Scalable and programmable topological transitions in plasmonic Moiré superlattices

Domain: physics
Published: 2026-01-22
Editorial outcome: minor_revision

Abstract
Topological transitions are fundamental phenomena in electronics, photonics, and quantum technologies. However, their scalability and tunability are constrained by material properties or structural rigidities. Here, we demonstrate that plasmonic Moiré superlattices offer a platform for programmable, large-range topological transitions via wavefront engineering. By tailoring the phases of elementary evanescent waves in hexagonal systems, we create Moiré lattices of optical skyrmions, whose topological invariants evolution is programmable and scalable. Theoretical calculations indicate that the topological invariants span a range of values going from −58 to +58, extendable by tuning the Moiré angle. Remarkably, these values are constrained by symmetry to exclude integer multiples of 3/2, revealing an intrinsic link between symmetry and topological quantization. Our work establishes a versatile control platform of real-space topology for exploring topological transitions mechanisms and studying critical topological phenomena, further promoting breakthroughs in structured light, photonic computing, and condensed matter physics.

Introduction
Topological transitions (TTs) characterize the transformations between distinct topological states1–3, playing a fundamental role in electronics4–6, photonics7–9, and quantum systems10. Conventionally, TTs have been extensively studied in terms of band structures in momentum-space4–9,11,12. Topological band theory describing the topological phase transitions of band structures advances theoretical physics and sheds light on edge states in topological materials, providing a foundation for low-energy electronic and photonic devices2,13. Related studies have also been focused on the topological structures in real-space, especially the well-known skyrmions, which open new avenues for high-density information storage, quantum computing, and structured light control14–29. However, in both scenarios, achieving scalable and programmable TTs remains a crucial challenge, as these systems are constrained by their reliance on intrinsic material properties or predefined architectures. The fundamentally limited dynamic tunability of the system has been the main challenge for advancing TT's theory and further comprehensive studies of critical topological phenomena.

In Moiré physics, Moiré superlattices, formed by the relative displacement or rotation of periodic structures, introduce a highly tunable degree of freedom that enables extensive structural modulation30–32. Initially explored in twisted bilayer graphene33, Moiré physics has unveiled several groundbreaking discoveries, including unconventional superconductivity33–35, the quantum anomalous Hall effect36,37, and controllable magnetism38. Subsequent advances have expanded Moiré platforms to plasmonic39–41, hydrodynamics42, and acoustic systems43, which exhibit exotic dispersion relations39, nontrivial vector field structures40–42, and skyrmion-like topological features40,41. These discoveries underscore their potential for applications in energy transfer, particle navigation, high-efficiency light manipulation and reconfigurable optical devices. In particular, the Moiré superlattices generated in plasmonic platforms40,41 can support arbitrarily large topological invariants. This capability not only markedly enriches the diversity of topological states but also facilitates topological transitions between these states.

Results
Formation and mechanism of optical TTs
Skyrmions are topological structures whose invariants count the number of times a vector field warps the unit sphere20. This topology has been established in optics to characterize the intrinsic topological properties of vector fields22–29. In two-dimensional real-space systems, the TI of a vector field can be quantified by ref. 22:
TI = 1/4\pi \int_D n \cdot (\partial_x n \times \partial_y n) dxdy, (1)
where n is a normalized vector, TI represents the number of times the normalized vector field wraps around the unit sphere (S2) as the 2D spatial domain is traversed. In optical systems, the unit vector n can be constructed from either the electric (E) or magnetic (H) field, yielding field skyrmions22, or from the spin density (S), leading to spin skyrmions23. In this work, to induce the TTs of field skyrmions, we synthesize the underlying electric field by tailoring the interference of six transverse magnetic (TM) evanescent modes, enabling precise control over the emerging topological textures. The out-of-plane electric field component of the vector field is given by (Supplementary Note 1):
Ez = E0 \sum_{\alpha=1}^3 cos(k_{s,\alpha} \cdot r + \phi_{\alpha}), (2)
where E0 denotes the amplitude, ks,\alpha is the in-plane wave vector, \phi_{\alpha} are the phases, and the relations among ks,\alpha and \phi_{\alpha} are illustrated in Fig. 1a. The in-plane field components, Ex and Ey, can be obtained with known Ez through Maxwell’s equations (Supplementary Note 1).

To characterize the influence of the phases \phi_{\alpha} on the resulting vector field described by Eq. (2), we introduce a synthetic phase parameter \beta = \phi_2 - \phi_1 - \phi_3, which not only reduces the dimensionality of parameter regulation but also encompasses all possible scenarios (Supplementary Note 1). In the numerical implementation, the relationship between the phase and \beta can be defined as \phi_2 = \beta/3 and \phi_1 = \phi_3 = -\beta/3. This choice not only better reflects the symmetry of the structure but also ensures that the center of the unit cell remains at the coordinate origin. The spatial distributions of Ez for \beta = 0 and \beta = \pi/2 are presented in Fig. 1b, c, respectively, accompanied by the corresponding normalized in-plane electric field vectors. These isolated topological states were previously obtained through methods equivalent to phase modulation44. Additional Ez field distributions for other \beta values are provided in Supplementary Fig. 1. The electric fields form two-dimensional hexagonal lattices, with unit cells outlined by black dashed lines. The TI, computed via Eq. (1), is found to be 1 for \beta = 0 and 0 for \beta = \pi/2. This observation indicates that the variations of \beta alter not only the morphology but also the topological state of the vector field. To systematically explore the dependence of TI on \beta, we calculated the TI for \beta values over a larger range as plotted in Fig. 1d (dashed line). This relation reveals a periodicity of 2\pi, within which three distinct topological states emerge, characterized by TIs of 1, 0, and −1. Specifically, the TI remains unchanged as \beta varies within several open intervals, such as (-\pi/2, \pi/2) and (\pi/2, 3\pi/2), indicating the robustness of the topological state. Besides, a discrete jump in the TI occurs as \beta crosses \pi/2, signifying a TT in the vector field configuration.

The mechanism underlying the TT of optical skyrmions mirrors that underlying topological phase transitions in topological insulators — both are driven by the emergence of singularities in the system. This commonality is elucidated by analyzing the topological invariants of skyrmions and comparing them to those arising in the two-band model used for topological insulators. Within topological insulators, the two-band model, which captures the evolution of the band structure across topological phase transitions, features energy bands of the form \pm|h|. The Chern number characterizing the topological nature of the band structure is given by a formula identical to that in Eq. (1). Here, h denotes a vector in the Brillouin zone parameterized by the hopping strengths. Consequently, the Chern number can be interpreted as the topological invariant of a skyrmion by treating h as a mapping from the first Brillouin zone to the real-space sphere S2. In topological insulators, a topological phase transition is triggered when the energy bands close and reopen. Band closure implies that the vector field is undefined at the point where |h| = 0. In the hexagonal lattice, we deduce that a necessary condition for a TT in such a system is similarly the emergence of singularities within the vector field. A detailed theoretical treatment is provided in Supplementary Note 2.

To verify the analogous role of singularities in the optical skyrmion system, we define the parameter E_{min} = |E|, representing the minimum absolute value of the vector field. The dependence of E_{min} on the phase parameter \beta is shown in Fig. 1d (solid line), which reveals that topological state transitions occur precisely at points where E_{min} = 0. This confirms that the emergence of a singularity is the necessary condition for a TT. In the absence of such singularities, the TI remains unchanged, despite continuous deformations of the vector field. To further reinforce this analogy, we define an energy band-like structure based on the commonality of Eq. (1) and the Chern number in the two-band model. This structure takes the form \pm|E|, exhibiting a well-defined gap for \beta = 0 or \pi (Fig. 1e) and a gapless configuration at \beta = \pi/2 (Fig. 1f). This gap closing coincides with an abrupt TT. Upon further increasing \beta, the gap reopens, and the system enters a different topological phase. This behavior mimics the gap-closing and reopening process in conventional topological band transitions, providing a clear real-space analogy for topological phase transitions.

Optical TT in Moiré superlattices
The properties of Moiré superlattices formed by interlayer twisting differ markedly from those of single-layer lattices43. The formation of a hexagonal Moiré superlattice (Fig. 2a) entails the interference of two identical hexagonal lattices with a relative angle \theta to generate a larger-scale hexagonal Moiré pattern. Given that the vector field described by Eq. (2) exhibits intrinsic hexagonal lattice symmetry, an optical Moiré superlattice can be constructed by coherently superimposing these two identical vector fields with a mutual rotation angle \theta. The resulting out-of-plane electric field component can be expressed as:
Ez = E0 \sum_{\alpha=1}^3 [cos(k_{s,\alpha,1} \cdot r + \phi_{\alpha,1}) + cos(k_{s,\alpha,2} \cdot r + \phi_{\alpha,2})], (3)
where the subscripts 1 and 2 distinguish the two hexagonal vector fields. The relation among the in-plane wave vectors ks,\alpha,1 and ks,\alpha,2, as well as the phase terms \phi_{\alpha,1} and \phi_{\alpha,2}, are shown schematically in Fig. 2b. Further details regarding the construction and interpretation of the Moiré vector fields can be found in Supplementary Note 3.

We introduce two synthetic phase parameters, \beta_1 = \phi_{2,1} - \phi_{1,1} - \phi_{3,1} and \beta_2 = \phi_{2,2} - \phi_{1,2} - \phi_{3,2}, to conveniently quantify the influence of the phase configurations on the resulting Moiré vector field. As we mentioned previously, each of these parameters can capture the essential phase modulation within each component vector field. Within the computational framework, the phases are related to the parameters (\beta_1, \beta_2) according to the convention of the hexagonal lattice and are selected as follows: \phi_{2,1} = \beta_1/3, \phi_{1,1} = \phi_{3,1} = -\beta_1/3, \phi_{2,2} = \beta_2/3, and \phi_{1,2} = \phi_{3,2} = -\beta_2/3. For the hexagonal Moiré superlattice, the Moiré angle \theta satisfies the relation: cos \theta = (n+m)^2 + 2mn / 2(n+m)^2 - 2mn, where m and n are positive integers45. The vector fields corresponding to m=2 and n=3 are presented for \beta_1 = \beta_2 = 0 (Fig. 2c) and \beta_1 = \beta_2 = \pi (Fig. 2d), with additional configurations presented in Supplementary Fig. 2. In all cases, black dashed lines indicate the unit cells of the hexagonal superlattices. The calculated TIs for these configurations are 7 (Fig. 2c) and −8 (Fig. 2d), indicating that the phase parameters \beta_1 and \beta_2 in the Moiré vector field described by Eq. (3) play a role analogous to the single \beta parameter in Eq. (2)—both modulate the topological states of the vector field. The dependence of TI on both \beta_1 and \beta_2 is shown in Fig. 2e. The parameter space is partitioned and color-coded into multiple regions with associated TI. Within the range from −8 to 8, the TIs span all integers and half-integers, except for \pm1.5, \pm3, \pm4.5 and \pm6. Notably, 0 and half-integer values are not prominently visible in Fig. 2e because they lie on the boundaries between adjacent regions. At these boundaries, the TIs assume the average value of the neighboring regions’ TIs (Supplementary Note 4). These results indicate that the topological state remains unchanged as long as \beta_1 and \beta_2 vary within a given region. However, a discrete TT occurs whenever (\beta_1, \beta_2) cross from one region to another, resulting in abrupt changes in the TI within the range of −8 to 8.

As we demonstrated in Fig. 1, singularities within the vector field serve as indispensable prerequisites for the TT. Consequently, the boundaries of the regions in Fig. 2e correspond to the zero contours of the vector field described by Eq. (3). The values of E_{min} = |E| as a function of \beta_1 and \beta_2 are presented in Fig. 2f. From the comparison between Fig. 2e, f one can notice that the region boundaries and the zero trajectories are perfectly aligned. Notably, at positions marked by black dashed lines in Fig. 2e, where singularities also appear, the TIs on either side remain unchanged. This underscores that the mere occurrence of singularities is an necessary yet insufficient criteria for triggering a TT. The transition requires not only that the field vanishes but that the vector field undergoes a topological reconfiguration in its local orientation. A more detailed analysis is provided in the Supplementary Note 4.

While the TI range is confined to −8 to 8 for the parameter set (m, n)=(2, 3), as illustrated in Fig. 2, the accessible TI range expands dramatically with increasing m and n. For example, when (m, n)=(3, 4), the range extends from −19 to 19; and for (m, n)=(4, 9), it further upscales from −58 to 58. This scaling behavior highlights the ability of Moiré structuring to support a broad range of topological states (Supplementary Note 5).

Symmetry-constrained discretization of TIs
An interesting yet crucial finding in Fig. 2e is that the TIs never take values that are integer multiples of 3/2. This phenomenon arises from the distribution and classification of singularities, as well as the symmetry of the structure. The Moiré superlattice described by Eq. (3) possesses C3 symmetry, where symmetric points are located either at the center or the vertices of the unit cell. Therefore, the center and vertices of the unit cell are termed high-symmetry points, while other positions are termed general-position points. Accordingly, singularities located at high-symmetry points are classified as high-symmetry singularities (red circles in Fig. 3a), whereas those occurring at general-position points are regarded as general-position singularities (black circles in Fig. 3a, b).

When high-symmetry singularities are present, they always force TI sign reversal (e.g., from –8 to 8, as shown with the solid arrow in Fig. 2e) whether or not generic-position singularities coexist. This effect originates from the antisymmetry of vector field response to phase parameters. In contrast, the generic-position singularities can only modify the TI value in discrete steps of 3 (e.g., from –8 to –5, as shown with the dashed arrow in Fig. 2e). This behavior emerges from the interplay between the winding numbers of singularities and the C3 symmetry of vector field (Supplementary Note 6).

These two mechanisms determine the permitted TT in the system. Based on this, we classify the system into two regimes: steady states and transient states. The steady states are regions of the parameter space (\beta_1, \beta_2) where no singularities are present, and the TI remains constant. The transient states are critical configurations where singularities emerge, and TTs occur between adjacent steady states. This helps to establish the fundamental constraint within the system, that is all steady-state TIs are forbidden from taking values that are integer multiples of 3. This restriction is followed from two observations. First, the extremal values of the TI spectrum, such as \pm8 in Fig. 2e, are not divisible by 3, and serve as boundary values for all reachable steady states. Second, the only allowed TI transitions—sign reversal and integer multiples of 3 changes—preserve the non-divisibility of TI by 3 throughout the entire parameter evolution. Therefore, the steady-state TI values form a discrete subset of the integers that exclude all multiples of 3. Furthermore, since transient states inherit their TI values as the arithmetic mean of adjacent steady states, they cannot take values that are integer multiples of 3/2 as well. As a result, the system exhibits a quantization rule that forbids integer multiples of 3/2 TI values, enforced by the topology and symmetry.

This exclusion rule is not restricted to a specific Moiré lattice configuration. It remains valid in a variety of systems with different (m, n) parameters. For instance, in configurations with (m, n)=(3, 4), the TI spans from –19 to +19, and in (m, n)=(4, 9), from –58 to +58. In both cases, neither the steady-state TIs nor their averages are divisible by 3/2, thereby confirming the universality of this topological selection rule across system scales and symmetry classes. Note that the pattern of topological states is governed by the structural symmetry, as demonstrated by our investigation of multilayer Moiré superlattices. In multilayer hexagonal superlattices (Supplementary Note 7), the preservation of C3 symmetry maintains a consistent pattern, even in complex multi-layer configurations. Conversely, the distinct symmetry of the square superlattices (Supplementary Note 8) yields a fundamentally different pattern of topological invariants.

Experimental demonstration of scalable and programmable TTs
We experimentally demonstrate scalable TTs using surface plasmon polaritons (SPPs)—TM-polarized evanescent waves propagating along a metal-dielectric interface46. The phase of each elementary SPP wave is dynamically controlled by a spatial light modulator, enabling the programmable construction of vector field topologies through control phase parameters. These code-like phase parameters function as effective degrees of freedom that map directly onto distinct topological states, thereby allowing flexible access to a wide range of TT. Detailed descriptions of the experimental setup and sample fabrication are provided in Methods.

The measured results for vector fields based on a hexagonal lattice configuration, governed by Eq. (2), are presented in Fig. 4a–c. The corresponding E-field distributions and reconstructed vector directions, obtained using the Gerchberg–Saxton (GS) algorithm47, are shown for three representative values of the encoded parameter: \beta = 0 (Fig. 4a), \pi/2 (Fig. 4b), and \pi (Fig. 4c). The GS algorithm is a phase retrieval technique that has been used to reconstruct SPP fields with topological characteristics48. Additional results are summarized in Supplementary Figs. 3 and 4. Figure 4d depicts the TIs as a function of \beta \in [0, 2\pi], with the dashed line indicating the theoretical predictions and the discrete points with standard deviations for experimental data. Clear TTs are observed at \beta = \pi/2 and 3\pi/2, demonstrating programmable switching between distinct topological phases.

We proceed to analyze the experimental results of a more complex and higher-order topological system, the plasmonic Moiré superlattice for (m, n)=(2, 3). Here, the topological state is jointly determined by two phase parameters, \beta_1 and \beta_2. As shown in Fig. 4e–k, this programmable approach allows continuous tuning of the system through a sequence of topological states with TIs of 8 (Fig. 4e), 7 (Fig. 4f), 5 (Fig. 4g), 4 (Fig. 4h), 2 (Fig. 4i), 1 (Fig. 4j), and 0 (Fig. 4k), respectively. Other cases are summarized in Supplementary Figs. 5 and 6. The extracted TIs (Fig. 4l), obtained for \beta_1 = \beta_2 swept from 0 to 6\pi, illustrate the programmable and scalable evolution of multiple TTs within a single modulation cycle.

It is worth noting that the topological states of skyrmion-like vector fields are identified solely by their TIs, not by their visual resemblance. In Fig. 4, despite apparent morphological similarities among some field distributions, their differing TIs confirm that they represent distinct topological states. Furthermore, while losses are inevitable in the plasmonic platform, the topological structures remain robust on the wavelength scale, and topological transitions are clearly realizable due to the sufficiently low loss levels; a detailed analysis is provided in Supplementary Note 9.

Methods
Experimental configurations
The experimental setup used to probe the topological structures formed by SPP at the gold/air interface is shown in Supplementary Fig. 7. An incident laser beam (\lambda = 632.8 nm, Thorlabs, HNL050LB) was directed through a series of optical elements to produce six or twelve excitation beams with specific phases. Among them, a spatial light modulator (Meadowlark Optics E19x12-500-1200-HDM8, 1920×1200, pixel size: 8×8 \mu m) is used to generate six or twelve linearly polarized lights with specific phases. A half-wave plate and a vortex wave plate are used to generate the input radially polarized lights. These beams were tightly focused onto the sample, which consisted of a three-layer structure with a 50 nm-thick gold film on a silica substrate and the air as the upper medium. This focusing was achieved using an oil-immersion objective (Olympus, 100×, NA=1.49), which excited the SPPs at the air/gold interface.

To map the local light field structure, silver nanoparticles with a diameter of 80 nm were immobilized on the gold surface. To spatially separate the weak SPP scattering from the nanoparticles from the directly transmitted background light, a circular opaque mask was inserted in the beam path to form a circular beam that illuminated the sample through a focusing objective. The intensity of the scattering radiation from the nanoparticle was measured using a photomultiplier tube (Hamamatsu, C12597-01), while the sample itself was mounted on a piezoelectric scanning stage (Physik Instrumente, P-727) with a resolution of 1 nm. The SPP field structure was obtained by scanning the nanoparticle across the beam profile. Due to the localized resonance properties of the silver nanoparticle-on-film gap structure, this probe reliably measures the out-of-plane electric field component of SPP49.

Sample preparation and data processing
The sample preparation followed these steps: first, a 5-nm-thick chromium layer was deposited onto glass coverslips (170 \mu m) by electron beam evaporation to improve adhesion. Subsequently, a 50-nm-thick gold film was deposited using the same method. Finally, a droplet of a fully diluted suspension of silver nanoparticles (80 nm in diameter) was placed on the surface of the gold film and allowed to evaporate naturally. While the dried sample contained numerous silver nanoparticles, measurements were exclusively performed on individual particles isolated by more than 20 \mu m from neighbors, ensuring that all scattered light originated from a single target nanoparticle. No additional fixation is required, as natural drying provides enough stability for measurement. The data was acquired at a sampling density of either 10 nm or 30 nm. This choice of sampling density is determined by the spatial variation rate of the vector field and experimental conditions. Theoretically, a higher density of data points provides a more accurate description of the vector field. However, denser data sampling is more susceptible to environmental influences. Even slight vibrations can introduce unintended frequency artifacts into the experimental data. Therefore, for rapidly varying fields, we typically use a 10 nm resolution, whereas a 30 nm resolution is generally sufficient for most common cases.

The original mapped intensity distributions of the out-of-plane electric field |Ez|2, excited on the surface of the gold film under six-beam and twelve-beam excitation light, are presented in Supplementary Figs. 3 and 5, respectively. After acquiring the experimental data, the real part of Ez and the corresponding phase patterns were reconstructed using the Gerchberg–Saxton (GS) algorithm47,48, as shown in Supplementary Figs. 4 and 6. Subsequently, the x-component Ex and y-component Ey were calculated according to Eq. (S4) of the supplementary information. The TIs of the vector field were then computed using Eq. (1) from the main text, and the discrete results are shown in Fig. 4d, l of the main text.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors report on an experimental and theoretical framework for achieving scalable and programmable topological phase transitions (TPTs) in plasmonic systems. The study utilizes the interference of six or twelve evanescent waves to form skyrmion vector fields. By manipulating the phases of the individual waves, the authors achieve tunable topological invariants (TIs). The selection rule of possible TI’s has been derived, revealing that changing the synthetic phase parameter in this system only allows for (integer and half-integer) TI’s that exclude integer multiples of 3/2. The work also aims to bridge momentum-space topological band theory with real-space skyrmion topologies.\nThe focus of this work on topological phase transitions in plasmonic Moiré superlattices sits at the intersection of several advanced fields—nanophotonics, topological physics, and solid-state physics. That said, for specialized communities—especially those working on topological photonics, metasurfaces, or plasmonics—the paper is both interesting and significant.\nThe authors are well-known for their expertise in nanophotonics research and present high-quality experimental results. The manuscript is well written, and the data presented well. Due to its novelty, we believe the manuscript could be suitable for Nature Communications.\nNevertheless, before making a final decision, we believe that certain points need to be addressed:\nMain concerns:\nNot only the concept of plasmonic skyrmion moiré superlattices, but also the fact that they enable arbitrarily large TIs in their super unit cells has recently been introduced in Refs. 37, 38.\nHowever, the ability to tune these TI’s continuously using a wisely defined synthetic phase parameter is novel. In our opinion, the authors could emphasize the similarities and differences more clearly to set themselves apart from these previous works.\nWe suggest broadening the study to additional configurations of moiré superlattices that have been introduced. In particular, different centers of rotation could change the range of accessible TIs and their selection rules. The authors could also discuss the effect of using three or more twisted lattices and the resulting influence on the TIs and selection rules.\nThe authors claim in the introduction that ‘the connection between the TPTs of skyrmions and the TPTs of energy bands has also been identified’ and in the conclusion that ‘the commonality between skyrmions and band structures provides the foundation for the construction of general theoretical system of TPTs, which opens new directions for developing next-generation electronic and photonic devices with enhanced compactness and performance.’\nUnfortunately, in the main text only little attention has been drawn to this connection and its implications.\nIt is commonly known that singularities in the field (i.e., |E|=0) are a necessary condition for TPTs. In the SI, the authors now introduce energy band-like structures, that follow ±√(E_x^2+E_y^2+E_z^2 )= ±|E|. From this definition of the bands, it is a straightforward conclusion that TPTs are connected to a gap-free structure as they require |E|=0.\nCan the authors state if such energy band-like structures have already been defined in other works? We think that the definition of these energy band-like structures should be included in a few sentences of the main text, as the authors highlight this finding as one of the main novelties of their work.\nIn addition, their connection to field singularities is straightforward and should not be hidden in the supplementary material.\nHow universal can these energy band-like structures be applied? Are there any applications and use cases? Is it, for example, possible to ‘simulate’ a two-band model using a plasmonic system and connect any parameters of such a solid-state system to the topology of the plasmonic system?\nThe authors also did not include such energy band-like structures for the moiré systems. Are there any conclusions that can be drawn from the topology of these bands, and are there any connections to moiré materials?\nApart from these points, we have some additional comments and questions:\nIn our opinion, it is necessary that the authors present the calculated Chern number density to support the calculated Chern numbers (i.e., TIs) from their analytical and experimental data. We suggest that the authors include these figures in the supplementary material. The field singularities should be connected to singularities and a change of the sign in the local Chern number density. The authors should also include a short explanation of how the Chern numbers are calculated from the Chern number density. Are they calculated by summing up all measured pixels of the Chern number density inside the super unit cell boundary, or are there any additional numerical methods involved?\nThe authors provide experimental confirmation of programmable TPTs using a nanoparticle-scattering technique, retrieving the amplitude of the out-of-plane field (Ez). The full vector fields are reconstructed using the Gerchberg-Saxton algorithm and Maxwell’s equations.\nIn our opinion, the authors could provide some more information about their experimental methods.\nWhat is the resolution of the measured field distribution? What are its limitations?\nHow many silver nanoparticles are located on the gold surface, and how are they immobilized? How many of these nanoparticles are involved in the measurement?\nThe GS algorithm is primarily applied to retrieve the phase from intensity measurements that are acquired in two different planes. Typically, the image plane and the far field plane. Can the authors provide more information on the usage of the GS algorithm to retrieve the phase of SPPs, or reference another work where this is explained? Does the GS algorithm always converge to the desired solution?\nThe authors state in line 120: ‘This relation reveals a periodicity of 2π, within which three distinct topological states emerge, characterized by TIs of 1, 0, and −1. Specifically, the TI remains unchanged as β varies within several open intervals, such as (−π/2, π/2) and (π/2, 3π/2), indicating the robustness of the topological state.’\nCould the authors elaborate on the robustness that is given by the half-open interval? Does this mean that the TI of 0 is more difficult to create or measure?\nMinor concerns:\nLine 81: remove ‘distint’.\nLine 90 and Supplementary Section 2: ‘warps’ should be replaced by ‘wraps’.\nSupplementary Section 2: ‘Soling’ should be replaced by ‘Solving’.\nLine 167: replace ‘simultaneous’ with ‘simultaneously’.\nLine 216: ‘This scaling behavior underscores the capacity of Moiré structuring to support a wide spectrum of topological states (Supplementary Information Section 5).’\nThis sentence should be rephrased to e.g:\n‘This scaling behavior highlights the ability of Moiré structuring to support a broad range of topological states (Supplementary Information, Section 5).’\nLine 285, 289: ‘The unit cell is outlined…’.\nLine 294: remove ‘nest’.", "reviewer_2": "I read with great interest the manuscript by B. Tian et al, about implementation of high-order topological invariants in plasmonic fields using Moiré superlattices. Although the idea itself is not new (first performed in Ref. 37 of the manuscript), the authors extend the known theory and understanding of this phenomenon, and showcase control over the topological invariant via an SLM. They characterize the reason for transitions between topological numbers and demonstrate their theory through an experiment.\nI find that the authors’ idea is innovative, interesting to a broad audience in photonics and topological physics, and – for the most part – performed rigorously (see my comments below for a few “loose-ends”). I am inclined to recommend publication in principle, given that the authors adequately address all comments below:\nMajor points\n- The authors call the transitions in their system “phase transitions”. It is not an accurate phrase, since it is not a many-body system, nor does it contain any form of nonlinearity. The cited works in topological photonics certainly do not show phase transitions, but rather – just “topological transitions”, which are the commonly agreed upon term. I request that the authors amend their manuscript in accordance with this point.\n- The claim of showing topological invariants between -58 and 58 is unsubstantiated. The authors show this in theory, while measuring only -8 to 8. I suggest that either the authors amend their claim, or properly measure topological invariants up to 58.\n- I greatly appreciate that the authors check whether or not their minimal field amplitudes tend to 0. In such points, the topological density is ill-defined, so the whole distribution cannot have a well-defined topological charge other than 0. In this respect, any fractional topological charge the authors measure is probably occurring in such a point, and therefore – is ill-defined. In addition, is it true that the dashed lines in fig. 2e coincide with zero minimal field amplitude lines shown in fig. 2f? this would mean that, along these lines, the topological constant is ill-defined, whereas right now it is defined via a particular color in fig. 2e. Therefore, the figure may have to change, and the authors should complete their check that indeed the topological density in cases they measure is well-defined.\n- In the same vein of checking that the distribution is well-defined, how do the authors make sure that the real part of the electric field is indeed much larger than the imaginary part? Of course, this can be defined up to a global phase, but to take only one part of the field, it must to be much larger than the other part, with the criterion defined in the supplementary information of Ref. 22. It seems, from looking at the supplementary, that this might not be the case for some of the measurements. I strongly recommend that the authors check this point to make sure that ther main claims are substantiated.\n- The authors claim to calculate (and measure) negative topological invariants. I find this to be somewhat confusing, since it seems the two different topological invariants only require that the whole field will have a global phase of pi. This, however, cannot uniquely define a field distribution since the global phase is arbitrary. Given this point, all the possible topological invariants will have been spanned within beta1<=2pi, instead of within another, more arbitrary number (as in figure 4). It would also mean that the initial transition the authors analyze, not within a Moiré lattice, is no transition at all. The authors should carefully address this point to make sure that their analysis is indeed valid.\n- Gerchberg-Saxton phase reconstruction is a tool that should be used very carefully, since it is agnostic of the requirements from Maxwell’s equations, as well as from the analytic nature of physical observables. This is particularly problematic in field distributions involving phase singularities, where the GS algorithm exhibits ambiguity. How do the authors make sure their reconstruction does not have artefacts? It is very important that this is addressed, since the entire experimental observation of the authors relies on the reconstruction being accurate.\n- The whole explanation the authors give about the effects of singularities in their system is very confusing. High-symmetry and generic-position singularities are never rigorously defined, so it is not possible to understand whether or not the authors explanation makes sense. I suggest that the authors revise their explanation or exclude it from the final manuscript.\nMinor points\n- The authors do not properly cite Ref. 37 (published recently in Nature Physics but under a different name), and also greatly minimize its contribution to the work they present. Whether they relied upon it or not, it is clear that Ref. 37 was the first to describe the majority of the theory the authors show in their work, and while the manuscript does contain many new advances, it does not properly point to the points Ref. 37 has already put forth. Similarly (but to a much smaller extent), the authors do not cite Tsesses et al, Nano Lett. 19 (2019), which observed and discussed some of the field distributions they present (in particular, the honeycomb lattice at beta=pi/2) or Frischwasser et al, Nat. Photon. 15 (2021), which was – to my knowledge – the first to use the GS algorithm to reproduce the phase of plasmonic fields with distinct topologies.\n- The authors should be very clear about their intention when saying they change beta. They write what this means only in the supplementary (i.e., phi2=beta/3,phi1=phi3=-beta/3), but this should be written in the manuscript itself and emphasized, since not every type of change in beta (for example, a change to only one of the phases) will have resulted in the authors’ findings.\n- The axes in fig. 1e,f are unclear, and the analogy to a band structure does not make much sense, since there are no bands in the authors’ system. I suggest removing them from the manuscript.\n- Although written very well for the most part, there are still some portions with awkward phrasing and typos, including (but not limited to) “the antisymmetric of vector field”, page 8 line 236.\n- The authors should add standard deviations on their measured topological invariants in fig. 4.\n- The rule the authors find for their topological invariants, while very interesting, only applies for C3 symmetry – i.e., it is not general for other types of symmetry (as is currently written in the manuscript)."}, "author_response": "Dear Editors and Reviewers：\nWe thank the reviewers for their time and comments regarding our manuscript entitled\n\"Scalable and Programmable Topological Phase Transitions in Plasmonic Moiré\nSuperlattices\"(ID: NCOMMS-25-44015). We appreciate their positive feedback and\nhave found the suggestions to be inspiring and helpful in improving the quality of our\nmanuscript. We have thoroughly looked through every comment and have revised the\nmanuscript accordingly. We hope our revisions have improved our manuscript to\nguarantee its publication in Nature Communications. All modifications have been\nhighlighted in the text, followed by a detailed point-by-point response for each\ncomment.\nResponse to Reviewer #1:\nComment: The authors report on an experimental and theoretical framework for\nachieving scalable and programmable topological phase transitions (TPTs) in\nplasmonic systems. The study utilizes the interference of six or twelve evanescent\nwaves to form skyrmion vector fields. By manipulating the phases of the individual\nwaves, the authors achieve tunable topological invariants (TIs). The selection rule of\npossible TI’s has been derived, revealing that changing the synthetic phase parameter\nin this system only allows for (integer and half-integer) TI’s that exclude integer\nmultiples of 3/2. The work also aims to bridge momentum-space topological band\ntheory with real-space skyrmion topologies.\nThe focus of this work on topological phase transitions in plasmonic Moiré superlattices\nsits at the intersection of several advanced fields—nanophotonics, topological physics,\nand solid-state physics. That said, for specialized communities—especially those\nworking on topological photonics, metasurfaces, or plasmonics—the paper is both\ninteresting and significant.\nThe authors are well-known for their expertise in nanophotonics research and present\nhigh-quality experimental results. The manuscript is well written, and the data\npresented well. Due to its novelty, we believe the manuscript could be suitable for\nNature Communications.\nNevertheless, before making a final decision, we believe that certain points need to be\naddressed:\nQ1: Not only the concept of plasmonic skyrmion Moiré superlattices, but also the fact\nthat they enable arbitrarily large TIs in their super unit cells has recently been\nintroduced in Refs. 37, 38.\nHowever, the ability to tune these TI’s continuously using a wisely defined synthetic\nphase parameter is novel. In our opinion, the authors could emphasize the similarities\n1 / 20\nand differences more clearly to set themselves apart from these previous works. We\nsuggest broadening the study to additional configurations of Moiré superlattices that\nhave been introduced. In particular, different centers of rotation could change the range\nof accessible TIs and their selection rules. The authors could also discuss the effect of\nusing three or more twisted lattices and the resulting influence on the TIs and selection\nrules.\nR1: Thank you for the constructive suggestion. We are sorry for the oversight in not\nemphasizing the importance of the cited Refs. 37 and 38, as both of them are\noutstanding works related to the development of plasmonic skyrmion Moiré\nsuperlattices. We certainly agree with the reviewer that clarifying the relationship\nbetween our work and the Refs. 37, 38 is crucial for highlighting the novel contribution\nof our study. We have now emphasized the impact of these two Refs in the introduction\nsection when discussing the plasmonic Moiré superlattice. The specific modifications\nmade are as follows:\nIn particular, the Moiré superlattices generated in plasmonic platforms40,41 can support\narbitrarily large topological invariants. This capability not only significantly enriches\nthe diversity of topological states but also facilitates topological transitions between\nthese states.\nThe detailed content can be found in lines 52 to 55 of the main text (Refs. 37 and 38 in\nthe initial version are now Refs. 40 and 41, respectively, due to structural adjustments).\nFurthermore, we sincerely thank the reviewer for the suggestion of extending the study\nto additional configurations, as this will certainly provide a deep insight into the\nrelationship between topological states and symmetry in such a system. Therefore, in\nthe Supplementary Information Sections 7 and 8, we present topological states in\nmultilayer hexagonal Moiré superlattices and square Moiré superlattices, respectively.\nThe results demonstrate that in multilayer hexagonal Moiré superlattices, even with\ncomplex structures, the pattern governing the existence of topological invariants\nremains unchanged due to the preservation of C symmetry. In contrast, the distinct\n3\nsymmetry of square Moiré superlattices leads to a different pattern of topological states,\nwhere phase variations lead to a zero value of the topological invariant. This\nphenomenon arises because the singularities generated in the square Moiré superlattice\nare created in pairs with opposite winding numbers. Consequently, these singularities\ncancel each other out, leading to no net change in the topological state. To further clarify\nthat the patterns of topological invariants in our work are specific to certain symmetries,\nwe have emphasized this point in the main text. The specific modifications made are as\nfollows:\nNote that the pattern of topological states is governed by the structural symmetry, as\ndemonstrated by our investigation of multilayer Moiré superlattices. In multilayer\nhexagonal superlattices (Supplementary Information Section 7), the preservation of C\n3\n2 / 20\nsymmetry maintains a consistent pattern, even in complex multi-layer configurations.\nConversely, the distinct symmetry of the square superlattices (Supplementary\nInformation Section 8) yields a fundamentally different pattern of topological\ninvariants.\nThe detailed content can be found in lines 294 to 300 of the main text.\nQ2: The authors claim in the introduction that ‘the connection between the TPTs of\nskyrmions and the TPTs of energy bands has also been identified’ and in the conclusion\nthat ‘the commonality between skyrmions and band structures provides the foundation\nfor the construction of general theoretical system of TPTs, which opens new directions\nfor developing next-generation electronic and photonic devices with enhanced\ncompactness and performance.’\nUnfortunately, in the main text only little attention has been drawn to this connection\nand its implications.\nIt is commonly known that singularities in the field (i.e., |E|=0) are a necessary\ncondition for TPTs. In the SI, the authors now introduce energy band-like structures,\nthat follow ±√𝐸2 +𝐸2 +𝐸2 = ±|𝐸| . From this definition of the bands, it is a\n𝑥 𝑥 𝑥\nstraightforward conclusion that TPTs are connected to a gap-free structure as they\nrequire |E|=0.\nCan the authors state if such energy band-like structures have already been defined in\nother works? We think that the definition of these energy band-like structures should be\nincluded in a few sentences of the main text, as the authors highlight this finding as one\nof the main novelties of their work.\nIn addition, their connection to field singularities is straightforward and should not be\nhidden in the supplementary material.\nHow universal can these energy band-like structures be applied? Are there any\napplications and use cases? Is it, for example, possible to ‘simulate’ a two-band model\nusing a plasmonic system and connect any parameters of such a solid-state system to\nthe topology of the plasmonic system?\nR2: We sincerely thank the reviewer for the insightful suggestions. We apologize for\nnot elaborating on the connection between our work on these aspects previously. We\nhave now incorporated a discussion and analysis of the energy band-like structure into\nthe main text to emphasize the significance of this feature in topological transitions.\nThe specific modifications made are as follows:\nThe mechanism underlying the TT of optical skyrmions mirrors that underlying\n3 / 20\ntopological phase transitions in topological insulators—both are driven by the\nemergence of singularities in the system. This commonality is elucidated by analyzing\nthe topological invariants of skyrmions and comparing them to those arising in the two-\nband model used for topological insulators. Within topological insulators, the two-band\nmodel, which captures the evolution of the band structure across topological phase\ntransitions, features energy bands of the form ±|h|. The Chern number characterizing\nthe topological nature of the band structure is given by a formula identical to that in\nEq. (1). Here, h denotes a vector in the Brillouin zone parameterized by the hopping\nstrengths. Consequently, the Chern number can be interpreted as the topological\ninvariant of a skyrmion by treating h as a mapping from the first Brillouin zone to the\nreal-space sphere S2. In topological insulators, a topological phase transition is\ntriggered when the energy bands close and reopen. Band closure implies that the vector\nh must vanish (h=0) at one or more points in the Brillouin zone. By extending this\nconcept to vector fields defined on a hexagonal lattice, we deduce that a necessary\ncondition for a TT in such a system is similarly the emergence of singularities within\nthe vector field. A detailed theoretical treatment is provided in Supplementary\nInformation Section 2.\nThe detailed content can be found in lines 138 to 154 of the main text.\nSpecifically, to the best of our knowledge, such an energy band-like structure has not\nbeen explicitly reported in the field of optics. We introduce this concept based on the\nanalogy between the skyrmion number and the Chern number in the two-band model.\nAccording to the topological phase transition mechanism of the two-band model, this\nband-like structure provides a generally applicable framework for describing changes\nin the topological states of vector fields in lattice systems. In other words, irrespective\nof the specific lattice configuration, this band-like structure can serve as a useful means\nto characterize the process of topological transition.\nHowever, it should be emphasized that this band-like structure shares only\nmathematical similarities with the energy band structure of the two-band model. In this\nmathematical analogy, the conduction band in the energy band structure corresponds to\nthe magnitude of the vector field |E| in the band-like structure, while the valence band\ncorresponds to the symmetric distribution of |E| with respect to the z=0 plane. Likewise,\nthe momentum space (k , k , k ) in the energy band model maps onto the real space (x,\nx y z\ny, z) in the band-like structure.\nThe underlying physical implications between band-like structure and band structure in\ntopological insulators are fundamentally distinct. Therefore, to date, we have not\nidentified a viable approach to utilize plasmonic systems for simulating the physical\nphenomena described by the two-band model. Nevertheless, we consider the reviewer’s\nsuggestion—to employ plasmonic systems in simulating the two-band model—a highly\nvaluable insight. We hypothesize that the mathematical analogy revealed in our\nmanuscript may lay the groundwork for unifying two distinct types of topological\n4 / 20\nstructures within a more generalized framework. Such unification could, in turn,\npotentially enable mutual simulation between these two topological systems.\nQ3: The authors also did not include such energy band-like structures for the Moiré\nsystems. Are there any conclusions that can be drawn from the topology of these bands,\nand are there any connections to Moiré materials?\nR3: We thank the reviewer for the keen observation and helpful suggestion. In the\nrevised Supplementary Fig. 2, we have included the energy band-like structures\nassociated with the hexagonal Moiré superlattices. Moreover, for the newly analyzed\nthree-layer hexagonal and square Moiré superlattices, the band-like structures have also\nbeen incorporated and illustrated in Supplementary Figs. 11 and 12, respectively.\nAnalysis of these frameworks, in both hexagonal lattices and hexagonal Moiré\nsuperlattices, reveals a fundamental principle of topological transitions: during the\nclosing and reopening of the energy band-like structure, both the number and the nature\nof the singularities collectively determine the topological invariant of the vector field\nafter the topological transition. Consequently, a greater number of singularities of the\nsame type leads to a more significant change of the topological invariant. Given that\nthe number of singularities in hexagonal Moiré superlattices is significantly higher than\nthat in hexagonal lattices, they enable a broader range of variation in topological\ninvariants.\nThe Moiré superlattices discussed in this work and Moiré metamaterials share a\nconceptual connection at the research level, even though their specific physical\nmechanisms and application scenarios differ from each other. Nevertheless, we note\nthat these two fields can mutually inspire and inform each other. For example, the\ndiscovery of unconventional superconductivity in Moiré graphene sparked tremendous\ninterest, subsequently fueling extensive research into both Moiré metamaterials and\nMoiré superlattices. Many later studies drew inspiration, to varying extents, from the\nfundamental ideas established in Moiré graphene. Similarly, the observation of diverse\ntopological transitions and multiple topological states in optical hexagonal Moiré\nsuperlattices suggests that this conceptual framework could be extended to other Moiré\nsystems and metamaterials. In particular, mapping the phase modulation strategies from\nMoiré superlattices onto Moiré metamaterials may enable the realization of rich\ntopological states in such systems, paving the way for the development of topological\nmaterials with enhanced or novel properties.\nQ4: In our opinion, it is necessary that the authors present the calculated Chern number\ndensity to support the calculated Chern numbers (i.e., TIs) from their analytical and\nexperimental data. We suggest that the authors include these figures in the\nsupplementary material. The field singularities should be connected to singularities and\na change of the sign in the local Chern number density. The authors should also include\na short explanation of how the Chern numbers are calculated from the Chern number\n5 / 20\ndensity. Are they calculated by summing up all measured pixels of the Chern number\ndensity inside the super unit cell boundary, or are there any additional numerical\nmethods involved?\nR4: Another great suggestion from the reviewer. We have now included the topological\ncharge density in Supplementary Fig. 4 and 6.\nHere, Supplementary Fig. R1 is attached to illustrate the topological invariants of\nskyrmions from experimental calculation. We note that Supplementary Fig. R1 presents\nthe data presented in the last column of Supplementary Fig. 4, namely, the topological\ncharge density of skyrmions in hexagonal lattices. The topological invariant is\ncomputed through a discrete approximation of the surface integral: the topological\ncharge density is summed over all points within a unit cell, multiplied by the area\nelement, and normalized by 4π. This operation represents a direct discretization of the\ncontinuous integral and involves no additional processing steps. To more clearly and\nconveniently illustrate the details of experimental data processing, we have additionally\ndescribed the method for calculating topological invariants using experimental data in\nthe caption of Supplementary Figs. 4 and 6.\nSupplementary Fig. R1| Topological charge density of skyrmions in hexagonal\nlattices.\nSupplementary Fig. R1 also reveals that even in the simplest hexagonal lattice (β=0),\nthe topological charge density exhibits a nonuniform distribution, taking both positive\nand negative values. The negative values are localized near the six vertices of the unit\ncell. As β increases, the regions of negative topological charge density around three\nnon-adjacent vertices expand, while those around the other three vertices gradually\nshrink. When β reaches π/2, the negative-charge regions associated with three non-\nadjacent vertices extend to occupy half of the unit cell, whereas the other three vanish\nentirely. The singularities are located at the three non-adjacent vertices where the\nnegative topological charge vanishes. Because the six vertices are shared by three\nadjacent unit cells, the unit cell under observation effectively contains only one\nsingularity, and the negative-charge regions occupy exactly 50% of the cell area. This\ncomplete balance between positive and negative contributions yields a topological\ninvariant of zero.\nAs β increases beyond π/2 and approaches π, the evolution of the topological charge\ndensity mirrors the behavior observed from β=0 to π/2, but with opposite sign. In other\n6 / 20\nwords, the evolution of the negative (positive) charge density regions for β increasing\nfrom 0 to π/2 corresponds precisely to that of the positive (negative) regions as β\nincreases from π down to π/2.\nIn more complex Moiré superlattices (Supplementary Fig. 6), the distribution of\ntopological charge density becomes more intricate, yet its evolution follows the same\ngeneral principle. Each topological state features regions of both positive and negative\ntopological charge density, which evolve continuously with variations in the phase\nparameters. The topological state of the system is jointly determined by the interplay\nbetween these regions. When a singularity emerges during the phase evolution, the\ncollective contribution of the charge density undergoes an abrupt change, thereby\ntriggering a transition in the topological state of the vector field.\nQ5: The authors provide experimental confirmation of programmable TPTs using a\nnanoparticle-scattering technique, retrieving the amplitude of the out-of-plane field (E ).\nz\nThe full vector fields are reconstructed using the Gerchberg-Saxton algorithm and\nMaxwell’s equations. In our opinion, the authors could provide some more information\nabout their experimental methods.\nWhat is the resolution of the measured field distribution? What are its limitations?\nHow many silver nanoparticles are located on the gold surface, and how are they\nimmobilized? How many of these nanoparticles are involved in the measurement?\nThe GS algorithm is primarily applied to retrieve the phase from intensity\nmeasurements that are acquired in two different planes. Typically, the image plane and\nthe far field plane. Can the authors provide more information on the usage of the GS\nalgorithm to retrieve the phase of SPPs, or reference another work where this is\nexplained? Does the GS algorithm always converge to the desired solution?\nR5: We thank the reviewer for the interest in our experimental details, and we would\nlike to take this opportunity to include necessary clarifications. We have expanded the\nsection detailing the experimental procedures in the main text. The specific\nmodifications made are as follows:\nThe sample preparation followed these steps: first, a 5-nm-thick chromium layer was\ndeposited onto glass coverslips (170 μm) by electron beam evaporation to improve\nadhesion. Subsequently, a 50-nm-thick gold film was deposited using the same method.\nFinally, a droplet of a fully diluted suspension of silver nanoparticles (80 nm in\ndiameter) was placed on the surface of the gold film and allowed to evaporate naturally.\nWhile the dried sample contained numerous silver nanoparticles, measurements were\nexclusively performed on individual particles isolated by more than 20 μm from\nneighbors, ensuring that all scattered light originated from a single target nanoparticle.\nNo additional fixation is required, as natural drying provides enough stability for\n7 / 20\nmeasurement. The data was acquired at a sampling density of either 10 nm or 30 nm.\nThis choice of sampling density is determined by the spatial variation rate of the vector\nfield and experimental conditions. Theoretically, a higher density of data points\nprovides a more accurate description of the vector field. However, denser data sampling\nis more susceptible to environmental influences. Even slight vibrations can introduce\nunintended frequency artifacts into the experimental data. Therefore, for rapidly\nvarying fields, we typically use a 10 nm resolution, whereas a 30 nm resolution is\ngenerally sufficient for most common cases.\nThe detailed content can be found in lines 385 to 401 of the main text.\nIn the phase retrieval process using the Gerchberg-Saxton algorithm, the initial phase\nguess is set to zero for all pixels. According to Eqs. (S3) and (S18), the phase retrieved\nfrom the measured intensity data can only take on values of 0 and ±π. Therefore, this\ndefault phase guess also aids the algorithm in converging to the correct solution.\nAdditionally, our experience shows that a more uniform intensity distribution leads to\nbetter phase retrieval performance. Thus, during experimental data acquisition,\ncalibrating the optical path and selecting stable particles with high responsiveness are\ncrucial for obtaining a recorded intensity distribution that is as uniform as possible.\nSince the structures proposed in this manuscript, whether hexagonal lattices or\nhexagonal Moiré superlattices, the vector field at any point exhibits continuously\ndistributed function values and derivatives. Consequently, the algorithm converges to\nthe target value when the experimental data exhibits no significant intensity distribution\nimbalance.\nWe also note that the Gerchberg-Saxton algorithm has been widely employed to\nreconstruct the phase in surface plasmon polaritons, for example, Nature Photonics, 15,\n442–448 (2021). The foundational principles of the algorithm are detailed in Optik 34,\n3 (1971); Optik 35, 2 (1972); and Optics Letters 3, 1 (1978).\nQ6: The authors state in line 120: ‘This relation reveals a periodicity of 2π, within which\nthree distinct topological states emerge, characterized by TIs of 1, 0, and −1.\nSpecifically, the TI remains unchanged as β varies within several open intervals, such\nas (−π/2, π/2) and (π/2, 3π/2), indicating the robustness of the topological state.’\nCould the authors elaborate on the robustness that is given by the half-open interval?\nDoes this mean that the TI of 0 is more difficult to create or measure?\nR6: We thank the reviewer for raising this and we would take the opportunity to clarify\nthis. Robustness refers to the property whereby a system’s topology remains invariant\nunder perturbations. For the skyrmions formed in the hexagonal lattice proposed in this\nwork, we find that the topological state of the vector field remains unchanged across\nthe phase parameter intervals (−π/2, π/2) or (π/2, 3π/2). The persistence of the\ntopological state under continuous variations of β within these ranges clearly\ndemonstrates its robustness against phase perturbations.\n8 / 20\nIn the hexagonal lattice, the topological state characterized by a TI = 0 represents a\ncritical point between two distinct topological phases, TI = +1 and TI = −1, occurring\nspecifically at β=π/2. Theoretically, each state can be uniquely defined by a specific\ninterval or discrete value of β. However, in practical implementations, the topological\nstates with TI = ±1 exhibit higher tolerance to experimental imperfections than the\ncritical state. This arises because, in real systems, the implemented phase is often\ninfluenced by external factors, such as inhomogeneities in the refractive index of the\nmedium. When such perturbations induce small deviations in phase parameters, the TI\n= ±1 states—corresponding to broad, stable phase intervals—remain unaffected by\nminor variations. In contrast, the critical TI = 0 state is inherently unstable: even slight\nperturbations can drive it to transition abruptly into one of the neighboring topological\nstates.\nQ7: Line 81: remove ‘distint’.\nLine 90 and Supplementary Section 2: ‘warps’ should be replaced by ‘wraps’.\nSupplementary Section 2: ‘Soling’ should be replaced by ‘Solving’.\nLine 167: replace ‘simultaneous’ with ‘simultaneously’.\nLine 216: ‘This scaling behavior underscores the capacity of Moiré structuring to\nsupport a wide spectrum of topological states (Supplementary Information Section 5).’\nThis sentence should be rephrased to e.g: ‘This scaling behavior highlights the ability\nof Moiré structuring to support a broad range of topological states (Supplementary\nInformation, Section 5).’\nLine 285, 289: ‘The unit cell is outlined…’.\nLine 294: remove ‘nest’.\nR7: Thank you very much for the careful and thoughtful review of our manuscript. We\nsincerely apologize for any unintentional errors and would like to confirm that we have\nthoroughly re-examined the entire manuscript and corrected all identified issues.\nResponse to Reviewer #2:\nComment: I read with great interest the manuscript by B. Tian et al, about\nimplementation of high-order topological invariants in plasmonic fields using Moiré\nsuperlattices. Although the idea itself is not new (first performed in Ref. 37 of the\nmanuscript), the authors extend the known theory and understanding of this\nphenomenon, and showcase control over the topological invariant via an SLM. They\ncharacterize the reason for transitions between topological numbers and demonstrate\ntheir theory through an experiment.\nI find that the authors’ idea is innovative, interesting to a broad audience in photonics\nand topological physics, and – for the most part –performed rigorously (see my\ncomments below for a few “loose-ends”). I am inclined to recommend publication in\nprinciple, given that the authors adequately address all comments below:\n9 / 20\nQ1: The authors call the transitions in their system “phase transitions”. It is not an\naccurate phrase, since it is not a many-body system, nor does it contain any form of\nnonlinearity. The cited works in topological photonics certainly do not show phase\ntransitions, but rather – just “topological transitions”, which are the commonly agreed\nupon term. I request that the authors amend their manuscript in accordance with this\npoint.\nR1: Thank you for the insightful suggestion. Our initial use of the term \"topological\nphase transition\" was inspired by condensed-matter physics, where it describes\ntransitions between distinct topological phases in electronic systems. This terminology\nwas adopted because the underlying mathematical framework—particularly the\nanalogy between the Chern number and the skyrmion number in the two-band model—\nis similar. However, we acknowledge that the physical mechanisms in our plasmonic\nMoiré system differ from those in condensed-matter systems. To ensure conceptual\naccuracy, we have therefore replaced \"topological phase transition\" with \"topological\ntransition\" throughout the whole manuscript, including in the title.\nQ2: The claim of showing topological invariants between −58 and 58 is unsubstantiated.\nThe authors show this in theory, while measuring only −8 to 8. I suggest that either the\nauthors amend their claim, or properly measure topological invariants up to 58.\nR2: We are grateful to the reviewer for the valuable suggestion. We have revised the\nmanuscript to clarify the distinction between theoretical predictions and experimental\nverification regarding the range of topological invariants. In theory, our calculations\nshow that very large-scale topological transitions can be achieved by changing the\nMoiré angle. Experimentally, we verified the occurrence of these transitions at a\nspecific Moiré angle (13.17°), confirming the predicted topological changes. The\nrelevant section now reads as follows:\nTheoretical calculations indicate that TTs occur between skyrmion configurations\ncarrying integer and half-integer topological invariants (TIs) within the range of −8 to\n+8, excluding TIs that are integer multiples of 3/2 at a Moiré angle of 13.17°. The\naccessible range of TIs further expands to −19 to +19 at 9.43°, and reaches −58 to\n+58 at 25.04°, representing one of the largest-scale TTs reported to date in physical\nsystems. This selection rule, governed by the discrete symmetry of the lattice, reflects\nan intrinsic constraint on real-space topological quantization. Experimentally, we\ninvestigated the evolution of the topological states of the vector field in the Moiré\nsuperlattice at a Moiré angle of 13.17°. Although measurements were conducted at this\nsingle angle, the experimental results clearly confirm the occurrence of the predicted\ntopological transitions, consistent with the theoretical calculations regarding the\npresence and distribution of these topological states.\nThe detailed content can be found in lines 62 to 73 of the main text.\n10 / 20\nQ3: I greatly appreciate that the authors check whether or not their minimal field\namplitudes tend to 0. In such points, the topological density is ill-defined, so the whole\ndistribution cannot have a well-defined topological charge other than 0. In this respect,\nany fractional topological charge the authors measure is probably occurring in such a\npoint, and therefore – is ill-defined. In addition, is it true that the dashed lines in Fig. 2e\ncoincide with zero minimal field amplitude lines shown in Fig. 2f ? this would mean that,\nalong these lines, the topological constant is ill-defined, whereas right now it is defined\nvia a particular color in Fig. 2e. Therefore, the figure may have to change, and the\nauthors should complete their check that indeed the topological density in cases they\nmeasure is well-defined.\nR3: We appreciate this particularly insightful comment from the reviewer. Our detailed\nanalysis of the vector field variation patterns shows that its amplitude can indeed reach\nzero. To illustrate this, we consider a singularity within a hexagonal lattice. The vector\nfield components in such a lattice are given by:\n𝐸 = 𝐸 ∑3 cos[𝑘 𝑥cos(𝜙 )+𝑘 𝑦sin(𝜙 )+𝜑 ], (R1a)\n𝑧 0 α=1 𝑠 α 𝑠 α α\n𝐸 = 𝐸 ∑3 𝑘𝑧cos(𝜙 )sin[𝑘 𝑥cos(𝜙 )+𝑘 𝑦sin(𝜙 )+𝜑 ], (R1b)\n𝑥 0 α=1 α 𝑠 α 𝑠 α α\n𝑘𝑠\n𝐸 = 𝐸 ∑3 𝑘𝑧sin(𝜙 )sin[𝑘 𝑥cos(𝜙 )+𝑘 𝑦sin(𝜙 )+𝜑 ], (R1c)\n𝑦 0 α=1 α 𝑠 α 𝑠 α α\n𝑘𝑠\nwhere 𝜑 = 𝛽/3 , 𝜑 = 𝜑 = −𝛽/3 , and the phases 𝜙 , 𝜙 , 𝜙 are successively\n2 1 3 1 2 3\nseparated by π/3. For the case of 𝛽=π/2, with 𝜙 = π/6, 𝜙 = π/2 and 𝜙 = 5π/6,\n1 2 3\nall three components of the vector field in Eq. (R1) vanish simultaneously at the\nposition (𝑘 𝑥=0, 𝑘 𝑦=4π/3). This confirms that the vector field amplitude reaches a\n𝑠 𝑠\nminimum of zero.\nDespite the presence of these singularities, a well-defined topological invariant exists\nand effectively characterizes the system. As shown in Supplementary Fig. R2a for\n𝛽=π/2, the vector field contains singularities. Length-normalized arrows indicate vector\ndirections, with points \"a,\" \"b,\" and \"c\" marking three singularities located at the\nvertices of the lattice unit cell. To better illustrate the variation trends near singularities,\nSupplementary Fig. R2b zooms in on the region around singularity \"a.\" At this\nsingularity, all three vector components vanish, leaving the vector direction undefined\nat that point. This implies that across the unit cell, the topological charge density is\nundefined only at these isolated singularities, while remaining well-defined elsewhere.\n11 / 20\nSupplementary Fig. R2| Distribution of Vector Fields and Singularities in\nhexagonal Lattice. a, The electric field distribution for 𝛽 =π/2. Length-normalized\narrows indicate the vector directions, with points \"a\", \"b\", and \"c\" marking three\nsingularities located at the vertices of the lattice unit cell. b, Zoomed-in view of the\nregion around singularity \"a\".\nThe topological invariant for skyrmions is computed via a surface integral. Importantly,\nthe presence of a finite number of singularities within the integration region does not\naffect the integral’s outcome. Therefore, the overall structure can still be robustly\ncharacterized by a topological invariant, even in the presence of singularities.\nFurthermore, our analysis shows that the dashed lines in Fig. 2e correspond precisely\nto the zero-value contours in Fig. 2f, indicating the presence of isolated singularities\nwithin the unit cell. As discussed above, these singularities do not influence the\ncalculation of the topological invariant. In summary, for any value of the phase\nparameter, a well-defined topological invariant exists to characterize the topological\nstate of the vector field.\nQ4: In the same vein of checking that the distribution is well-defined, how do the\nauthors make sure that the real part of the electric field is indeed much larger than the\nimaginary part? Of course, this can be defined up to a global phase, but to take only one\npart of the field, it must to be much larger than the other part, with the criterion defined\nin the supplementary information of Ref. 22. It seems, from looking at the\nsupplementary, that this might not be the case for some of the measurements. I strongly\nrecommend that the authors check this point to make sure that their main claims are\nsubstantiated.\nR4: We thank the reviewer for providing us with the opportunity to clarify this\nimportant point. As shown in Figs. 1a and 2b of the main text, the phase relationships\nin both the hexagonal lattice and the hexagonal Moiré lattice indicate that a pair of\ncounter-propagating monochromatic waves exhibits opposite phases. This specific\nphase relationship results in a purely real-valued vector field distribution, as confirmed\nmathematically by Eqs. (S3) and (S18).\nHowever, this idealized result applies only under lossless conditions. In practice,\nelectromagnetic vector fields experience propagation losses, leading to a complex-\n12 / 20\nvalued field, as investigated and discussed in Ref. 22. In Supplementary Information\nSection 9, we analyze the topological vector field distribution under such lossy\nconditions. Our analysis shows that, under weak-loss conditions, the real part of the\nvector field dominates over the imaginary part, and the topological properties are well\npreserved. This provides a more comprehensive and rigorous validation of the\ntopological state transitions in optical vector fields.\nWe sincerely thank the reviewer again for this valuable question, which has helped to\nenhance the completeness and clarity of our work.\nQ5: The authors claim to calculate (and measure) negative topological invariants. I find\nthis to be somewhat confusing, since it seems the two different topological invariants\nonly require that the whole field will have a global phase of π. This, however, cannot\nuniquely define a field distribution since the global phase is arbitrary. Given this point,\nall the possible topological invariants will have been spanned within β ≤2π, instead of\n1\nwithin another, more arbitrary number (as in figure 4). It would also mean that the initial\ntransition the authors analyze, not within a Moiré lattice, is no transition at all. The\nauthors should carefully address this point to make sure that their analysis is indeed\nvalid.\nR5: We greatly appreciate the reviewer’s rigorous comment and the opportunity to\nelaborate on this important point. The key to resolving the apparent ambiguity regarding\nthe definition of negative topological invariants lies in understanding the time-\ndependent nature of the electromagnetic vector field and the instantaneous definition of\nthe field skyrmion.\nUnlike optical spin skyrmions—which are defined through the time-averaged spin\nangular momentum and therefore represent stationary field configurations—the field\nskyrmions discussed in this work are defined from the instantaneous spatial distribution\nof the electric-field vector. The electromagnetic field is inherently time-varying, and at\nany given instant, the spatial orientation of the local vector field determines the\nskyrmion structure at that moment. Consequently, applying a global phase shift to the\nentire field (e.g., multiplying by eiϕ) corresponds physically to observing the field at a\ndifferent instant in its temporal evolution.\nIf the vector orientations at two instants are reversed—meaning that every local vector\npoints in the opposite direction—the corresponding skyrmion numbers will take\nopposite signs (e.g., +1 and −1). These two distributions, while connected through a\nglobal temporal phase, are not topologically equivalent, because their real-space vector\norientations trace oppositely oriented mappings onto the Poincaré sphere. Therefore,\nthey represent distinct topological configurations of the vector field.\nAn interesting and relevant example is the work of Giessen et al. (Science 368,\neaba6415, 2020), which demonstrated a hexagonal field skyrmion lattice—\n13 / 20\ncorresponding to the β = 0 case in our Fig. 1 with the addition of the full temporal factor\neiωt. In this structure, the field distributions during the first and second halves of an\noptical cycle become exact inverses of each other, corresponding respectively to\nskyrmion numbers of +1 and −1. The transition point at t=π/2ω marks a singular\nconfiguration where the real-part field amplitude vanishes everywhere, signifying a\ntopological inversion through a null-field state.\nBuilding upon this concept, our work introduces a phase-control parameter β that\nenables a generalized and continuous realization of such topological inversions at a\nfixed moment in time. As β varies from 0 to π, the vector field evolves continuously,\nwith its skyrmion number transitioning smoothly from +1 to −1 (see Fig. R3). Within\neach interval—β ∈ [0, π/2) and β ∈ (π/2, π]—the field maintains a well-defined\ntopological invariant (TI = +1 or −1, respectively), and the transition between them\noccurs through the emergence of discrete singularities rather than through global field\nextinction.\nSupplementary Fig. R3| Variation of the electric field distribution (E ) in\nz\nhexagonal lattices. a-l, Distribution of the out-of-plane electric field component E\nz\nwhen β varies from 0 (a) to π (g) in intervals of π/6.\nThis process provides a physically meaningful and mathematically rigorous route for\ndefining both positive and negative topological invariants in time-varying optical vector\nfields. It generalizes the previously reported time-driven skyrmion inversions to a\nbroader framework in which the evolution of the field topology is governed by a\ncontrollable phase parameter. In this sense, the transitions we report are not artifacts of\narbitrary phase choices, but rather represent intrinsic topological transformations that\nare fully consistent with the vector-field mapping and its associated topological\ninvariant.\nFurthermore, we regard the +1 and −1 states as distinct topological configurations not\nonly because of their opposite vector-field orientations, but also because the negative\nskyrmion number plays an essential role in the broader Moiré superlattice framework.\nIn more complex Moiré systems, negative topological invariants naturally emerge as\n14 / 20\nintermediate states in the continuous evolution of the system’s topology. If one\nacknowledges transitions between two positive topological states within the Moiré\nframework, it necessarily follows that negative skyrmion numbers must also be\nincluded, as they represent indispensable intermediate configurations along the\ncomplete topological evolution pathway. Therefore, incorporating both positive and\nnegative skyrmion numbers provides a consistent and comprehensive description of the\nfull range of topological transitions occurring in Moiré superlattices.\nQ6: Gerchberg-Saxton phase reconstruction is a tool that should be used very carefully,\nsince it is agnostic of the requirements from Maxwell’s equations, as well as from the\nanalytic nature of physical observables. This is particularly problematic in field\ndistributions involving phase singularities, where the GS algorithm exhibits ambiguity.\nHow do the authors make sure their reconstruction does not have artefacts? It is very\nimportant that this is addressed, since the entire experimental observation of the authors\nrelies on the reconstruction being accurate.\nR6: We appreciate the professional comment from the reviewer. We fully agree that the\naccuracy of the results recovered by the GS algorithm is crucial for the experiment.\nThe vector field constructed in our experiment, formed by the interference of TM mode\nevanescent waves, resembles the one described by Eq. (R1). In such a vector field,\nwhether in a hexagonal lattice or a hexagonal Moiré superlattice, the structure exhibits\nnot only continuous function values but also derivatives of any order. Furthermore,\nalthough singularities emerge during topological transitions, these refer specifically to\npoints where the vector direction is undefined, as illustrated in Fig. R3b. Such\nsingularities do not compromise the continuity and smoothness of the vector field itself,\nand the corresponding phase values remain strictly 0 or ±π. Consequently, the GS\nalgorithm converges relatively easily to the target results for this type of structure.\nFinally, the inherent robustness of the topological structure helps to mitigate biases\nintroduced during experimental data post-processing. Even in the most ideal case, the\nexperimental outcomes still cannot perfectly match the theoretical predictions.\nNevertheless, our research establishes that a necessary condition for the change of\ntopological states is the emergence of new singularities. Therefore, the deviations in the\nvector field distribution within local regions of the lattice unit cell will not alter the\ntopological properties, provided they do not introduce new singularities. These\nproperties serve as the foundation for the reliable convergence of our experimental\nresults to the target topological state.\nQ7: The whole explanation the authors give about the effects of singularities in their\nsystem is very confusing. High-symmetry and generic-position singularities are never\nrigorously defined, so it is not possible to understand whether or not the authors\nexplanation makes sense. I suggest that the authors revise their explanation or exclude\nit from the final manuscript.\n15 / 20\nR7: We thank the reviewer for this pertinent comment, and we sincerely apologize for\nany lack of clarity in our previous description. We have revised the manuscript to\nprovide a clear description of these points. The specific modifications are as follows:\nAn interesting yet crucial finding in Fig. 2e is that the TIs never take values that are\ninteger multiples of 3/2. This phenomenon arises from the distribution and\nclassification of singularities, as well as the C symmetry of the structure. The Moiré\n3\nsuperlattice described by Eq. (3) possesses C symmetry, where symmetric points are\n3\nlocated either at the center or the vertices of the unit cell. Therefore, the center and\nvertices of the unit cell are termed high-symmetry points, while other positions are\ntermed general-position points. Accordingly, singularities located at high-symmetry\npoints are classified as high-symmetry singularities (red circles in Fig. 3a), whereas\nthose occurring at general positions are regarded as general-position singularities\n(black circles in Fig. 3a and 3b).\nThe detailed content can be found in lines 255 to 264 of the main text.\nWe adopt this classification because the two types of singularities play fundamentally\ndistinct roles in topological transitions. To clarify their respective roles, Supplementary\nFig. R4 shows the configurations of systems containing each type of singularity,\nalongside the corresponding evolution of topological invariants (Supplementary Fig.\nR4 corresponds to Figs. 3 and 2e in the main text).\nSupplementary Fig. R4| Symmetry-Constrained Discretization of Topological\nInvariants. a, b, The unit cells with (a) and without (b) high-symmetry singularities.\nc, Topological invariants as β and β change independently. High-symmetry\n1 2\nsingularities and general-position singularities are marked with red and black circles,\nrespectively.\nWhen only general-position singularities are generated (Supplementary Fig. R4b), the\ntopological invariant changes by a multiple of 3. For example, along the process marked\nby the black dashed arrow in Supplementary Fig. R4c, the invariant changes from −8\nto −5. In contrast, when a high-symmetry singularity is generated (Supplementary Fig.\nR4a), whether or not accompanied by general-position singularities, the topological\n16 / 20\ninvariant undergoes sign reversal. This is illustrated by the black solid arrow in\nSupplementary Fig. R4c, where the invariant changes from −8 to +8.\nIn Supplementary Fig. R4c, the maximum attainable value of the topological invariant\nis 8. Given that the topological invariant can only reverse its sign or change in integer\nmultiples of 3, it follows that its value in Supplementary Fig. R4c cannot be a multiple\nof 3. Moreover, along the boundaries between adjacent regions in Supplementary Fig.\nR4c, the topological invariant equals the average of the values on both sides. As a result,\nnone of these boundary values can be integer multiples of 3/2.\nQ8: The authors do not properly cite Ref. 37 (published recently in Nature Physics but\nunder a different name), and also greatly minimize its contribution to the work they\npresent. Whether they relied upon it or not, it is clear that Ref. 37 was the first to\ndescribe the majority of the theory the authors show in their work, and while the\nmanuscript does contain many new advances, it does not properly point to the points\nRef. 37 has already put forth. Similarly (but to a much smaller extent), the authors do\nnot cite Tsesses et al, Nano Lett. 19 (2019), which observed and discussed some of the\nfield distributions they present (in particular, the honeycomb lattice at beta=pi/2) or\nFrischwasser et al, Nat. Photon. 15 (2021), which was – to my knowledge – the first to\nuse the GS algorithm to reproduce the phase of plasmonic fields with distinct topologies.\nR8: We thank the reviewer for carefully pointing this out. Indeed, the references\nmentioned by the reviewer, particularly Ref. 37, are extremely inspiring to the work we\nreported here. We apologize for missing this and have carefully revised the main text\nto conduct a more comprehensive discussion of their relevance and specific\ncontributions. The specific modifications made are as follows:\nIn particular, the Moiré superlattices generated in plasmonic platforms40,41 can support\narbitrarily large topological invariants. This capability not only significantly enriches\nthe diversity of topological states but also facilitates topological transitions between\nthese states.\nFigs. 1b and 1c display the spatial distributions of E for β=0 (Fig. 1b) and β=π/2 (Fig.\nz\n1c), respectively, accompanied by the corresponding normalized in-plane electric field\nvectors. These isolated topological states were previously obtained through methods\nequivalent to phase modulation44. Supplementary Fig. 1 shows the additional E field\nz\ndistributions for other β values.\nThe corresponding E -field distributions and reconstructed vector directions, obtained\nz\nusing the Gerchberg–Saxton (GS) algorithm47, are shown for three representative\nvalues of the encoded parameter: β=0, π/2, and π. The GS algorithm is a phase retrieval\ntechnique that has been used to reconstruct SPP fields with topological\ncharacteristics48.\n17 / 20\nThe detailed content can be found in lines 52-55, 123-127, and 311-315 of the main\ntext. (Refs. 37 and 38 in the initial version are now Refs. 40 and 41, respectively, due\nto structural adjustments).\nQ9: The authors should be very clear about their intention when saying they change\nbeta. They write what this means only in the supplementary (i.e., φ =β/3, φ =φ =−β/3),\n2 1 3\nbut this should be written in the manuscript itself and emphasized, since not every type\nof change in beta (for example, a change to only one of the phases) will have resulted\nin the authors’ findings.\nR9: We thank the reviewer for the thoughtful suggestion. We agree that the quantitative\nrelationship between β and (φ , φ , φ ) is fundamental to understanding the existential\n1 2 3\npatterns of the topological invariants. Accordingly, we have moved this analysis from\nthe Supplementary Information to the main text and have emphasized its significance\nin two key locations. The specific modifications made are as follows:\nIn the numerical implementation, the relationship between the phase and β can be\ndefined as 𝜑 = 𝛽/3 and 𝜑 = 𝜑 = −𝛽/3. This choice not only better reflects the\n2 1 3\nsymmetry of the structure but also ensures that the center of the unit cell remains at the\ncoordinate origin.\nWithin the computational framework, the phases are related to the parameters (β , β )\n1 2\naccording to the convention of the hexagonal lattice and are selected as follows:\n𝜑 = 𝛽 /3, 𝜑 = 𝜑 = −𝛽 /3, and 𝜑 = 𝛽 /3, 𝜑 = 𝜑 = −𝛽 /3.\n2,1 1 1,1 3,1 1 2,2 2 1,2 3,2 2\nThe detailed content can be found in lines 119-123 and 207-210 of the main text.\nQ10: The axes in Figs. 1e, f are unclear, and the analogy to a band structure does not\nmake much sense, since there are no bands in the authors’ system. I suggest removing\nthem from the manuscript.\nR10: We thank the reviewer for the astute observation. We sincerely apologize for the\npreviously unclear descriptions of the axes in Figs. 1e and 1f. We have revised this\nsection to provide greater clarity for readers in the main text. The specific modifications\nmade are as follows:\nDuring the TTs, the evolution of skyrmions is analogous to that of energy bands.\nConsequently, the energy band-like structure undergoes a change consistent with that\nof energy bands: it transitions from a gapped structure (e) to a gapless structure (f) and\nback to a gapped structure (e).\nThe detailed content can be found in lines 91 to 94 of the main text.\nWe certainly agree that the band-like structure proposed in our work shares only a\n18 / 20\nmathematical formalism with electronic band structures, while their physical origins\nand interpretations are fundamentally distinct.\nWe introduced this structure primarily for two reasons. First, it provides a clear and\nintuitive framework for demonstrating the conditions underlying topological transitions\nin optical vector fields. Given that topological phase transitions in electronic band\nstructures are well-established, adopting a similar mathematical language within the\nskyrmion context helps to elucidate the mechanisms driving skyrmion topological\ntransitions. Second, by leveraging this shared mathematical description, we hope to\nencourage future explorations of deeper connections between skyrmion physics and\nband theory. We believe this formal analogy may inspire further discoveries of novel\nphysical phenomena at their intersection.\nIt is for these purposes that we have included the band-like structure in our manuscript.\nWe would be very grateful for any additional suggestions or comments you might have,\nand we will endeavor to address them thoroughly.\nQ11: Although written very well for the most part, there are still some portions with\nawkward phrasing and typos, including (but not limited to) “the antisymmetric of vector\nfield”, page 8 line 236.\nR11: We are grateful for your meticulous review and for identifying the errors in our\nmanuscript. We have conducted a thorough proofreading of the paper and ensured that\nall minor spelling mistakes are now corrected.\nQ12: The authors should add standard deviations on their measured topological\ninvariants in fig. 4.\nR12: We thank the reviewer for the valuable suggestion on this. We have revised Fig.\n4 in accordance with your recommendation by incorporating error bars that indicate the\nstandard deviation for all applicable data points. This addition provides a more\ncomprehensive representation of the data dispersion and strengthens the statistical\nvalidity of our results.\nQ13: The rule the authors find for their topological invariants, while very interesting,\nonly applies for C symmetry – i.e., it is not general for other types of symmetry (as is\n3\ncurrently written in the manuscript).\nR13: Your understanding precisely captures our key finding. In this study, we\ndiscovered that the topological states of the vector field in hexagonal Moiré\nsuperlattices cannot take integer multiples of 3/2. The restriction arises from the\ninherent C symmetry of hexagonal Moiré superlattices. When structural symmetry\n3\nchanges, the existence rules of topological states also change.\n19 / 20\nTo emphasize this fact more clearly, we present topological states in multilayer\nhexagonal Moiré superlattices and square Moiré superlattices in the Supplementary\nInformation Sections 7 and 8. The results demonstrate that in multilayer hexagonal\nMoiré superlattices, even with complex structures, the pattern governing the existence\nof topological invariants remains unchanged due to the preservation of C symmetry.\npattern of topological states, where phase variations result in the topological invariant\nalways remaining zero. This phenomenon arises because the singularities generated in\nthe square Moiré superlattice are typically created in pairs with opposite winding\nnumbers. Consequently, these singularities cancel each other out, leading to no net\nchange in the topological state.\nTo further clarify that the patterns of topological invariants in our work are specific to\ncertain symmetries, we have emphasized this point in the main text. The specific\nmodifications made are as follows:\nNote that the pattern of topological states is governed by the structural symmetry, as\ndemonstrated by our investigation of multilayer Moiré superlattices. In multilayer\nhexagonal superlattices (Supplementary Information Section 7), the preservation of C3\nsymmetry maintains a consistent pattern, even in complex multi-layer configurations.\nConversely, the distinct symmetry of the square superlattices (Supplementary\nInformation Section 8) yields a fundamentally different pattern of topological\ninvariants.\nThe detailed content can be found in lines 294 to 300 of the main text.\n20 / 20"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have improved the manuscript accordingly and it can now be accepted for publication.", "reviewer_2": "I read with great pleasure the response of the authors to my comments and to the comments of the other reviewer. I believe they have adequately addressed our concerns, and I recommend that their manuscript be published in Nature Communications.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": ""}]

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