Chiral valley edge states based on Dirac mass engineering

Domain: physics
Published: 2025-11-27
Editorial outcome: minor_revision

Abstract
Valleytronics has emerged as a promising platform for energy-efficient and high-speed information processing. However, backscattering-induced valley depolarization remains a fundamental limitation, making valley-polarized waves with backscattering-free propagation highly desirable. Here, we demonstrate a general approach to realize chiral valley edge states in topological photonic crystals, which integrates the robust chiral edge states with valley degree of freedom. By controlling the valley Dirac masses, we selectively confine the chiral edge band around a single valley, enabling backscattering-free propagation while inducing valley polarization. Via engineering Dirac masses in both momentum and real spaces, we propose valley multiplexing as a novel functionality that enables independent and arbitrary control over waves with distinct valley polarizations. Moreover, two key components for valley multiplexing are demonstrated: a valley (de-)multiplexer and a valley-locked waveguide crossing, facilitating low-crosstalk signal routing. By establishing the interplay between topological quantum Hall and valley Hall phases, our work offers a new framework for robust valley-based information processing.

Introduction
Owing to advantages in energy efficiency, robustness, and high processing speed, valley has been highlighted as a promising degree of freedom (DoF) for information encoding and storage, catalyzing the birth of valleytronics1–3 and valley photonics4,5. In certain crystalline materials like graphene and transition metal dichalcogenides (TMDCs), valleys refer to a set of local extrema in energy bands, possessing the same energy but inequivalent momenta1. To process information via valley DoF, it is crucial to enable valley wave transport and control valley polarization6–9. However, since information is encoded in momentum space, the inevitable defect-induced backscattering is a persistent obstacle. This limited robustness of valley polarization mainly originates from its inherent reciprocity. In valleytronics, backscattering leads to inter-valley scattering even with perfect valley initialization10, causing severe valley depolarization and limiting valley lifetime; while in valley photonics, it gives rise to Anderson localization and propagation loss11,12.

Conversely, the quantum (anomalous) Hall phase has long been recognized as a strong topological phase, supporting unidirectional chiral edge states in accordance with the bulk-boundary correspondence13–15. Due to the broken time-reversal symmetry and the resulting non-reciprocity, these chiral edge states are entirely immune to backscattering and remain unaffected by defects and interface deformations of any kind. This novel property of backscattering-free transmission offers the potential for addressing the power consumption issue inherent in traditional complementary metal-oxide-semiconductor (CMOS) device technologies, spurring research efforts into the Chern networks16–18. Nonetheless, while the existence of chiral edge states is guaranteed by topology, their dispersion configuration in momentum space lacks specific constraints. Consequently, these chiral edge states typically span the Brillouin zone (BZ), without exhibiting valley-dependent characteristics.

Intuitively, integrating chiral edge states with the valley DoF offers a promising solution to not only solve the longstanding valley depolarization issue in valleytronics but also expand the usable DoFs in Chern networks. Although previous theoretical studies have suggested that chiral valley edge states can function as valley filters19–21, experimental confirmation remains absent. Beyond chiral valley edge states, the interplay between quantum Hall and valley Hall phases has also been explored22,23, revealing that chiral and valley edge states can coexist within a single system—either in separate band gaps or within the same band gap but spatially segregated.

Here, we propose a general approach to realize chiral valley edge states through Dirac mass engineering. In crystalline lattices, valleys can be effectively considered as low-energy massive Dirac quasiparticles, with their topological properties determined by the Dirac masses24,25. Thus, by means of Dirac mass engineering, the chiral edge band can be selectively confined around a single valley, thereby imparting valley-dependent characteristics. Due to the non-reciprocity and backscattering-free propagation, the chiral valley edge states can robustly preserve valley polarization during transmission.

We demonstrate the realization of chiral valley edge states by leveraging hybrid topological photonic crystal systems composed of Chern photonic crystals (CPCs) and valley photonic crystals (VPCs). The resulting unidirectional propagation and robust valley polarization are verified both numerically and experimentally. Furthermore, via precisely engineering Dirac masses in both momentum and real spaces, we introduce a novel functionality termed valley multiplexing, which enables independent and arbitrary control over waves associated with distinct valley polarizations. Two key components for photonic valley multiplexing are exhibited: a photonic valley (de-)multiplexer, serving as the core component enabling valley multiplexing; and a photonic valley-locked waveguide crossing, which facilitates low-crosstalk inter-crossing between waves with different valley-polarizations—a functionality that is difficult to achieve using either pure valleytronic or pure Chern systems.

Results
Chiral valley edge states
It is well established that topological quantum Hall insulators host gapless unidirectional chiral edge states, in accordance with the bulk-boundary correspondence (Fig. 1a)26–28. Nonetheless, these chiral edge states typically span both valleys in momentum space and thus lack valley-dependent characteristics. On the other hand, owing to the time-reversal symmetry, valley Hall insulators support “helical” valley edge states29–31, which manifest as two counter-propagating edge modes with distinct valley polarizations (Fig. 1b). Related by time-reversal symmetry, their inherent reciprocity makes the backscattering unavoidable.

Our primary objective in realizing the chiral valley edge state is to confine the chiral edge band around a single valley (Fig. 1c). In the following, we present a general approach based on valley Dirac mass engineering, showing that chiral edge bands can be selectively restricted to a specific valley. These states integrate unidirectional propagation with valley-dependent characteristics, inheriting the advantages of both quantum Hall and valley Hall phases.

Usually, in two-dimensional (2D) graphene-like lattices, valleys refer to a set of inequivalent local extrema in energy bands, located at BZ corners (i.e., the K and K’ points). The eigenstates near a given valley (say, the K valley) are governed by a local Dirac Hamiltonian H K (k) = v x k x \sigma x + v y k y \sigma y + m K \sigma z. As such, a valley can be effectively deemed as a massive Dirac quasiparticle, with its topological features characterized by its Dirac mass—m K. At a domain wall between two lattices with opposite-sign m K (see Fig. 1d), a unidirectional K-valley Jackiw-Rebbi mode is topologically ensured, whose propagation direction is determined by the signs of two m K 32.

For a single 2D lattice, the Dirac masses on different valleys (m K and m K') can be separately engineered. This allows us to mold the flows of different valley states independently by spatially encoding the distributions of m K (r) and m K' (r), thereby enabling precise control over the valley polarization of chiral edge states. For instance, as shown in Fig. 1e, by engineering m K (r) but keeping a homogeneous m K' (r), we can construct a Z-shaped bending waveguide supporting chiral K-valley waves. Conversely, by keeping m K (r) uniform and engineering m K' (r), a backscattering-immune power splitter for K’-valley waves can be created (Fig. 1f). More significantly, since the Dirac masses can be discretionarily engineered in both momentum and real spaces and thereby the valley polarization of each chiral edge mode can be independently controlled, it is possible to simultaneously implement a unidirectional K-polarized bending waveguide and a backscattering-immune K’-polarized power splitter within the same structure by simply superposing the distributions of K- and K’-valley masses (Fig. 1g). It is worth emphasizing that, this flexible and comprehensive utilization of both valleys—referred to as valley multiplexing—not only multiplies the transmission capacity but also enables diverse functionalities on a single chip, offering significant potential for high-density integration.

We demonstrate the realization of chiral valley edge states in hybrid topological photonic crystal systems composed of topological CPCs and VPCs, which together provides sufficient DoFs for Dirac mass engineering. For instance, Fig. 2a plots the band structure of a Chern photonic crystal (CPC-1), composed of a honeycomb array of gyromagnetic rods (yttrium iron garnet, YIG). Figure 2b presents the band structure of VPC-1, made from two sets of dielectric rods with different radii. Berry curvatures of the two crystals are encoded in color, reflecting their different valley Dirac masses. Here, we consider the transverse magnetic (TM) polarization. Detailed material information can be found in Method and Supplementary Note 1.

Because their K (K’) valley masses take opposite (same) signs, when we combine these two crystals together (Fig. 2c), the domain-wall between them supports only a chiral K-valley mode propagating to the right (see the red in-gap state in Fig. 2d). Figure 2c exhibits the experimental setup for measuring the chiral valley edge states. To mimic the 2D condition, the photonic crystals are sandwiched between two metallic plates. An external magnetic field of B ext = 0.042 T is applied to YIG rods (along ±z direction) by positioning two magnets above and beneath each rod. A dielectric foam board (n = 1.04) is used to position the rods (see inset in Fig. 2c). The dispersion of chiral K-valley edge state is experimentally probed by performing Fourier transform on the measured complex E z field along the domain wall, as represented by the color plaques in Fig. 2d [see Method and Supplementary Note 2 for detailed experimental setup]. Similarly, we can selectively endow the chiral edge state with K’-valley polarization by utilizing CPC-2 (time-reversal counterpart of CPC-1, featuring positive masses at both valleys) and VPC-1, as shown in Fig. 2e. Notably, since the orientation of B ext is reversed when changing from CPC-1 to CPC-2, the propagation direction of the corresponding edge states flips [see Supplementary Note 3 for more detailed numerical and experimental edge dispersion]. We also validate the robustness of our chiral valley edge states by considering wave transport against position perturbations and several types of obstacles (Supplementary Figs. S10 and S11).

To further present the unique valley multiplexing functionality, two representative photonic devices based on the chiral valley edge states are demonstrated: a photonic valley (de-)multiplexer, serving as the core component enabling valley multiplexing; and a photonic valley-locked waveguide crossing, which facilitates low-crosstalk inter-crossing between waves with different valley polarizations.

Photonic valley (de-)multiplexer
Multiplexer, as a core component of multiplexing techniques33,34, is designed to combine individual signals into a composite one. Transmitting such a composite signal through a shared bus channel not only enhances data transmission capacity but also improves chip space utilization. In the context of valley multiplexing, which aims to merge K- and K’-polarized waves into a single waveguide (i.e., the bus channel), the Dirac mass distributions for two valleys can be deduced as in Fig. 3a. According to the superposed Dirac mass distribution, we construct the photonic valley multiplexer using VPC-1, CPC-1, and CPC-2 (yellow, blue and pink domain in Fig. 3b, respectively).

As illustrated in Fig. 3b, a Y-junction multiplexer is formed between three domain-wall waveguides. As we mentioned above, a chiral K-valley (K’-valley) edge state is supported at the VPC-1@CPC-1 (VPC-1@CPC-2) domain wall. In addition, since CPC-1 and CPC-2 have opposite masses at both valleys, the CPC-1@CPC-2 domain wall supports two chiral valley edge states, each localized at one specific valley (see inset of Fig. 3b and Supplementary Fig. S9). Thus, the CPC-1@CPC-2 domain wall serves as the bus waveguide, carrying signals with both valley polarizations. Consequently, the downward-propagating K-polarized wave and upward-propagating K’-polarized wave converge into the horizontal bus waveguide. Simulated field profiles in Fig. 3c, d show smooth transmission with no observable scattering into the opposite valley, as confirmed by the Fourier spectra (insets of Fig. 3c, d). Moreover, the simulated (solid lines) and measured (dashed lines) transmission spectra in Fig. 3e, f also verify both the functionality of valley multiplexer and the one-way characteristic of valley chiral edge states. Given that obstacles at the junction point may induce inter-valley scattering, we further provide simulations of the multiplexing process against position perturbation and cavity obstacles (Supplementary Fig. S12).

For independent manipulation of valley-polarized modes, a valley de-multiplexer is essential to split different valley waves into designated channels. Notably, since all three channels in Fig. 3b are one-way waveguides, this structure cannot be used as a de-multiplexer by simply impinging waves from the right-hand side. Instead, a photonic valley de-multiplexer can be implemented by its reversed structure as depicted in Fig. 4a. When electromagnetic waves are incident from the left, the two valley components are routed into different channels at the Y-junction (Fig. 4b), demonstrating de-multiplexing functionality. To verify this valley-dependent splitting effect, we compare the E z-field profiles with the simulated eigen edge modes. As shown in Fig. 4c, d, the upper CPC-1@VPC-1 waveguide field is asymmetric about the edge with a clockwise energy vortex, while the field in lower CPC-2@VPC-1 waveguide is symmetric with two energy vortexes. These features align well with the eigenmodes at K/K’ valley, confirming their valley polarization purity (see the eigenmodes in Supplementary Fig. S7). Moreover, we measure the field profiles in the two domain-wall waveguides, respectively, as shown in the right panels of Fig. 4c, d. The measured profiles match well with the simulation results. In addition, we directly measure the field profiles along the CPC-1@VPC-1 and CPC-2@VPC-1 channels and extract their corresponding edge dispersions (Supplementary Figs. S4d and S4e). All our experimental results confirm the pure valley polarization in both channels.

In valley photonics, valley edge states exhibit topologically protected refraction at the valley-preserving zigzag termination31,35. Since the outgoing beams conserve tangential components of the wavevector, the direction of the refracted beam is determined by the valley polarization of valley edge state. Therefore, this characteristic allows further experimental verification of two channels’ pure valley polarizations, by observing the refraction direction at the waveguide exits. As shown in Fig. 4e, we place an E z-polarized point source at the CPC-1@CPC-2 waveguide. The excited field is routed by the de-multiplexer and subsequently refracts into the upper and lower empty-waveguide regions. Notably, since the frequencies of our chiral valley edge states lie below the air’s light cone, two acrylic semi-circular lenses are placed at the exits to efficiently couple the beams outward [see supplementary Note 2 for details of the experimental setup]. The resulting output beams exhibit valley-dependent refraction: the measured output beam at the upper-right CPC-1@VPC-1 exit is refracted to the left while the beam at the lower-right CPC-2@VPC-1 exit is refracted to the right. These observed field patterns are consistent with both the k-space analyses and the numerically simulated results. An additional advantage of our approach is the reconfigurability of propagation paths by engineering the Dirac mass distribution. As the simulated and measured results shown in Fig. 4f, by reversing the valley Dirac masses of VPC domain (orange domain, denoted as VPC-2), the propagation paths of the K- and K’-polarized modes are flipped, enabling flexible routing. We also analyze the robustness of valley de-multiplexer, as presented in Supplementary Fig. S13.

Valley-locked waveguide crossing
Apart from multiplexer and de-multiplexer, waveguide crossing is another critical component for high-density integrated in photonic chips. As device density and circuit complexity increase, optical waveguide crossings become essential elements that significantly influence the overall performance of photonic chips36–40. Here, we propose a valley-locked waveguide crossing based on valley-dependent Dirac mass engineering. To our best knowledge, crossing between topological waveguides has yet to be proposed or realized so far, whose performance can be effectively improved by topological aspects of robust edge states.

In order to have a K-valley edge mode propagating horizontally and a K’-valley edge mode propagating vertically, we design the Dirac mass distributions for two valleys separately (Fig. 5a), and then construct the valley-locked waveguide crossing via the superposition scheme. As a result, the valley Dirac mass configuration required for each domain is illustrated in Fig. 5b: CPC-1 (blue) and CPC-2 (pink) occupy the upper-left and lower-right regions, while VPC-1 (yellow) and VPC-2 (orange) occupy the lower-left and upper-right. Due to the distinct valley polarizations of edge modes, the two channels are decoupled from each other even at the intersection point, resulting in minimal crosstalk. Furthermore, inheriting the topological protection of chiral edge states, this type of waveguide crossing can effectively minimize both backscattering loss and crosstalk.

We validate the performance of the valley-locked waveguide crossing by analyzing its transmission spectra and electromagnetic field profiles. First, we launch electromagnetic waves from port 1, then probe the transmission spectra from all other ports, as shown in Fig. 5c. The pink line plots the crosstalk from port 1 to port 4, which is lower than −16.2 dB within the edge-mode-supported frequency range (highlighted in gray, from 5.20 to 5.67 GHz). Notably, the orange line shows even lower crosstalk from port 1 to port 3, confirming the unidirectional nature of our waveguide channels. Corresponding experimental measurements of transmission spectra are shown in Fig. 5d, indicating an average crosstalk level of approximately −9.8 dB. We scan the near-field pattern around the crossing region hole-by-hole to observe the cross transmission of two valley waves. The measured field profile at 5.57 GHz demonstrates a smooth propagation with negligible crosstalk (Fig. 5e). Besides, a 2D Fourier transform is applied to the measured field profiles to obtain its momentum information. As demonstrated in Fig. 5f, the momentum is concentrated around the K valley, confirming the valley-locked feature of topological waveguide crossing. Similar behaviors are observed when electromagnetic waves are excited from port 3, showing a crosstalk level below −15.7 dB in simulation (Fig. 5g) and an average crosstalk about −11.8 dB in experiment (Fig. 5h). Its momentum distribution concentrates around K’ valley (Fig. 5j). Although there exists some insertion loss and degradation of the crosstalk level in experiment, which may result from the reserved air holes and fabrication imperfections, our findings demonstrate that signals in the two channels maintain distinct valley polarizations, effectively ensuring the low crosstalk in this topological waveguide crossing. We also perform analyses on the robustness of this valley-locked waveguide crossing (Supplementary Fig. S14).

Methods
Materials
The valley photonic crystals in this work are realized by two kinds of dielectric rods with radii d 1 = 6 mm and d 2 = 7.5 mm, arranged in hexagonal lattice with lattice constant a = 18 mm. The material of the rod is ceramic, with relative permittivity of 8.8 and a relative permeability of 1. The Chern photonic crystals in this work are realized by using gyromagnetic rods biased by an external magnetic field. The gyromagnetic rods have radii of d 0 = 6 mm, and also are arranged in a hexagonal lattice with lattice constant a = 18 m. The gyromagnetic material is yttrium iron garnet (YIG), a ferrite with relative permittivity 14.6. The relative magnetic permeability of the YIG has the form of

\mu = \begin{pmatrix} \mu r & i\mu k & 0 \\ -i\mu k & \mu r & 0 \\ 0 & 0 & 1 \end{pmatrix}, (1)

where \mu r = 1 + (\omega 0 + i\alpha\omega)\omega m / ((\omega 0 + i\alpha\omega)^2 - \omega^2), \mu k = \omega\omega m / ((\omega 0 + i\alpha\omega)^2 - \omega^2), \omega m = 4\pi\gamma M s, \omega 0 = \gamma\mu 0 H 0, 4\pi M s = 0.195 T is the saturation magnetization, \mu 0 H 0 = 0.042 T is the external magnetic field, \gamma = 1.76 \times 10^{11} s^{-1} T^{-1} is the gyromagnetic ratio, \alpha = 0.0001 is the damping coefficient, and \omega is the operating frequency. The Dirac masses of the Chern photonic crystals can be controlled by the direction of the external magnetic field, which can be tuned by the direction of magnets in the experiment. Both the ceramic and YIG rods have a height of 5 mm, and are sandwiched between two metallic plates to mimic the two-dimensional condition. Photonic modes with transverse magnetic polarization are considered in this work.

Experiments
In our microwave experiments, monopole antennas connected to a vector network analyzer (Keysight E5071C) via coaxial cables were employed to excite and probe electromagnetic waves. The antennas used had a length of 13 mm. For all measurements, electromagnetic fields were excited by inserting a source antenna into the sample through reserved air holes.

For edge dispersion measurements, the probe antennas scanned the E z-field profiles sequentially along the domain walls between photonic crystals through reserved air holes. Edge dispersions (Figs. 2, S4 and S8) were obtained via Fourier transformation of the measured E z fields. Similarly, E z fields in Fig. 5e, i were measured by probing each hole individually around the waveguide crossing region, with the corresponding momentum information in Fig. 5f, j extracted through Fourier transformation. The measured FFT spectra in Figs. 2, S4 and S8 are obtained by summing up the k-components in both the first and second BZs. The 2D FFT maps plotted in Figs. 3 and 5 are original spectra in 2D k-space, without BZ folding.

In Fig. 4e, f, electromagnetic fields were excited at the CPC-1@CPC-2 domain wall, directing the waves according to valley polarization into free space, where refracted E z fields were scanned. To maintain quasi-two-dimensional conditions, a metallic parallel-plate waveguide was constructed in free space to support guided modes. Additionally, since edge mode frequencies lie below the air’s light cone, two acrylic semi-circular lenses were placed at the outputs to facilitate coupling into free space.

Simulations
The dispersion relations and field patterns were simulated using the finite element software COMSOL Multiphysics. The unit cell of the Chern photonic crystals and valley photonic crystals are hexagonal lattice, of which we applied Floquet boundary conditions on the outer boundaries when calculating the bulk band structures in Fig. 2a, b, S5, and S6. The edge band structure in Fig. 2d, e, S7-S9 was calculated using a supercell with 1 \times 14 periods.

Data availability
All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Information. Additional data related to this paper may be requested from the corresponding authors.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In this manuscript, J.-W. Liu et al report on a novel type of photonic topological states found at the interface between a Chern and a valley-Hall topological insulators. These inherit the valley degree of freedom properties of valley-Hall kink states and the non-reciprocal nature of Chern-type edge states, i.e., they are chiral edge states at and around a specific K-valley in momentum space. The rationale behind these waveguides is that the Dirac masses are engineered both in real and momentum space in an adequate way. Following the very same rationale, efficient components for valley-based photonic circuitry –such as (de-)multiplexers or waveguide crossings—are devised and theoretically/experimentally demonstrated. Overall, the proposed and demonstrated approach for backscatter-immune valley transport is very sound and constitutes an important step in the field of topological photonics. A future implementation of such configuration at technologically-relevant wavelengths would be groundbreaking. Unfortunately, the results are not presented in the most convincing manner, and the authors do not provide enough information about the experiments for the complete understanding of the reader. I therefore cannot recommend publication in Nature Communications for now, but I am quite certain I can be in the future provided that the following comments and suggestions—from the most general to the most specific—are considered by the authors.\n- In the abstract, the authors should mention earlier that the implementation is done in a photonic crystal. Otherwise, it can read confusing, to the point that I was unsure it was within my area of expertise until I read considerably far into it.\n- The authors mention that all devices are characterized experimentally, but I do not see any experimental data for the multiplexer and for the de-multiplexer the only experimental evidence is the refraction of the field outside the structure. From other experiments, it seems that they can visualize the near field, e.g., they show dispersion diagrams via FFTs, so why not do that?\n- Connected to the previous point, absolutely no description of the experimental methods is given. The authors need to give at least a succinct description of the methodology (how the modes are excited, how the near fields are measured, how transmission is measured, etc.).\n- It would be nice to have Fig. S5 or part of it brought into the main text. Having a visualization of the actual structure at some point always helps the reader, especially the more generic readership. Papers in topological photonics, especially those at microwave frequencies, are now often taking the visualization of the fabricated structure into supplementary materials, which is making them obscure to read as the distinction between what is simulation and what is experiment becomes unclear. I would really encourage the authors to consider this.\n- The mentioning of references 28 and 29 needs to come earlier and in a more organic way. Otherwise, it seems like big claims are made prior to that without acknowledgement of the relevant state-of-the-art.\n- For context, the following publications need to be cited as examples of dual topological Valley Hall and Chern insulator.\n--> Wang, Y., Wang, HX., Liang, L. et al. Hybrid topological photonic crystals. Nat Commun 14, 4457 (2023). I understand it is achieved for two different band gaps, but I find it relevant to the context of the submitted manuscript.\n-->Gao, P., Zhang, Z. and Christensen, J. Phys. Rev. B 101, 020301(R) (2020). Very similar ideas from the ones proposed for sonic crystals.\n- Can the authors plot, for the multiplexer and de-multiplexer, the scattering matrix amplitudes versus wavelength/frequency? They mention this happens “without any valley flipping”, but being more quantitative is important. Field maps are useful, but getting the absolute numbers would be appreciated to understand the validity of such devices for large-scale circuits. Actually, I believe a more quantitative analysis of the scattering parameters is important, similar to what is done with the waveguide crossing.\n- Have the authors studied the effects of disorder? I understand that the non-reciprocal nature of the waveguide talks for itself in terms of disorder-induced backscattering, but I would be interested in understanding at least the effect of fabrication imperfections on the scattering matrices of (de-)multiplexer and waveguide crossing.\n- Line 55: The sentence “the limited robustness of valley polarization mainly originates from its association with a relatively weak topological phase—the valley Hall phase—which is protected solely by crystalline symmetry” requires a reference.\n- Can the authors at least discuss the difference between simulation and experiment for the valley-locked waveguide crossing? There are a few dB in difference for both insertion loss and crosstalk.\n- Typo in line 87: “exponentially” is probably “experimentally”.\n- Line 91: the authors say that it is difficult to make a non-interfering crossing between waves with different valley-polarizations. Is it difficult or not possible? If \"difficult\", do other papers exist on the topic?\nOverall, my feeling is that re-organizing the way the data is presented, with more emphasis on and details of the experimental results will already substantially improve the manuscript. I hope this can be implemented by the authors.", "reviewer_2": "In this work, the authors develop a microwave system that they report to host chiral valley edge states. The majority of the paper is a theoretical study that describes the requirements of a physical system to host such states. Specifically they develop a metamaterial system comprised of magnetic (YIG) few mm diameter rods in a quasi 2D system. They study the propagation of microwave radiation through the devices. The study includes fabrication of the structure and experiments on microwave propagation through the devices. The theoretical work is fairly well described and appears reasonable. The fabrication of the structure is not described in sufficient detail. Experimental results are presented, but there is no detailed description of the experimental technique or diagram of the measurement. The description of the fabrication and experimental techniques needs to be improved before publication can be considered.", "reviewer_3": "In the written response to the reviewers and through the updated version of the manuscript, the authors have carefully addressed most of my previous points, some to an extent larger than I would have expected (the effect of disorder). I also feel that important recommendations by the other reviewers have been addressed adequately. In addition, I find that, given the previously missing important references and the need to give credit to previous work sooner in the text—and the related comments of Reviewers 3&4—the new title is more suitable for the manuscript. I am happy to say that I can now recommend publication in Nature Communications and only want to raise a few minor points/comments that I hope the authors can and will consider.\n\n- If not constrained by editorial reasons, I would suggest that Fig. 2 is split into a figure discussing the chiral valley waveguide and another on the multiplexer. It would structure the manuscript better and seems natural given the flow of the manuscript. It would also give the authors the opportunity to make the part on the waveguide design to be more generic by including some of the measured dispersion from Fig. S8.\n\n- I find that the comparison in Fig. 3(c) and (d) of the field that propagates after the demultiplexer and the respective eigenmodes is nearly tautological. It is the only mode that can propagate there at that frequency, so the only possible outcome is that the two simulated fields look the same. I would encourage the authors to prioritize the experimental result on the field probed on the top and bottom segment rather than including something that looks more valuable as a double-check to validate that the simulations are well set.", "reviewer_4": "The manuscript 'Chiral Valley Edge States' by J.-W. Liu, G.-G. Liu, B. Zhang et al. reports a study of imparting valley-polarization to chiral edge states by assembling an interface between a Chern insulating and valley-polarized phase in photonic crystals. Geometric variations of this configuration are subsequently proposed for achieving valley multiplexing, demultiplexing, and valley-locked waveguiding.\nThe underlying idea is described clearly and succinctly, and the design approach for the valley-selective waveguiding and (de)multiplexers is elegant and simple. The experimental results are similarly convincing and well-substantiated by measurements and analysis. That said, the underlying idea is not very novel or, in our assessment, likely to be very impactful. Unfortunately, key references to prior and highly relevant work are missing (see below). Concretely, several works have previously proposed and demonstrated valley-polarized chiral edge states: as such, the main novelty here lies in the specific proposals for waveguiding and (de)multiplexing. While conceptually neat, the likely impact of these \"device\" designs – considering that time-reversal broken topology is highly challenging to scale to technologically relevant frequencies due to very considerable material challenges – is likely to be rather low.\nWith this limited impact and novelty in mind, we cannot recommend the manuscript for publication in a high-impact, broad-readership journal like Nature Communications, and we recommend that the authors seek publication elsewhere.\nIn addition to the above assessment, we add two comments below, the first of which is more critical than the second. We also add some minor comments for the consideration of the authors.\nMain comments\n1. Proper contextualization relative to existing literature on chiral valley edge states is missing. For example, the following papers - probably an incomplete list of work in this area - each demonstrate a valley-polarized chiral edge state:\n- [R1] Y. Wang et al., Hybrid topological photonic crystals, Nat. Commun. 14, 4457 (2023) [https://doi.org/10.1038/s41467-023-40172-6].\n- [R2] J. Chen et al., One-Way Valley-Robust Transport in Edge-Tailored Photonic Crystals, Phys. Rev. Lett. 134, 203803 (2025) [https://doi.org/10.1103/PhysRevLett.134.203803].\n- [R3] J. Liu et al., Coexisting valley and chiral edge states in topological photonic crystals, Opt. Express 33, 13887 (2025) [https://doi.org/10.1364/OE.551936].\n[R1] demonstrates a valley-polarized chiral edge state in Fig. 3b. Although the valley-polarization may not have been the primary goals of these papers, they nevertheless amount to prior art and calls into question the claims of novelty in the introduction, abstract, and discussion of the present manuscript (e.g., \"[...] proposing the concept of chiral valley edge states\", \"Here, we propose and demonstrate the concept of chiral valley edge states\", \"Our results establish a novel interplay between [..] quantum Hall and valley Hall phases\", \"In this work, we propose the concept of chiral valley edge states\"). If that novelty is not there, it should not be suggested that it is; if the novelty is there, it should be more clearly contextualized relative to existing literature. The authors briefly note some related work, namely [28, 29] but do so rather late and quite briefly (lines 149-151).\n2. To better elucidate the origin of there being no \"valley flipping\" in Fig. 2fgh, would the authors please discuss this aspect in the context of the edge dispersion plots for each of the three interfaces featured in Fig. 2f (i.e., CPC-1 vs. CPC-2 (in Fig. S4), CPC-2 vs. VPC-1 (in Fig. S2), and CPC-1 vs. VPC-1 (in Fig. 2d)) and discuss the phase-matching of edge states going from the CPC-1/2 vs. VPC-1 configurations into CPC-1 vs. CPC-2 configuration. We found it difficult to understand how the CPC-1 vs. CPC-2 waveguide could support propagation of valley-polarized modes, until we realized that the CPC-1 vs. CPC-2 waveguide (of course) had two edge chiral modes. We suggest that the authors more clearly explain that the robustness against valley-flipping is achieved by the CPC-1 vs. CPC-2 configuration supporting two edge modes, each of which is individually valley-polarized.\nMinor comments\n1. Fig. 1efg: the hashing is too translucent/thin to be clearly discernible in print.\n2. The insets of Fig. 3ef and the associated text in lines 196-199 attempt to summarize a k-space analysis of the out-coupling. But this summary is very terse, and it is difficult to understand in this limited form. Please either expand this into something more substantial or relegate it to the Supplementary Information.\n3. The authors state (lines 55-57) that the valley Hall phase is protected solely by crystalline symmetry. This is too generous a characterization for two reasons: (1) the valley Hall phase is only associated with a genuinely quantized topological invariant in the limit of vanishing Dirac masses. For larger Dirac masses, the valley Chern number is only approximately quantized. (2) the topology of the valley Hall phase, to the degree that it is well-defined cf. (1), does not require symmetry. Rather, it requires its absence (inversion breaking). It is true that the valley degree of freedom could be argued as requiring symmetry (i.e., a symmetry connecting the K and K’ valleys) – but this symmetry has no bearing on topology, only the presence of valleys). We suggest the authors be more precise about these nuances to reduce the chance of confusion.\n4. On lines 110-111, the authors describe valleys as \"Usually [...], valleys refer to a set of local extrema in the bands\". This seems to be unnecessarily imprecise and surely includes circumstances that would not be thought of as valleys. Valleys correspond to a set of inequivalent k-points that are related by a broken symmetry, which, if restored, is associated with a band-degeneracy at these k-points. Other definitions are possible, but the present one seems too vague and generic.\n5. Fig. 4d,h: please indicate an estimated cross-talk level, as done for the theoretical results (already in the text, but please include in figures also).\n6. Fig. 4f,j: the Fourier density resolution is very limited and the contrast very poor: currently, it is essentially only possible to distinguish near-zero values (black) from a single maximum-valued pixel (white). Please adjust the colormap to something more legible (e.g., with a minimum value not corresponding to black).\n7. In the methods section, the authors note that the Chern photonic crystals use ferrite disks with ε = 14.6, while the valley photonic crystals use ceramic disks with ε = 8.8. The sizes of the disks are roughly similar (~6 mm) and the lattice constants are identical. Given the large discrepancy in permittivity, what is the explanation for the very similar bulk band structures and mid-gap frequencies (Figs. 2ab and S1b) between the Chern and valley crystals? Relatedly, would the authors kindly also indicate the numerical values of the permeability tensor at the midgap frequency (5.5 GHz).\n8. The authors should indicate what the considered field polarization is. Presumably, it is transverse magnetic (TM) for the permeability to have any effect."}, "author_response": "Response Letter for the Manuscript\n“Chiral Valley Edge States” (NCOMMS-25-33922)\nDear Reviewers,\nWe sincerely appreciate your constructive and detailed feedback on our work. In this\nresponse letter, we will address all your questions and explain in detail how they have\nhelped us to improve our work and manuscript accordingly.\nWe thank Reviewer #1 for describing our idea that is “very sound” and “constitutes an\nimportant step”. More importantly, we thank Reviewer #1 for the very helpful\nsuggestion that have helped us to substantially improve our work and manuscript. In\nthe following, we provide detailed responses to all your insightful questions.\nWe thank Reviewer #2 for recognizing our theoretical work as “fairly well described”\nand “reasonable”. We have taken the insufficiency pointed out by Reviewer #2 that\n“The description of the fabrication and experimental techniques needs to be improved”\nand we have carefully improved our manuscript accordingly. Below, we will exhibit\nour improvement in detailed.\nWe have seriously taken the objection from Reviewers #3 and #4, against the novelty\nof our work that “the underlying idea is not very novel or, in our assessment, likely to\nbe very impactful”. This insightful comment has prompted us to rethink our idea, and\nto conduct a more systematic comparison between our work and the existing literature\non chiral valley edge states. As a result, we have found out more distinction in our work.\nWe will explain, as below, the distinction between our work and the literature in detail.\nWe hope that we can convince you of the novelty of our work. In addition, we will\nresponse detailly to all your helpful suggestions.\nIn the text below, your comments are quoted in Italics and followed by our detailed\nresponse. We have also revised the main text and the Supplementary Informations based\non\nyour comments, and these updates are highlighted in blue in the manuscript. In the\ntext below, the references to these updates are highlighted in blue and by a vertical red\nline in the left margin simultaneously\nBest regards,\nAll the authors\nResponse to Reviewer #1: ... [omitted for brevity, see full text]"}, {"round": 2, "reviewer_comments": {"reviewer_1": "My comments to the authors are attached.", "reviewer_2": "The authors have addressed my concerns. The paper paper is improved and suitable for publication.", "reviewer_4": "The authors have made extensive efforts to answer the concerns highlighted in the first review. While some of the concerns effectively cannot be alleviated (e.g., relating to likely technological impact), the work is improved in several respects (especially relating to better contextualization relative to earlier work). The title of the work is now also more fitting. Similarly, several minor issues or minor points of confusion or imprecision have been addressed.\nOverall, the work is much improved. Our previous \"Main comments 1 & 2\" are now well-addressed. The novelty of the work - and what is not novel in it - is now more accurately reflected. The question of possible technological impact remains: I maintain that the technological obstacles in lifting device-proposals such as these to relevant frequencies are very tall. Still, the work is an instructive account of an elegant idea, supported by both convincing, comprehensive experiments and by pedagogical, well-described theory and computation. On balance, my conclusion is a qualified recommendation for publication in Nature Communications, provided the two minor comments below are addressed.\n# Minor comments\nPoint 1.\nThe authors claim that past work on Floquet engineering suggests a promising path for time-reversal breaking, and a\nnumber of such past papers as \"enabl[ing] the observation of quantum Hall phases\":\n- Nature 496, 196–200 (2013): this is not a genuine Floquet system, but one using a synthetic dimension. In fact, time-reversal is not genuinely broken, but only broken if the synthetic dimension is abstracted away.\n- arXiv:2412.00619 (2024), Nat. Commun. 15, 9642 (2024), & Phys. Rev. Lett. 120, 087401 (2018): these are all strictly theoretical/computational studies. As such, they give little hint as to the feasibility of genuine Floquet engineering in photonic crystals.\nIf the authors wish to properly substantiate a Floquet-enabled \"fix\" to the time-reversal breaking question, they should cite more relevant literature. To the best of my knowledge, there is only one genuine experimental demonstration of time-modulated Floquet topology in \"real\" photonic crystals: namely https://www.nature.com/articles/s41565-025-02003-1\n(previously https://arxiv.org/abs/2304.09385). Still, even this work confirms that time-reversal breaking at high frequencies by time modulation remains extremely challenging (cf. achievable Floquet gap sizes).\nThus, the claim in the Discussion that \"Our main results can be extended to optical frequencies by leveraging Floquet topological insulators [42-45]\" is too strong. I suggest the authors tone down this claim to better reflect that this _might_ be doable.\nPoint 2.\nIn response to our earlier \"minor comment 4\", the authors have added the word \"inequivalent\" to the sentence \"[...] valleys refer to a set of inequivalent local extrema in energy bands, [...]\". As described in the previous report, this is still an unnecessarily inaccurate and loose definition of valleys. Very many local band extrema at inequivalent k-points do not correspond to valleys (must such cases, in fact)."}, "author_response": "Response Letter for the Manuscript\n“Chiral Valley Edge States based on Dirac Mass Engineering”\n(NCOMMS-25-33922A-Z)\nDear Reviewers,\nWe sincerely thank you for your valuable recognition of our work and your\nrecommendation for publication. We are also deeply grateful for the time and effort you\ninvested in reviewing our manuscript and for the constructive suggestions provided.\nPlease find our detailed, point-by-point responses below. Each of your comments is\nquoted in italics, followed by our reply. We have revised both the main text and the\nSupplementary Information in accordance with your suggestions, and these updates are\nhighlighted in blue in the manuscript. In the text below, the references to these updates\nare highlighted in blue and by a vertical red\nline in the left margin simultaneously.\nBest regards,\nAll the authors\nResponse to Reviewer #1: ... [omitted for brevity, see full text]"}]

Decision letter