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REVIEW 3 major objections 4 minor 75 references

Polariton Chern Bands in 2D Photonic Crystals Beyond Dirac Cones

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper shows that 2D photonic-crystal polaritons can form Chern bands from symmetry-protected bound states in the continuum and quadratic touching bands, not just Dirac cones, and that a practical GaP/TMD design can open a topological…

desk verdict Solid quadratic-touching platform with a credible 12 meV gap; the BIC higher-Chern scheme needs one concrete dark-exciton separation calculation. read the letter →

arxiv 2506.01240 v1 pith:4BTSTLFM submitted 2025-06-02 physics.optics cond-mat.mes-hall

Xin Xie , Kai Sun , Hui Deng This is my paper
classification physics.opticscond-mat.mes-hall
keywords polaritonCherninsulatorphotoniccrystalboundstatesinthecontinuumquadratictouchingbandstopologicalphotonicsBerrycurvaturetime-reversalsymmetrybreakingtransitionmetaldichalcogenides
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Most topological-polariton research has relied on hexagonal lattices with Dirac cones, which give sub-meV gaps. This paper proposes two alternative photonic-crystal band structures: bands with symmetry-protected bound states in the continuum (BICs) and bands with symmetry-protected quadratic touching at the Γ point. The paper argues that, when strongly coupled to excitons whose time-reversal symmetry is broken, both produce gapped polariton bands with nonzero Chern numbers, including higher Chern numbers (±2) and a ring-shaped Berry curvature distribution. It supports this with an effective spinor model, general symmetry arguments, and FDTD simulations of a gallium-phosphide/TMD platform showing a topological gap of about 12 meV—roughly two orders of magnitude larger than the previous experimental polariton Chern insulator. If correct, this would shift the design space for topological photonic devices toward common lattice geometries and Γ-point physics.

What carries the argument

The engine of the argument is the winding coupling $g_k e^{im\phi_k}$ between a photonic mode and an exciton mode, where $\phi_k$ is the azimuthal angle of the in-plane wave vector and $m$ is an integer fixed by the lattice point-group symmetry. Around a symmetry-protected BIC, the polarization vortex of the photonic band supplies this winding; around a quadratic touching point, the two rotational eigenmodes of the lattice carry a winding phase $e^{\pm 2i\phi_k}$ with $m=2$. A nonzero exciton splitting $\Delta\omega$ breaks time-reversal symmetry, and the spinor eigenstates of the two-band Hamiltonian trace a trajectory on the Bloch sphere whose winding number determines the Chern number. For the quadratic-touching platform the paper upgrades this to a 6×6 Hamiltonian with two photon modes and two spin-exciton modes, and uses it to map the topological gaps as a function of detuning, total coupling $g=\sqrt{|\alpha|^2+|\beta|^2}$, coupling imbalance $|\alpha|/|\beta|$, and Zeeman splitting $\Delta\omega$, finding that optimal gaps reach roughly the total coupling strength. This machinery also yields the ring-shaped Berry curvature distribution around Γ that the paper connects with favorable conditions for quantum Hall physics.

What would settle it

Compute the full band structure of a BIC-coupled polariton system with a realistic set of dark exciton modes included but no bright-dark separation; if the Chern numbers vanish (bands trivialize) and no energy shift of the bright exciton reopens the topological gap without also destroying the winding coupling, the BIC platform is refuted.

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Extended reading notes

Core claim

The central claim is that polariton Chern bands are not tied to the Dirac-cone framework. The authors show that two symmetry-protected photonic bands commonly found in 2D photonic crystals—a non-degenerate band with a bound state in the continuum surrounded by a polarization vortex, and a pair of bands with a quadratic touching degeneracy at Γ—can each be turned into a Chern insulator by strong coupling to a single bright exciton or to spin-split exciton states. In both cases the winding coupling $g_k e^{im\phi_k}$ between photon and exciton modes, together with a finite time-reversal-symmetry-breaking splitting $\Delta\omega$, makes the pseudospin eigenstates trace a loop on the Bloch sphere; the loop is full for the BIC case and half for the quadratic-touching case, yielding Chern numbers $\pm m$ and $\pm m/2$, respectively. The paper validates the picture with a 6×6 effective Hamiltonian for the quadratic-touching platform, showing three topological gaps whose sizes depend on detuning, coupling strength, and splitting, and with FDTD simulations of a triangular-lattice GaP photonic crystal coupled to a TMD monolayer. The simulated device exhibits six polariton Chern bands with Chern numbers $\pm1$ and a middle topological gap near 12 meV, plus chiral edge states consistent with bulk–edge correspondence.

Load-bearing premise

The load-bearing premise is that, in the BIC scheme, the bright exciton can be energetically separated from the many dark exciton modes without closing the topological gap or destroying the winding coupling; the paper states this separation can be done (e.g., by an optical Stark shift) but provides no simulation showing it works.

Editorial extensions

If this is right

  • Polariton Chern insulators can be designed at the Γ point in lattices with C3, C4, and C6 symmetry, greatly expanding the material and geometry choices beyond hexagonal Dirac-cone lattices.
  • Higher Chern numbers (up to ±2 in the shown examples) become accessible through high-order BICs, which can support multiple chiral edge channels.
  • The demonstrated ~12 meV middle gap in the GaP/TMD design is roughly two orders of magnitude larger than the 0.1 meV gap of the previous micropillar experiment, opening the way to robust topological devices at elevated temperatures.
  • The quasi-uniform ring-shaped Berry curvature around Γ may facilitate observation of quantum Hall effects and is argued in the paper to help stabilize fractional excitations.
  • Chiral edge states appear in each topological gap, with opposite chirality in the middle gap compared with the upper and lower gaps, exactly as bulk–edge correspondence demands.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A concrete next step the paper leaves open is to simulate the mode-selective optical Stark shift on the bright exciton while explicitly including all dark exciton modes; such a calculation would confirm whether the reconstructed Chern bands survive in the full exciton manifold.
  • Because the Berry curvature forms a broad ring around Γ rather than a sharp Dirac peak, these polariton bands may be favorable for observing the anomalous Hall drift of polariton wavepackets over a wide momentum range.
  • Applying the same design rules to moiré excitons, whose valley splittings can exceed 40 meV, could push the achievable topological gap in the quadratic-touching platform beyond the 12 meV reported here.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes two general mechanisms for realizing polariton Chern bands in 2D photonic crystals (PhCs) beyond the usual Dirac-cone framework: (i) bands with symmetry-protected bound states in the continuum (BICs) whose surrounding polarization vortex provides a winding coupling to excitons, and (ii) symmetry-protected quadratic touching bands at the Brillouin-zone center. For each mechanism the authors derive effective Hamiltonians, compute Chern numbers (|C|=1 and |C|=2), analyze the dependence of the topological gap on detuning, coupling, and time-reversal breaking, and propose a concrete GaP/TMD design. The quadratic-touching platform is validated by FDTD band-structure calculations, a 12 meV middle gap, and chiral edge-state simulations, whereas the BIC platform is illustrated with model band structures.

Significance. The work is significant because it broadens the design space for polariton Chern insulators, which have so far been limited to Dirac-cone-based hexagonal lattices with sub-meV gaps. The quadratic-touching platform with a predicted ~12 meV gap, supported by FDTD and edge-state simulations, is a concrete, experimentally accessible step toward topological polaritonics at optical frequencies. The BIC-based proposal, if realized, would enable higher Chern numbers (|C|=2) with ring-shaped Berry curvature. The effective-Hamiltonian analysis is standard and the parameters are drawn from realistic ranges or extracted from FDTD spectra, and the 12 meV gap is an output rather than a fitted target, which strengthens the credibility of the quadratic-touching results. However, the BIC platform rests on an unquantified bright–dark exciton separation step that is not yet demonstrated, and the full-Brillouin-zone gap of the practical design is not completely mapped.

major comments (3)
  1. [Sec. III, Fig. 2(b)-(c)] The central claim for the BIC-based platform, including the higher-Chern (|C|=2) example, depends on the assertion that 'Chern bands can be reconstructed by energetically separating bright excitons from dark ones' via the mode-selective optical Stark effect. No simulation, model, or quantitative estimate is provided for the achievable Stark shift relative to the exciton-photon coupling, the dark-exciton bandwidth, or the required energy separation. Because the topological gap must open near the crossing of the photon and exciton bands, where a dense manifold of dark exciton modes resides within the PhC Brillouin zone, the residual coupling to dark excitons could close the topological gap or open trivial gaps. A concrete multi-mode calculation including the dark-exciton continuum is required to support the claim that the BIC platform produces the advertised Chern bands.
  2. [Sec. VI, Fig. 5(c)] The paper claims a 'large topological gap across the BZ' (Introduction and Sec. VI), but the FDTD band structure and the middle-gap size are shown only along the Γ–K–M path. The global gap in the full 2D Brillouin zone is not mapped, leaving open the possibility that bands approach elsewhere in the BZ and reduce the global topological gap. The authors should either compute the gap over the full BZ or provide a symmetry-based argument, together with checks at all high-symmetry points and along the BZ boundary, to establish the global gap.
  3. [Sec. IV, paragraph on dark excitons] For the quadratic-touching platform, the statement that 'dark excitons in this system are inside the trivial gaps and would not affect the topological gaps or edge states' is made without quantitative support. The actual energy range of dark-exciton modes relative to the topological gaps is not estimated for the GaP/TMD parameters. If dark excitons overlap in energy with the topological gaps, they could hybridize with the polariton modes and alter the gap and edge states. A quantitative estimate or a calculation including a dark-exciton continuum would substantiate this assumption.
minor comments (4)
  1. [Sec. IV] There is a typographical error: 'quratic dispersions' should be 'quadratic dispersions' in the paragraph on quadratic touching bands.
  2. [Sec. II, Eq. (1)] The basis of the two-component Hamiltonian in Eq. (1) is not explicitly defined. Clarifying which states (e.g., the two circular polarizations of the exciton or photon and exciton) form the spinor would improve readability.
  3. [Sec. III, Eq. (5)] The relationship between the general Hamiltonian in Eq. (1) and the BIC coupling Hamiltonian in Eq. (5) is not spelled out, in particular how Δω, g_k, and Φ_k map to ω_c, ω_x, and the winding coupling. A short sentence or a parameter mapping would help the reader.
  4. [Sec. VI, Fig. 5(b)] The extraction of ħg = 22 meV and |α|/|β| = 1.73 from the FDTD polariton splittings is described briefly; providing the fitting procedure or the underlying spectra would make the parameter extraction more reproducible.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation chain; Chern numbers and the 12 meV gap are genuine outputs of a self-contained model, though the BIC bright-dark exciton separation is asserted without simulation.

full rationale

The central derivation is self-contained. In Sec. II, the Hamiltonian H_k in Eq. (1) with winding coupling e^{imφ} is solved explicitly for eigenstates (2)-(3), and the Chern numbers m or m/2 follow from the spinor covering of the Bloch sphere; this is a direct model calculation, not a restatement of an input. The BIC case (Eq. (5)) and quadratic-touching case (Eq. (6)) similarly produce band structures and Berry curvatures (Figs. 2-3) from stated symmetry properties, with the winding coupling cited to external literature on polarization vortices and quadratic degeneracies [43-45,63-65], not to a self-citation. Section VI's practical design is also non-circular: ħg=22 meV and |α|/|β|=1.73 are extracted from FDTD polariton splittings, Δω=42.5 meV is chosen, and the ~12 meV middle gap is then computed and confirmed by FDTD bandstructure and edge modes; the gap is not used as a fitting target. The paper does contain self-citations, notably refs. [68], [70], and [71], but they support general experimental capabilities (TMD-PhC integration, valley-selective optical Stark effect) rather than the topological claim, and equivalent external references ([69]) are also cited. The one genuine weakness is an omitted proof, not circularity: Sec. III admits that dark excitons trivialize the BIC polariton bands and asserts that 'Chern bands can be reconstructed by energetically separating bright excitons from dark ones' via the optical Stark effect, but provides no simulation or quantitative estimate. That gap undercuts the BIC platform's practicality, especially the |C|=2 example, but it is not a step in which a prediction reduces to its own input. Hence no significant circularity; score 2 for minor non-load-bearing self-citations and the explicit feasibility caveat.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard symmetry-based winding models and on several chosen physical parameters. No new entities are introduced.

free parameters (4)
  • Zeeman splitting Δω = 40 meV (model), 42.5 meV (FDTD)
    Chosen to maximize the topological gap; physical TR-breaking knob, not derived from first principles.
  • Total coupling strength ħg = 30 meV (model), 22 meV (FDTD)
    Chosen in model; extracted from FDTD polariton splittings.
  • Coupling ratio |α|/|β| = 1.56 (model), 1.73 (FDTD)
    Set to optimize gap in model; extracted from FDTD.
  • Exciton-photon detuning δ = 0 (optimal)
    Set to zero to maximize hybridization; a design condition.
assumptions (4)
  • domain assumption A symmetry-protected BIC at Γ is surrounded by a polarization vortex with a conserved topological charge m.
    Used in Sec. III to set winding coupling Φ = mφ; taken from prior BIC literature (refs 43-45).
  • domain assumption Quadratic-touching bands in 2D PhCs have a winding coupling of the form ν k^2 e^{-i2φ} with winding number m=2.
    Used in Sec. IV to construct the six-band Hamiltonian; from refs 63-65.
  • domain assumption The six-band effective Hamiltonian (Eq. 6) fully captures the coupled PhC-exciton system, including the two photonic modes and two spin exciton branches.
    Central model of Sec. IV; validated at Γ by FDTD in Sec. VI but the winding form away from Γ is assumed.
  • domain assumption TMD excitons can be represented as Lorentz holes in the permittivity tensor in FDTD simulations.
    Used in Sec. VI practical realization; based on ref 75.

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Cite this review

Pith. "Pith review of Polariton Chern Bands in 2D Photonic Crystals Beyond Dirac Cones." pith.science (2026). https://pith.science/paper/4BTSTLFM

@misc{pith2026250601240,
  author       = {Pith},
  title        = {Pith review of: Polariton Chern Bands in 2D Photonic Crystals Beyond Dirac Cones},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4BTSTLFM}},
  note         = {Machine review of arXiv:2506.01240}
}
abstract

Polaritons, formed by strong light-matter interactions, open new avenues for studying topological phases, where the spatial and time symmetries can be controlled via the light and matter components, respectively. However, most research on topological polaritons has been confined to hexagonal photonic lattices featuring Dirac cones at large wavenumbers. This restricts key topological properties and device performance, including sub-meV gap sizes that hinder further experimental investigations and future applications of polariton Chern insulator systems. In this study, we move beyond the traditional Dirac cone framework and introduce two alternative band structures in photonic crystals (PhCs) as promising platforms for realizing polariton Chern bands: bands with symmetry-protected bound states in the continuum (BICs) and bands with symmetry-protected degeneracies at the $\Gamma$ points. These band structures are prevalent in various PhC lattices and have features crucial for experimental studies. We show examples of higher Chern number bands, more uniform Berry curvature distributions, and an experimentally feasible system capable of achieving a large topological gap. Our findings show the broad applicability of polariton Chern bands in 2D PhCs, provide design principles for enhancing the functionality and performance of topological photonic devices, opening up exciting possibilities for better understanding and using topological physics.

Figures

Figures reproduced from arXiv: 2506.01240 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Spinor character of eigenstates parameterized [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Topological polariton formed by coupling the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) Polariton band structure with TR symmetry, [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a-c) Eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Numerical simulation of band structures of a 2D [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Schematic of 1D interfaces with [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

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Works this paper leans on

75 extracted references · 46 canonical work pages

  1. [1]

    L. Lu, J. D. Joannopoulos, and M. Soljaˇ ci´ c, Topological photonics, Nat. Photonics8, 821 (2014)

    2014

  2. [2]

    A. B. Khanikaev and G. Shvets, Two-dimensional topo- logical photonics, Nat. Photonics11, 763 (2017)

    work page 2017

  3. [3]

    Ozawa, H

    T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zil- berberg, and I. Carusotto, Topological photonics, Rev. Mod. Phys.91, 015006 (2019)

    2019

  4. [4]

    Smirnova, D

    D. Smirnova, D. Leykam, Y. Chong, and Y. Kivshar, Nonlinear topological photonics, Appl. Phys. Rev.7, 021306 (2020)

    work page 2020

  5. [5]

    M. Kim, Z. Jacob, and J. Rho, Recent advances in 2D, 3D and higher-order topological photonics, Light Sci. Appl. 9, 130 (2020)

    work page 2020

  6. [6]

    R. O. Umucalılar and I. Carusotto, Artificial gauge field for photons in coupled cavity arrays, Phys. Rev. A84, 043804 (2011)

    work page 2011

  7. [7]

    Hafezi, E

    M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, Robust optical delay lines with topological protection, Nat. Phys.7, 907 (2011)

    work page 2011

  8. [8]

    Hafezi, S

    M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, Imaging topological edge states in silicon photonics, Nat. Photonics7, 1001 (2013)

    work page 2013

Show all 75 references
  1. [9]

    Wu and X

    L.-H. Wu and X. Hu, Scheme for Achieving a Topologi- cal Photonic Crystal by Using Dielectric Material, Phys. Rev. Lett.114, 223901 (2015)

    2015

  2. [10]

    Dong, X.-D

    J.-W. Dong, X.-D. Chen, H. Zhu, Y. Wang, and X. Zhang, Valley photonic crystals for control of spin and topology, Nat. Mater.16, 298 (2017)

    2017

  3. [11]

    M. I. Shalaev, W. Walasik, A. Tsukernik, Y. Xu, and N. M. Litchinitser, Robust topologically protected trans- port in photonic crystals at telecommunication wave- lengths, Nat. Nanotechnol.14, 31 (2019)

    2019

  4. [12]

    W. Liu, Z. Ji, Y. Wang, G. Modi, M. Hwang, B. Zheng, V. J. Sorger, A. Pan, and R. Agarwal, Generation of helical topological exciton-polaritons, Science370, 600 (2020)

    2020

  5. [13]

    M. Li, I. Sinev, F. Benimetskiy, T. Ivanova, E. Khes- tanova, S. Kiriushechkina, A. Vakulenko, S. Guddala, M. Skolnick, V. M. Menon, D. Krizhanovskii, A. Al` u, A. Samusev, and A. B. Khanikaev, Experimental ob- servation of topological Z2 exciton-polaritons in transi- tion met...

    2021

  6. [14]

    Arregui, J

    G. Arregui, J. Gomis-Bresco, C. M. Sotomayor-Torres, and P. D. Garcia, Quantifying the Robustness of Topo- logical Slow Light, Phys. Rev. Lett.126, 027403 (2021)

    2021

  7. [15]

    C. A. Rosiek, G. Arregui, A. Vladimirova, M. Albrecht- sen, B. Vosoughi Lahijani, R. E. Christiansen, and S. Sto- bbe, Observation of strong backscattering in valley-Hall photonic topological interface modes, Nat. Photonics17, 386 (2023)

    2023

  8. [16]

    A. B. Khanikaev and A. Al` u, Topological photonics: Ro- bustness and beyond, Nat. Commun.15, 931 (2024)

    2024

  9. [17]

    F. D. M. Haldane and S. Raghu, Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry, Phys. Rev. Lett. 100, 013904 (2008)

    2008

  10. [18]

    Raghu and F

    S. Raghu and F. D. M. Haldane, Analogs of quantum- Hall-effect edge states in photonic crystals, Phys. Rev. A 78, 033834 (2008)

    2008

  11. [19]

    Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljaˇ ci´ c, Observation of unidirectional backscattering-immune topological electromagnetic states, Nature461, 772 (2009)

    2009

  12. [20]

    M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Sza- meit, Photonic Floquet topological insulators, Nature 496, 196 (2013)

    2013

  13. [21]

    Lustig, S

    E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, Photonic topolog- ical insulator in synthetic dimensions, Nature567, 356 (2019)

    2019

  14. [22]

    N. H. Lindner, G. Refael, and V. Galitski, Floquet topo- logical insulator in semiconductor quantum wells, Nat. Phys.7, 490 (2011)

    2011

  15. [23]

    K. Fang, Z. Yu, and S. Fan, Realizing effective magnetic field for photons by controlling the phase of dynamic modulation, Nat. Photonics6, 782 (2012)

    2012

  16. [24]

    L. J. Maczewsky, J. M. Zeuner, S. Nolte, and A. Szameit, Observation of photonic anomalous Floquet topological insulators, Nat. Commun.8, 13756 (2017)

    2017

  17. [25]

    H. Zhao, X. Qiao, T. Wu, B. Midya, S. Longhi, and L. Feng, Non-Hermitian topological light steering, Sci- ence365, 1163 (2019)

    2019

  18. [26]

    L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, and P. Xue, Non-Hermitian bulk–boundary correspondence in quantum dynamics, Nat. Phys.16, 761 (2020)

    2020

  19. [27]

    T. Dai, Y. Ao, J. Mao, Y. Yang, Y. Zheng, C. Zhai, Y. Li, J. Yuan, B. Tang, Z. Li, J. Luo, W. Wang, X. Hu, Q. Gong, and J. Wang, Non-Hermitian topological phase transitions controlled by nonlinearity, Nat. Phys.20, 101 (2024)

    2024

  20. [28]

    Weisbuch, M

    C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, Observation of the coupled exciton-photon mode split- ting in a semiconductor quantum microcavity, Phys. Rev. Lett.69, 3314 (1992)

    1992

  21. [29]

    Karzig, C.-E

    T. Karzig, C.-E. Bardyn, N. H. Lindner, and G. Refael, Topological Polaritons, Phys. Rev. X5, 031001 (2015)

    2015

  22. [30]

    Gianfrate, O

    A. Gianfrate, O. Bleu, L. Dominici, V. Ardizzone, M. De Giorgi, D. Ballarini, G. Lerario, K. W. West, L. N. Pfeiffer, D. D. Solnyshkov, D. Sanvitto, and G. Malpuech, Measurement of the quantum geometric tensor and of the anomalous Hall drift, Nature578, 381 (2020)

    2020

  23. [31]

    Polimeno, G

    L. Polimeno, G. Lerario, M. De Giorgi, L. De Marco, L. Dominici, F. Todisco, A. Coriolano, V. Ardizzone, M. Pugliese, C. T. Prontera, V. Maiorano, A. Moliterni, C. Giannini, V. Olieric, G. Gigli, D. Ballarini, Q. Xiong, A. Fieramosca, D. D. Solnyshkov, G. Malpuech, and D. Sanv...

    2021

  24. [32]

    A. V. Nalitov, D. D. Solnyshkov, and G. Malpuech, Po- laritonZTopological Insulator, Phys. Rev. Lett.114, 116401 (2015)

    2015

  25. [33]

    Bardyn, T

    C.-E. Bardyn, T. Karzig, G. Refael, and T. C. H. Liew, Topological polaritons and excitons in garden-variety sys- tems, Phys. Rev. B91, 161413 (2015)

    2015

  26. [34]

    Yi and T

    K. Yi and T. Karzig, Topological polaritons from pho- tonic Dirac cones coupled to excitons in a magnetic field, Phys. Rev. B93, 104303 (2016)

    2016

  27. [35]

    L. He, J. Wu, J. Jin, E. J. Mele, and B. Zhen, Po- laritonic Chern Insulators in Monolayer Semiconductors, Phys. Rev. Lett.130, 043801 (2023)

    2023

  28. [36]

    Klembt, T

    S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. H¨ ofling, Exciton-polariton topological insulator, Nature562, 552 (2018)

    2018

  29. [37]

    von Neumann and E

    J. von Neumann and E. P. Wigner, ¨Uber merkw¨ urdige diskrete Eigenwerte, inThe Collected Works of Eu- gene Paul Wigner: Part A, edited by A. S. Wightman (Springer, Berlin, Heidelberg, 1993) pp. 291–293

    1993

  30. [38]

    C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljaˇ ci´ c, Bound states in the continuum, Nat. Rev. Mater.1, 1 (2016)

    2016

  31. [39]

    M. Kang, T. Liu, C. T. Chan, and M. Xiao, Applications of bound states in the continuum in photonics, Nat. Rev. Phys.5, 659 (2023)

    2023

  32. [40]

    C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. John- son, J. D. Joannopoulos, and M. Soljaˇ ci´ c, Observation of trapped light within the radiation continuum, Nature 499, 188 (2013)

    2013

  33. [41]

    J. Jin, X. Yin, L. Ni, M. Soljaˇ ci´ c, B. Zhen, and C. Peng, Topologically enabled ultrahigh-Q guided resonances ro- bust to out-of-plane scattering, Nature574, 501 (2019)

    2019

  34. [42]

    M. Kang, L. Mao, S. Zhang, M. Xiao, H. Xu, and C. T. Chan, Merging bound states in the continuum by har- nessing higher-order topological charges, Light Sci. Appl. 11, 228 (2022)

    2022

  35. [43]

    B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljaˇ ci´ c, Topological Nature of Optical Bound States in the Con- tinuum, Phys. Rev. Lett.113, 257401 (2014)

    2014

  36. [44]

    Zhang, A

    Y. Zhang, A. Chen, W. Liu, C. W. Hsu, B. Wang, F. Guan, X. Liu, L. Shi, L. Lu, and J. Zi, Observation of Polarization Vortices in Momentum Space, Phys. Rev. Lett.120, 186103 (2018)

    2018

  37. [45]

    Yoda and M

    T. Yoda and M. Notomi, Generation and Annihilation of Topologically Protected Bound States in the Continuum and Circularly Polarized States by Symmetry Breaking, Phys. Rev. Lett.125, 053902 (2020)

    2020

  38. [46]

    B. Wang, W. Liu, M. Zhao, J. Wang, Y. Zhang, A. Chen, F. Guan, X. Liu, L. Shi, and J. Zi, Generating optical vortex beams by momentum-space polarization vortices centred at bound states in the continuum, Nat. Photonics 14, 623 (2020)

    2020

  39. [47]

    Huang, C

    C. Huang, C. Zhang, S. Xiao, Y. Wang, Y. Fan, Y. Liu, N. Zhang, G. Qu, H. Ji, J. Han, L. Ge, Y. Kivshar, and Q. Song, Ultrafast control of vortex microlasers, Science 367, 1018 (2020)

    2020

  40. [48]

    Ardizzone, F

    V. Ardizzone, F. Riminucci, S. Zanotti, A. Gianfrate, M. Efthymiou-Tsironi, D. G. Su` arez-Forero, F. Todisco, 9 M. De Giorgi, D. Trypogeorgos, G. Gigli, K. Baldwin, L. Pfeiffer, D. Ballarini, H. S. Nguyen, D. Gerace, and D. Sanvitto, Polariton Bose–Einstein condensate from a ...

    2022

  41. [49]

    X. Wu, S. Zhang, J. Song, X. Deng, W. Du, X. Zeng, Y. Zhang, Z. Zhang, Y. Chen, Y. Wang, C. Jiang, Y. Zhong, B. Wu, Z. Zhu, Y. Liang, Q. Zhang, Q. Xiong, and X. Liu, Exciton polariton condensation from bound states in the continuum at room temperature, Nat. Com- mun.15, 3345 (2024)

    2024

  42. [50]

    W. Liu, B. Wang, Y. Zhang, J. Wang, M. Zhao, F. Guan, X. Liu, L. Shi, and J. Zi, Circularly Polarized States Spawning from Bound States in the Continuum, Phys. Rev. Lett.123, 116104 (2019)

    2019

  43. [51]

    Y. Chen, H. Deng, X. Sha, W. Chen, R. Wang, Y.-H. Chen, D. Wu, J. Chu, Y. S. Kivshar, S. Xiao, and C.-W. Qiu, Observation of intrinsic chiral bound states in the continuum, Nature613, 474 (2023)

    2023

  44. [52]

    Kodigala, T

    A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kant´ e, Lasing action from photonic bound states in continuum, Nature541, 196 (2017)

    2017

  45. [53]

    Hwang, H.-C

    M.-S. Hwang, H.-C. Lee, K.-H. Kim, K.-Y. Jeong, S.- H. Kwon, K. Koshelev, Y. Kivshar, and H.-G. Park, Ultralow-threshold laser using super-bound states in the continuum, Nat. Commun.12, 4135 (2021)

    2021

  46. [54]

    Y. Yu, A. Sakanas, A. R. Zali, E. Semenova, K. Yvind, and J. Mørk, Ultra-coherent Fano laser based on a bound state in the continuum, Nat. Photonics15, 758 (2021)

    2021

  47. [55]

    Y. Ren, P. Li, Z. Liu, Z. Chen, Y.-L. Chen, C. Peng, and J. Liu, Low-threshold nanolasers based on miniaturized bound states in the continuum, Sci. Adv.8, eade8817 (2022)

    2022

  48. [56]

    Z. Liu, Y. Xu, Y. Lin, J. Xiang, T. Feng, Q. Cao, J. Li, S. Lan, and J. Liu, High-Q Quasibound States in the Continuum for Nonlinear Metasurfaces, Phys. Rev. Lett. 123, 253901 (2019)

    2019

  49. [57]

    Koshelev, S

    K. Koshelev, S. Kruk, E. Melik-Gaykazyan, J.-H. Choi, A. Bogdanov, H.-G. Park, and Y. Kivshar, Subwave- length dielectric resonators for nonlinear nanophotonics, Science367, 288 (2020)

    2020

  50. [58]

    Schiattarella, S

    C. Schiattarella, S. Romano, L. Sirleto, V. Mocella, I. Rendina, V. Lanzio, F. Riminucci, A. Schwartzberg, S. Cabrini, J. Chen, L. Liang, X. Liu, and G. Zito, Di- rective giant upconversion by supercritical bound states in the continuum, Nature626, 765 (2024)

    2024

  51. [59]

    Z. Liu, T. Guo, Q. Tan, Z. Hu, Y. Sun, H. Fan, Z. Zhang, Y. Jin, and S. He, Phase Interrogation Sensor Based on All-Dielectric BIC Metasurface, Nano Lett.23, 10441 (2023)

    2023

  52. [60]

    S. A. Skirlo, L. Lu, and M. Soljaˇ ci´ c, Multimode One-Way Waveguides of Large Chern Numbers, Phys. Rev. Lett. 113, 113904 (2014)

    2014

  53. [61]

    L. Lu, J. D. Joannopoulos, and M. Soljaˇ ci´ c, Topological states in photonic systems, Nat. Phys.12, 626 (2016)

    2016

  54. [62]

    Y. Wu, C. Li, X. Hu, Y. Ao, Y. Zhao, and Q. Gong, Applications of Topological Photonics in Integrated Pho- tonic Devices, Adv. Opt. Mater.5, 1700357 (2017)

    2017

  55. [63]

    Y. D. Chong, X.-G. Wen, and M. Soljaˇ ci´ c, Effective the- ory of quadratic degeneracies, Phys. Rev. B77, 235125 (2008)

    2008

  56. [64]

    J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, First- principles study of Dirac and Dirac-like cones in phononic and photonic crystals, Phys. Rev. B86, 035141 (2012)

    2012

  57. [65]

    Z. Gao, H. Zhao, T. Wu, X. Feng, Z. Zhang, X. Qiao, C.-K. Chiu, and L. Feng, Topological quadratic-node semimetal in a photonic microring lattice, Nat Commun. 14, 3206 (2023)

    2023

  58. [66]

    Ochiai and M

    T. Ochiai and M. Onoda, Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states, Phys. Rev. B80, 155103 (2009)

    2009

  59. [67]

    M. J. Collins, F. Zhang, R. Bojko, L. Chrostowski, and M. C. Rechtsman, Integrated optical Dirac physics via inversion symmetry breaking, Phys. Rev. A94, 063827 (2016)

    2016

  60. [68]

    Zhang, R

    L. Zhang, R. Gogna, W. Burg, E. Tutuc, and H. Deng, Photonic-crystal exciton-polaritons in monolayer semi- conductors, Nat. Commun.9, 713 (2018)

    2018

  61. [69]

    P. D. Cunningham, A. T. Hanbicki, T. L. Reinecke, K. M. McCreary, and B. T. Jonker, Resonant optical Stark ef- fect in monolayer WS 2, Nat. Commun.10, 5539 (2019)

    2019

  62. [70]

    C.-K. Yong, M. I. B. Utama, C. S. Ong, T. Cao, E. C. Regan, J. Horng, Y. Shen, H. Cai, K. Watan- abe, T. Taniguchi, S. Tongay, H. Deng, A. Zettl, S. G. Louie, and F. Wang, Valley-dependent exciton fine struc- ture and Autler–Townes doublets from Berry phases in monolayer MoSe2...

    2019

  63. [71]

    L. Zhou, B. Liu, Y. Liu, Y. Lu, Q. Li, X. Xie, N. Lydick, R. Hao, C. Liu, K. Watanabe, T. Taniguchi, Y.-H. Chou, S. R. Forrest, and H. Deng, Cavity Floquet engineering, Nat. Commun.15, 7782 (2024)

    2024

  64. [72]

    K. F. Mak and J. Shan, Semiconductor moir´ e materials, Nat. Nanotechnol.17, 686 (2022)

    2022

  65. [73]

    Chang, C.-X

    C.-Z. Chang, C.-X. Liu, and A. H. MacDonald, Collo- quium: Quantum anomalous Hall effect, Rev. Mod. Phys. 95, 011002 (2023)

    2023

  66. [74]

    W. Zhao, K. Kang, Y. Zhang, P. Kn¨ uppel, Z. Tao, L. Li, C. L. Tschirhart, E. Redekop, K. Watanabe, T. Taniguchi, A. F. Young, J. Shan, and K. F. Mak, Realization of the Haldane Chern insulator in a moir´ e lattice, Nat. Phys.20, 275 (2024)

    2024

  67. [75]

    Y. Li, A. Chernikov, X. Zhang, A. Rigosi, H. M. Hill, A. M. van der Zande, D. A. Chenet, E.-M. Shih, J. Hone, and T. F. Heinz, Measurement of the optical dielectric function of monolayer transition-metal dichalcogenides: MoS2, MoSe2, WS2, and WSe2, Phys. Rev. B90, 205422 (2014)

    2014

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