those used in the\nexperiments in Figs. 3 and 4 of the main text. During the time evolution, the initial single-frequency\nexcitation and the generated harmonics produce interference textures, a characteristic feature of\nFloquet-driven systems. For the unipolar harmonic NHSE, the hybridized harmonics can collectively\naccumulate toward the same boundary (Figs. R1a–R1c). By contrast, for the bipolar harmonic NHSE,\nthese mixed harmonic components could dynamically localize toward opposite boundaries (Figs.\nR2a–R2c). However, because different harmonics continuously convert into one another in the\ntime domain, it is difficult to disentangle the localization behavior of individual harmonics directly\nfrom the temporal dynamics alone. To resolve this, we perform a fast Fourier transform (FFT) of\nthe steady time-domain signals. The resulting frequency-domain spectra clearly reveal the unipolar\n(Figs. R1d–R1f) and bipolar (Figs. R2d–R2f) skin localization of different harmonics. Remarkably,\nthe harmonic NHSEs extracted from the FFT analysis are consistent with the results obtained\ndirectly from frequency-domain simulation presented in the main text, and both agree well with\nthe experimental observations (Figs. R1g–R1i and R2g–R2i).\nwith the full analysis provided in Supplementary Note 6. The added sentences in the main text read:\n“In addition, under continuous cosinoidal excitation, our systems exhibit interference textures in\nits temporal wave dynamics, a hallmark of Floquet-driven systems. The harmonic NHSEs extracted\nfrom these temporal dynamics agree well with both frequency-domain simulations and\nexperimental observations (see Supplementary Note 6).”\nFig. R1. a–c Temporal wave dynamics of a cosinoidal signal incident from the middle site of the\nFHN lattice, under different excitation frequencies f and modulation frequencies Ω . d–f\nresults based on frequency-domain framework, accompanied by the experimental results. Stars\ndenote the source excitations.\nlong-range FHN lattice, under different excitation frequencies f and modulation frequencies Ω. d–\nf Corresponding FFT spectra obtained from the steady time-domain signals in a–c. g–i Simulation\nresults based on frequency-domain framework, accompanied by the experimental results. tars\ndenote the source excitations.\n7. For multi-band models, does the coupling between different Floquet branches lead to\nphenomena distinct from those in single-band models, or affect the formation of the Harmonic\nNHSE? Can the topological phases still be distinguished by the low-order Floquet spectrum of the\nHamiltonian Hk? Does it introduce new topological phenomena? Please provide a brief discussion.\nReply: This is an insightful question. For multi-band models, the topological phases (or NHSE\nmorphologies) of different harmonics can still be distinguished by the zeroth-order Floquet\nspectrum of the enlarged Hamiltonian ℋ , since only the zeroth-order component in the excitation\n(cid:3038)\nvector can be directly activated under realistic excitation conditions [see Eq. (2) in the main text].\nWithin this framework, the coupling between different Floquet branches can indeed affect the\nformation of the harmonic NHSE, as it may induce new spectral topologies for ℋ that differ from\n(cid:3038)\nthose of the time-averaged Hamiltonian 𝐻(cid:3365) . However, it should be noted that for single-band\n(cid:3038)\nsystems, different Floquet branches remain non-interacting even when they overlap, so the zeroth-\norder Floquet spectrum of ℋ is identical to that of 𝐻(cid:3365) . By contrast, in multi-band systems,\n(cid:3038) (cid:3038)\n5\ndifferent Floquet branches can interact when they come close to each other, making the zeroth-\norder Floquet spectrum of ℋ distinct from that of 𝐻(cid:3365) .\n(cid:3038) (cid:3038)\nFollowing your constructive suggestion, we have provided a brief discussion in the revised\nmanuscript (Page 5), i.e., “Consequently, the harmonic NHSE under realistic excitation can be fully\ncharacterized by examining solely on the zeroth-order Floquet spectrum of ℋ , or equivalently,\n(cid:3038)\nthe spectrum of 𝐻(cid:3365) for single-band models. (Remarkably, for two- or multi-band models, the\n(cid:3038)\nzeroth-order Floquet spectrum of ℋ can differ fundamentally from the spectrum of 𝐻(cid:3365) , due to\n(cid:3038) (cid:3038)\nthe interaction between different Floquet branches upon closing46.) The above theoretical\nframework provides a general route to account for harmonic effects in Floquet non-Hermitian\nsystems.”\n8. In figure3c, 4b, only s21 and s31 are considered to extract the coupling strength. If s12/s21,\ns13/s31 are all considered, can the coupling still fit well with the experiment data?\nReply: Yes. Even when the transmissions s12/s21 and s13/s31 are all considered, the extracted\ncouplings remain in good agreement with the experimental data. This consistency has already been\nvalidated in our previous works [Phys. Rev. Lett. 130, 017201 (2023); Phys. Rev. Lett. 134, 126603\n(2025)]. Therefore, in the present study, it is justified to use only the representative transmission\nresponses s21 and s31 to reliably fit the experimental data.\n9. How about the energy distribution at the other frequencies? Can you provide the spectra\nresponse at single stie (at edge / middle of the model)?\nReply: In our Floquet system, we did not use a broadband sweep-frequency excitation and\ntherefore do not have broadband spectra responses at individual sites. The reason is that different\nharmonics would interfere with each other under broadband excitation. As illustrated in Fig. R3,\nan excitation at frequency 𝑓 generates three harmonics at 𝑓 (cid:3398)Ω , 𝑓 , and 𝑓 (cid:3397)Ω (see yellow\n(cid:2868) (cid:2868) (cid:2868) (cid:2868)\narrows); an excitation at 𝑓 (cid:3398)Ω produces harmonics at 𝑓 (cid:3398)2Ω, 𝑓 (cid:3398)Ω, and 𝑓 (see red arrows);\n(cid:2868) (cid:2868) (cid:2868) (cid:2868)\nand an excitation at 𝑓 (cid:3397)Ω produces harmonics at 𝑓 , 𝑓 (cid:3397)Ω , and 𝑓 (cid:3397)2Ω (see blue arrows).\n(cid:2868) (cid:2868) (cid:2868) (cid:2868)\nWhen these excitations coexist under a broadband input, the measured response at any site\nbecomes a superposition of multiple harmonic components, making it impossible to isolate the\nNHSE of a specific harmonic.\nFig. R3. Illustration of harmonic responses under broadband excitation.\nThis submitted manuscript by Zhang et al. presented a novel and compelling study on the topic of\nharmonic non-Hermitian skin effect (NHSE). The authors successfully bridged the fields of time-\ndependent Floquet engineering and non-Hermitian topology by introducing the harmonic waves,\nwhich were experimentally generated by time-periodic driving. The distinct and controllable skin\neffect morphologies were then exhibited. Overall, I think the concept is innovative, the theoretical\nframework is solid, and the experimental realization in an acoustic platform with real-time\nmodulations is impressive and provides clear, convincing evidence. This work may open a new\ndirection in the study of dynamic non-Hermitian systems and is likely to be of significant interest to\nthe broader physics community. I believe this paper is well-suited for publication in Nature\nCommunications after the following suggested revisions are addressed.\nReply: We sincerely thank you for the high evaluation of our work. We also greatly appreciate your\nconstructive suggestions and insightful questions, all of which have been carefully considered and\naddressed below.\n1. The main text presented simulated results. It would be beneficial for the authors to describe the\nsimulation methods, particularly explaining how the active unidirectional couplers were realized in\nthe simulations.\nReply: We would like to explain that our simulated results are obtained by numerically solving the\nexcitation equation [Eq. (2) in the main text] with experimentally calibrated parameters, rather\nthan by using numerical simulation softwares. In this framework, the role of the active\nunidirectional couplers is achieved by explicitly imposing unidirectional coupling, which faithfully\nreproduces their functionality in the experiment.\n2. I am interested in several experimental details. Specifically, more information on the feed-back\nmicrophones and loudspeakers (such as types or models) used for the real-time unidirectional\ncouplers. Additionally, how to ensure the consistent performance characteristics of these compact\nacoustic components?\nReply: In the Methods section, we provide the specific types of the preamplifier, voltage-controlled\namplifier, and phase shifter. However, we do not list type numbers for the microphones and\nloudspeakers because these components are custom-fabricated and thus have no commercial\nequivalents. To ensure consistency, we calibrate a large batch of devices and select those with the\nmost uniform performance characteristics for use in the experiments.\n3. In the Supplementary Information Note 3, the authors employed the transmission amplitude\nspectra |𝑆_21| and |𝑆_31| to fit the relevant parameters. Since it is challenging to directly\nmeasure the acoustic pressure at the source, I am interested in the measurement methodology for\nthe individual transmission coefficients. Additionally, the fitting equation in Eq. (S6) differs from the\none used in the authors' prior work (10.1103/PhysRevLett.130.017201). Could the authors\nelaborate on the consideration behind this modification in the methodology please?\nReply: This is an insightful question. First, regarding the measurement of the source response,\nwe\nfabricated a dedicated device that places the loudspeaker and microphone in close proximity,\nallowing us to directly measure the source pressure amplitude S . Accordingly, each experimental\n0\n7\ntransmission coefficient, e.g., |S |, is obtained by measuring the pressure in cavity 2 under\n21\nexcitation at cavity 1 and further normalizing the result by S . Alternatively, one may introduce a\n0\nfree fitting parameter accounting for the unknown source amplitude, which would still yield\naccurate fitting results even if the absolute source pressure is not directly measured.\nSecond, the fitting formula used in this work [e.g., Eq. (S2)] is derived from the Green’s function\nmethod, whereas the fitting formula in our previous work [Phys. Rev. Lett. 130. 017201 (2023)] is\nbased on coupled-mode theory. The key difference is that coupled-mode theory explicitly accounts\nfor the additional insertion losses introduced by the loudspeaker and microphone, while the\nGreen’s function method does not. In our experiments, these insertion losses are very small and\ncan be absorbed into the intrinsic loss of each cavity, allowing us to use the Green’s function\napproach without loss of accuracy. In practice, both methods remain valid.\n4. Other minor suggestions:\n4.1 Some figure captions (e.g., Figs. 2, 4) are very dense with parameter lists. Consider moving some\nof these details to the main text or methods for improved readability.\nReply: We appreciate this suggestion. We have made appropriate revisions to the figure captions\nto improve readability.\n4.2 On Page 7, lines 128-130, the text \"...the +1st- and −1st-order harmonics undergo unidirectional\naccumulation toward the right and left boundaries, respectively...\" seems to contradict the caption\nof Fig. 2f, which states the +1st harmonic goes left and the -1st goes right. Please verify and correct\nthis for consistency.\nReply: Thank you for pointing this out. We have corrected the direction of accumulation to ensure\nfull consistency between the main text and the caption of Fig. 2f.\nIn this manuscript, Zhang et al. introduce and experimentally demonstrate the concept of a\n\"harmonic non-Hermitian skin effect (NHSE)\" in a Floquet-engineered acoustic lattice. The core idea\nis that when a monochromatic wave excites a time-periodically modulated non-Hermitian lattice,\nit can generate multiple harmonics, each of which can exhibit a distinct skin localization\nphenotype—unipolar or bipolar—dictated by the spectral winding topology at its respective\nfrequency. The authors provide a theoretical framework based on a frequency-domain analysis and\npresent compelling experimental evidence using a reconfigurable acoustic platform with active,\nreal-time modulations. The ability to selectively control and spatially separate different harmonic\ncomponents via a single-frequency excitation is a key experimental achievement.\nI find this work to be of high quality, timely, and highly suitable for publication in Nature\nCommunications. The concept of \"harmonic NHSE\" is novel and represents a significant extension\nof the non-Hermitian skin effect into the realm of Floquet systems with genuine time modulation.\nThe study successfully bridges a gap in the field by focusing on the harmonic generation aspect,\nwhich is a fundamental feature of Floquet systems but had been overlooked in previous studies of\nFloquet NHSE, particularly those relying on synthetic dimensions (e.g., waveguide arrays). The\ncombination of a clear theoretical proposal with a sophisticated and convincing experimental\ndemonstration in acoustics is a major strength. The observation of both unipolar and, more\nimpressively, bipolar harmonic NHSE, where different harmonics localize at opposite ends of the\nsystem, is a striking result. The findings are likely to stimulate immediate interest across various\nplatforms in photonics, circuit QED, and cold atoms.\nI am therefore strongly in favor of publication, provided that the authors can address the following\npoints to further enhance the clarity and theoretical rigor of their otherwise excellent manuscript.\nReply: We sincerely thank you for the high evaluation and strong support of our work. We also\ngreatly appreciate your constructive suggestions and insightful questions, all of which have been\ncarefully considered and addressed below.\n1. GBZ Calculation for Floquet Model: The topological origin of the skin effect for each harmonic is\ncentral to the present approach. To make this connection more transparent and rigorous, I\nencourage the authors to calculate and present the Generalized Brillouin Zone (GBZ) for their\nFloquet model, following the established non-Bloch band theory (e.g., Yao and Wang, PRL, 2018).\nDemonstrating how the GBZ and the associated complex wavevectors dictate the skin localization\ndirection for the fundamental wave and its harmonics would significantly strengthen the\ntopological foundation of the \"harmonic NHSE\" concept.\nReply: Thank you for your constructive suggestion. As demonstrated in the main text, for the single-\nband Floquet model considered in this work, the harmonic NHSE is governed by the spectral\ntopology of the time-averaged Hamiltonian 𝐻(cid:3365) . Accordingly, we compute the Generalized Brillouin\n(cid:3038)\nZone (GBZ) of 𝐻(cid:3365) under open boundary conditions (OBCs), following the non-Bloch band theory.\n(cid:3038)\nAs shown in Fig. R4a, the GBZ of the Floquet Hatano–Nelson (FHN) model forms a circular loop\nlocated entirely inside the unit circle, which corresponds to complex wave vector 𝑘(cid:3398)𝑖ln𝑟 with\nln𝑟 (cid:3407)0 and non-Bloch phase factor 𝛽 (cid:3404)𝑟𝑒(cid:3036)(cid:3038) with 𝑟 (cid:3407)1. This indicates that all eigenstates (and\n9\nthus the harmonics) exhibit leftward skin localization, yielding a unipolar harmonic non-Hermitian\nskin effect (NHSE). By contrast, for the long-range FHN model, the GBZ becomes an irregular closed\ntrajectory, as shown in Fig. R4b. Eigenstates (or harmonics) associated with GBZ segments inside\nthe unit circle localize toward the left boundary, while those associated with segments outside the\nunit circle localize toward the right boundary (corresponding to complex wave vector 𝑘(cid:3398)𝑖ln𝑟 with\nln𝑟 (cid:3408)0 and non-Bloch phase factor 𝛽 (cid:3404)𝑟𝑒(cid:3036)(cid:3038) with 𝑟 (cid:3408)1 ). The coexistence of these two\nlocalization directions gives rise to a bipolar harmonic NHSE. Remarkably, the intersection points\nbetween the GBZ and the unit circle host purely real Bloch wave vectors, corresponding to\nextended eigenstates (or harmonics). These GBZ results explicitly demonstrate how the complex\nwave vectors underlying 𝐻(cid:3365) determine the localization directions of the harmonics, thereby\n(cid:3038)\nproviding a direct and rigorous topological foundation for the harmonic NHSE observed in our\nFloquet system.\nFollowing your suggestion, we have added the GBZ description in the revised main text (Pages 5\nand 6), with the above interpretations incorporated into revised Supplementary Note 1. The\nrevisions in the main text reads “We further calculate the generalized Brillouin zone (GBZ) for 𝐻(cid:3365)\n(cid:3038)\nfollowing the established non-Bloch band theory4, which forms a circular loop inside the unit circle\nand thereby confirms the unipolar harmonic NHSE (see Supplementary Note 1).” and “The GBZ\ncontaining segments both inside and outside the unit circle is presented in Supplementary Note 1.”\nFig. R4. a GBZ of the time-averaged Hamiltonian of the FHN lattice. The unit circle corresponds to\nthe conventional BZ. b Similar to a but for long-range FHN lattice. Blue dots mark the Bloch-like\npoints.\n2. Clarification on the Extended Nature of the Zeroth Harmonic (f₀): In Figure 1 and the\ncorresponding discussion of the bipolar NHSE, the zeroth-order harmonic (f₀) is depicted as\nextended. This appears counterintuitive because the underlying static lattice, with its nonreciprocal\nstatic hoppings (κ_L, κ_R), should itself exhibit clear features of NHSE, localizing all eigenstates.\nThe authors should provide a clearer explanation for this observation. Is this extended behavior a result\nof a specific parameter choice (e.g., f₀ being exactly at a Bloch-like point in the spectrum)? A\ndetailed discussion on why f₀ resists skin localization in this specific scenario, while the system\nremains non-Hermitian, is necessary to resolve this apparent contradiction for the reader.\nReply: This is an insightful question. First, we confirm that the zeroth-order harmonic f₀ in bipolar\nharmonic NHSE is indeed chosen to lie exactly at a Bloch-like point of the spectrum. Below we\nexplain why a Bloch-like point corresponds to an extended state despite the underlying\nnonreciprocal couplings. In simple settings such as the Hatano–Nelson model with nearest-\nneighbor nonreciprocal couplings, all eigenstates localize toward the direction of the dominant\ncoupling, giving the well-known unipolar NHSE [Phys. Rev. Lett. 77, 570 (1996)]. However, in more\ncomplex settings—e.g., when long-range nonreciprocal couplings are introduced—the situation\nbecomes richer. Besides the majority of eigenstates that exhibit leftward and rightward NHSE\n(bipolar NHSE), there can exist isolated extended eigenstates, whose eigenenergies exactly\ncorrespond to the Bloch-like points [Phys. Rev. Lett. 123, 246801 (2019); Phys. Rev. Res. 1, 023013\n(2019)]. These points and the associated bipolar NHSE defy the oversimplified picture in which all\neigenstates localize toward the dominant hopping direction. These Bloch-like extended states can\nbe understood from two complementary perspectives. 1) Spectral winding perspective: A Bloch-\nlike point is an eigenenergy at which the periodic boundary condition (PBC) spectrum intersects\nthe open boundary condition (OBC) spectrum (see Fig. R5a). Its PBC spectral winding number is\ntherefore zero, unlike the majority of eigenenergies that carry a nonzero winding number.\nAccording to the topological origin of the NHSE [Phys. Rev. Lett. 124, 086801 (2020); Phys. Rev. Lett.\n125, 126402 (2020)], only eigenstates associated with nonzero winding undergo skin localization.\nThe Bloch-like eigenstate, having zero winding, remains extended. 2) GBZ perspective: At a Bloch-\nlike point, the GBZ intersects the unit circle, meaning that the corresponding Bloch wave vector is\npurely real (see Fig. R5b). As a result, the wave function is spatially extended [Phys. Rev. Lett. 121,\n086803 (2018)]. In contrast, the bipolar NHSE emerges because points inside (outside) the unit\ncircle correspond to exponential localization toward the left (right) boundary.\nFollowing your suggestion, we have clarified the above points in the revised main text (Page 7),\ni.e., “Notably, the extended nature at the Bloch-like point can be interpreted in terms of both the\nvanishing spectral winding number and the specific GBZ with purely real Bloch wave vectors4,8.”\nFig. R5. a PBC and OBC spectra of the time-averaged Hamiltonian of the long-rang FHN lattice. The\neigenenergy with zero spectral winding number denotes the Bloch-like point (blue dot). b GBZ of\nthe time-averaged Hamiltonian of the long-rang FHN lattice. The point where GBZ intersects with\nunit cell corresponds to the Bloch-like point (blue dots).\n3. Justification for the Excitation Problem Framework: The authors state that \"the harmonic NHSE\ncan be adequately characterized by focusing solely on the zeroth-order Floquet spectrum of\nmathcal{H}_k, or equivalently, the spectrum of Ḥ_k\", justifying this by the orthogonality of Fourier\nharmonics in the excitation vector P_m. While this simplification is crucial for the analysis, the\nreasoning provided is somewhat obscure. I find it difficult to fully grasp why the topology of the\ntime-averaged Hamiltonian \\bar{H}_k alone determines the skin effect for all higher harmonics\ngenerated in the excitation problem. The authors should elaborate on this point, perhaps with a\nmore detailed derivation or a physical argument in the Methods or Supplementary Information,\nto\nclarify the validity of this approach.\nReply: First, we clarify why only the zeroth-order harmonic component is directly excited in realistic\nexcitation settings. For an N-site Floquet non-Hermitian lattice described by the real-space\nHamiltonian 𝑯 (cid:4666)𝑡(cid:4667), the system’s dynamics under excitation is governed by the nonhomogeneous\n𝑟\nSchrödinger equation:\nd\n𝑖 𝐐(cid:4666)𝑡(cid:4667)(cid:3404)2𝜋𝑯 (cid:4666)𝑡(cid:4667)𝐐(cid:4666)𝑡(cid:4667)(cid:3397)𝐏(cid:4666)𝑡(cid:4667), (cid:4666)R1(cid:4667)\nd𝑡 (cid:3045)\nwhere\n𝐏(cid:4666)𝑡(cid:4667)(cid:3404)2𝜋𝑒(cid:2879)(cid:3036)(cid:2870)(cid:3095)(cid:3033)(cid:3116)(cid:3047)(cid:4666)𝛿 ,𝛿 ,…,𝛿 ,…,𝛿 (cid:4667)(cid:2904), (cid:4666)R2(cid:4667)\n(cid:2869)(cid:3046) (cid:2870)(cid:3046) (cid:3041)(cid:3046) (cid:3015)(cid:3046)\nrepresents a time-domain excitation at a specific site 𝑛 (cid:3404)𝑠 (only 𝛿 (cid:3404) 1) with monochromatic\n𝑠𝑠\nfrequency 𝑓 , and 𝐐(cid:4666)𝑡(cid:4667) represents the time-domain response across all sites. 𝑯 (cid:4666)𝑡(cid:4667) and 𝐐(cid:4666)𝑡(cid:4667) can\n0 𝑟\nbe written as Fourier series:\n(cid:2878)(cid:2998)\n𝑯 (cid:4666)𝑡(cid:4667)(cid:3404) (cid:3533) 𝚫 𝑒(cid:2879)(cid:3036)(cid:2870)(cid:3095)(cid:3040)(cid:2960)(cid:3047), (cid:4666)R3(cid:4667)\n(cid:3045) (cid:3040)\n(cid:3040)(cid:2880)(cid:2879)(cid:2998)\n(cid:2878)(cid:2998) (cid:2878)(cid:2998) (cid:2878)(cid:2998) (cid:2904)\n𝐐(cid:4666)𝑡(cid:4667)(cid:3404)(cid:3437) (cid:3533) 𝑎(cid:2869) 𝑒(cid:2879)(cid:3036)(cid:2870)(cid:3095)(cid:4666)(cid:3033)(cid:3116)(cid:2878)(cid:3040)(cid:2960)(cid:4667)(cid:3047),…, (cid:3533) 𝑎(cid:3041)𝑒(cid:2879)(cid:3036)(cid:2870)(cid:3095)(cid:4666)(cid:3033)(cid:3116)(cid:2878)(cid:3040)(cid:2960)(cid:4667)(cid:3047),…, (cid:3533) 𝑎(cid:3015)𝑒(cid:2879)(cid:3036)(cid:2870)(cid:3095)(cid:4666)(cid:3033)(cid:3116)(cid:2878)(cid:3040)(cid:2960)(cid:4667)(cid:3047)(cid:3441) ,(cid:4666)R4(cid:4667)\n(cid:3040) (cid:3040) (cid:3040)\n(cid:3040)(cid:2880)(cid:2879)(cid:2998) (cid:3040)(cid:2880)(cid:2879)(cid:2998) (cid:3040)(cid:2880)(cid:2879)(cid:2998)\nwhere 𝚫 (cid:3404) (cid:2869) (cid:1516) (cid:3021) 𝑯 (cid:4666)𝑡(cid:4667)𝑒(cid:3036)(cid:2870)(cid:3040)(cid:3095)(cid:2960)(cid:3047)d𝑡 is the 𝑚th-order Fourier components of 𝑯 (cid:4666)𝑡(cid:4667), and 𝑛 (cid:3404)1:𝑁\n(cid:3040) (cid:3021) (cid:2868) (cid:3045) (cid:3045)\ndenot the site index. By plugging Eqs. (R2-R4) into Eq. (R1), and using the orthogonality properties\nof Fourier harmonics:\n〈𝑒(cid:2879)(cid:3036)(cid:2870)(cid:3095)(cid:3040)(cid:2960)(cid:3047),𝑒(cid:2879)(cid:3036)(cid:2870)(cid:3095)(cid:3040)(cid:4594)(cid:2960)(cid:3047)〉(cid:3404)𝛿\n(cid:3040)(cid:3040)(cid:4594), we get\n𝚫 (cid:3397)𝜁 𝐈 𝚫 𝚫 ⋯ ⋯ 𝚫 𝐐 𝐏\n(cid:2868) (cid:3040) (cid:2879)(cid:2869) (cid:2879)(cid:2870) (cid:2879)(cid:2870)(cid:3040) (cid:2879)(cid:3040) (cid:2879)(cid:3040)\n⎛ ⎞⎛ ⎞ ⎛ ⎞\n𝚫 ⋱ ⋱ ⋱ ⋯ ⋮ ⋮ ⋮\n⎜ (cid:2869) ⎟⎜ ⎟ ⎜ ⎟\n⎜ ⎟⎜ ⎟ ⎜ ⎟\n⎜ 𝚫 (cid:2870) ⋱ 𝚫 (cid:2868) (cid:3397)𝜁 (cid:2869) 𝐈 𝚫 (cid:2879)(cid:2869) 𝚫 (cid:2879)(cid:2870) ⋮ ⎟⎜ 𝐐 (cid:2879)(cid:2869)⎟ ⎜ 𝐏 (cid:2879)(cid:2869)⎟\n⎜ ⎟\n⎜ ⎟ ⎜ ⎟\n⎜ ⋮ ⋱ 𝚫 𝚫 (cid:3397)𝜁 𝐈 𝚫 ⋱ ⋮ ⎟ 𝐐 (cid:3404) 𝐏 ,(cid:4666)R5(cid:4667)\n(cid:2869) (cid:2868) (cid:2868) (cid:2879)(cid:2869) ⎜ (cid:2868) ⎟ ⎜ (cid:2868) ⎟\n⎜ ⎟\n⎜ ⎟ ⎜ ⎟\n⎜ ⎟\n⎜ ⋮ 𝚫 (cid:2870) 𝚫 (cid:2869) 𝚫 (cid:2868) (cid:3398)𝜁 (cid:2869) 𝐈 ⋱ 𝚫 (cid:2879)(cid:2870) ⎟⎜ 𝐐 (cid:2869) ⎟ ⎜ 𝐏 (cid:2869) ⎟\n⎜ ⎟⎜ ⎟ ⎜ ⎟\n⎜ ⋮ … ⋱ ⋱ ⋱ 𝚫 ⎟⎜ ⋮ ⎟ ⎜ ⋮ ⎟\n(cid:2879)(cid:2869)\n⎝ 𝚫 ⋯ ⋯ 𝚫 𝚫 𝚫 (cid:3398)𝜁 𝐈⎠⎝𝐐 ⎠ ⎝𝐏 ⎠\n(cid:2870)(cid:3040) (cid:2870) (cid:2869) (cid:2868) (cid:3040) (cid:3040) (cid:3040)\nwhere 𝜁 (cid:3404)𝑓 (cid:3397)𝑚Ω denotes the 𝑚 th-order harmonic frequency generated by the\n(cid:3040) (cid:2868)\nmonochromatic excitation, 𝐐 (cid:3404)(cid:4666)𝑎(cid:2869) ,𝑎(cid:2870),…,𝑎(cid:3041),…𝑎(cid:3015)(cid:4667)(cid:2904) represents the frequency-domain\n(cid:3040) (cid:3040) (cid:3040) (cid:3040) (cid:3040)\nresponse of the 𝑚 th-order harmonic across the lattice, 𝐏 (cid:3404)\n(cid:3040)\n2𝜋(cid:4666)𝛿 𝛿 ,𝛿 𝛿 ,…,𝛿 𝛿 ,…,𝛿 𝛿 (cid:4667)(cid:2904) describes the excitation applied at site 𝑠 . Notably,\n(cid:3040)(cid:2868) (cid:2869)(cid:3046) (cid:3040)(cid:2868) (cid:2870)(cid:3046) (cid:3040)(cid:2868) (cid:2871)(cid:3046) (cid:3040)(cid:2868) (cid:3015)(cid:3046)\nonly the component with 𝑚 (cid:3404)0 and 𝑛 (cid:3404)𝑠 satisfies 𝛿 𝛿 (cid:3404)1 , while all other components\n(cid:3040)(cid:2868) (cid:2871)(cid:3046)\nvanish. This reflects the fact that, in realistic settings, only the zeroth-order harmonic component\nat site s can be directly excited.\nSecond, we explain why the spectral topology of the time-averaged Hamiltonian 𝐻(cid:3365)\n(cid:3038)\ndetermines the NHSE for all harmonics. For our single-band Floquet system described by the\nfrequency-domain Hamiltonian\n𝐻(cid:3365) (cid:3397)𝑚Ω Δ Δ ⋯ ⋯ Δ\n(cid:3038) (cid:2879)(cid:2869) (cid:2879)(cid:2870) (cid:2879)(cid:2870)(cid:3040)\n⎛ Δ ⋱ ⋱ ⋱ ⋯ ⋮ ⎞\n(cid:2869)\n⎜ ⎟\nΔ ⋱ 𝐻(cid:3365) (cid:3397)Ω Δ Δ ⋮\n⎜ (cid:2870) (cid:3038) (cid:2879)(cid:2869) (cid:2879)(cid:2870) ⎟\nℋ (cid:3404)⎜ ⋮ ⋱ Δ 𝐻(cid:3365) Δ ⋱ ⋮ ⎟, (cid:4666)R6(cid:4667)\n(cid:3038) (cid:2869) (cid:3038) (cid:2879)(cid:2869)\n⎜ ⎟\n⎜ ⋮ Δ (cid:2870) Δ (cid:2869) 𝐻(cid:3365) (cid:3038) (cid:3398)Ω ⋱ Δ (cid:2879)(cid:2870) ⎟\n⎜ ⎟\n⋮ … ⋱ ⋱ ⋱ Δ\n(cid:2879)(cid:2869)\n⎝ Δ ⋯ ⋯ Δ Δ 𝐻(cid:3365) (cid:3398)𝑚Ω⎠\n(cid:2870)(cid:3040) (cid:2870) (cid:2869) (cid:3038)\nthere exists a unitary transformation 𝑈(cid:2993)ℋ 𝑈 (cid:3404)ℋ(cid:3561) , under which ℋ becomes block-diagonal:\n(cid:3038) (cid:3038) (cid:3038)\n𝐻(cid:3365) (cid:3397)𝑚Ω 0 0 ⋯ ⋯ 0\n(cid:3038)\n⎛ 0 ⋱ ⋱ ⋱ ⋯ ⋮ ⎞\n⎜ ⎟\n0 ⋱ 𝐻(cid:3365) (cid:3397)Ω 0 0 ⋮\n⎜ (cid:3038) ⎟\nℋ(cid:3561) (cid:3404)⎜ ⋮ ⋱ 0 𝐻(cid:3365) 0 ⋱ ⋮ ⎟. (cid:4666)R7(cid:4667)\n(cid:3038) (cid:3038)\n⎜ ⎟\n⎜ ⋮ 0 0 𝐻(cid:3365) (cid:3038) (cid:3398)Ω ⋱ 0 ⎟\n⎜ ⎟\n⋮ … ⋱ ⋱ ⋱ 0\n⎝ 0 ⋯ ⋯ 0 0 𝐻(cid:3365) (cid:3398)𝑚Ω⎠\n(cid:3038)\nThis block-diagonalization indicates that the PBC spectrum of ℋ is simply an infinite collection of\n(cid:3038)\ncopies of the PBC spectrum of 𝐻(cid:3365) , shifted by the Floquet modulation frequency Ω. Although the\n(cid:3038)\noriginal Hamiltonian ℋ contains couplings between different Floquet replicas (i.e., Δ with 𝑚 (cid:3405)\n(cid:3038) (cid:3040)\n0 ), these couplings do not modify the underlying spectral topology. Specifically, the 𝑚 th-order\nFloquet spectrum of ℋ corresponds to that of 𝐻(cid:3365) (cid:3398)Ω, and its winding topology governs the NHSE\n(cid:3038) (cid:3038)\nwithin the associated Floquet replica. (This is in sharp contrast to two- or multi-band Floquet\nsystems, where Floquet modulation not only creates new replicas but also alters the original\nspectral topology and hence the NHSE [Phys. Rev. B 101, 045415 (2020)].) Under this condition,\nwhen the zeroth-order harmonic component 2𝜋𝛿 𝛿 at frequency 𝑓 is excited, the role of\n(cid:3040)(cid:2868) (cid:2871)(cid:3046) (cid:2868)\nFloquet modulation is merely to generate higher harmonics at 𝑓 (cid:3399)𝑚Ω. These harmonics can be\n(cid:2868)\nregarded as secondary excitations of the zeroth-order component at the corresponding\nfrequencies. Naturally, their NHSEs remain dictated by the spectral topology of 𝐻(cid:3365) .\n(cid:3038)\nInspired by your constructive suggestion, we have revised the relevant paragraphs in the main\ntext (Page 5), with the above detailed discussions incorporated into Methods section (Page 17), to\nclarify the above points. The revisions in the main text reads “It is noted that only one component\nin 𝐏 satisfies 𝛿 𝛿 (cid:3404)1 (i.e., 𝑚 (cid:3404)0 and 𝑛 (cid:3404)𝑠), whereas all other components vanish, owing to\n(cid:2868) (cid:3040)(cid:2868) (cid:2871)(cid:3046)\nthe orthogonality of Fourier harmonics (see Methods). This reflects the fact that, in realistic\nexcitation settings, only the zeroth-order harmonic component at site 𝑠 can be directly excited.\nMoreover, as demonstrated above, the NHSE within the 𝑚th-order Floquet replica is governed by\nthe 𝑚 th-order Floquet spectrum of ℋ , or equivalently by the spectrum of 𝐻(cid:3365) (cid:3398)𝑚Ω .\n(cid:3038) (cid:3038)\nConsequently, the harmonic NHSE under realistic excitation can be fully characterized by examining\nsolely on the zeroth-order Floquet spectrum of ℋ , or equivalently, the spectrum of 𝐻(cid:3365) for single-\n(cid:3038) (cid:3038)\nband models. (Remarkably, for two- or multi-band models, the zeroth-order Floquet spectrum of\nℋ can differ fundamentally from the spectrum of 𝐻(cid:3365) , due to the interaction between different\n(cid:3038) (cid:3038)\nFloquet branches upon closing46.) The above theoretical framework provides a general route to\naccount for harmonic effects in Floquet non-Hermitian systems.”\n14"}, {"round": 2, "reviewer_comments": {"reviewer_1": "I'm satisfied with the revision. No more comments.", "reviewer_2": "I would like to appreciate the authors' efforts during the revision of the submitted manuscript and all the issues raised in the last round of review have been well addressed I think. Thus, I am happy to recommend it for publication in the current form.", "reviewer_3": "I have read the revised manuscript and the responses of authors. I think that all the important raised issues have been satisfactorily addressed, and the revised manuscript can now be published."}, "author_response": ""}]

Decision letter