A Group-Theoretical Framework for Local k-Space Topology and Berry Phase in 2D Photonic Systems

Grazia Salerno; Xinyi Yuan

REVIEW 4 major objections 4 minor 61 references

A Group-Theoretical Framework for Local k-Space Topology and Berry Phase in 2D Photonic Systems

T0 review · 4 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper argues that in C6v-symmetric 2D photonic crystals, only E1 in-plane modes couple to the far field; A2, B2, and E2 are symmetry-protected bound states in the continuum and E1 modes generate optical vortex beams.

desk verdict A useful but uneven framework: the core C6v BIC classification is known, and the genuinely new parts—Table I, transversality-adapted PWE basis, and the mirror-breaking leff analysis—are good enough to referee, but the central dark/bright claim needs a diffraction-order caveat and the quantitative extensions are under-derived. read the letter →

arxiv 2607.29356 v1 pith:WZCHSEJD submitted 2026-07-31 physics.optics cond-mat.mes-hall

Xinyi Yuan , Grazia Salerno This is my paper

classification physics.opticscond-mat.mes-hall

keywords photoniccrystalsboundstatesinthecontinuumopticalvortexbeamsBerryphaseirreduciblerepresentationspoint-groupsymmetryC6vChernnumbers

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

The paper adapts the irreducible-representation (IR) machinery of solid-state physics to photonic Bloch modes, adding the transversality condition and radiative coupling that Maxwell's equations impose. Its central result is a selection rule at the Γ point of a triangular lattice with C6v symmetry: the far field transforms as E1, so only E1 in-plane modes can radiate; A2, B2, and E2 modes are symmetry-protected bound states in the continuum, while E1 modes act as sources of optical vortex beams. The same decomposition is tabulated for other point groups, giving a dark/bright inventory for any high-symmetry point above the light cone. For Berry phase, the paper shows that a four-band tight-binding model on the same lattice block-diagonalizes as A1⊕A1⊕E2, that tuning the A1 coefficient reverses inter-band connectivity and redistributes Berry curvature, and that adding a time-reversal-breaking σz term produces Chern numbers 0,±1,0. A sympathetic reader would care because this yields a symmetry-only, parameter-free route to predicting dark and bright modes and to engineering topological phases in photonic crystals.

What carries the argument

The load-bearing object is the irreducible-representation (IR) decomposition of Bloch modes at high-symmetry k-points, implemented through projection operators onto symmetry-adapted bases. For the local-k-space part, the key is a first-shell plane-wave basis with the transverse condition enforced (ϕn = t_n e^{i G_n·r}, t_n = ẑ×Ĝ_n), which yields the reducible representation A2⊕B2⊕E1⊕E2 for C6v; the far-field vector (E_x,E_y) transforms as E1. Schur's lemma / the mode-hybridizing condition Γ_mode⊗Γ_rad ⊃ A1 then produces the dark/bright assignment. For the Berry-phase part, the machinery is the block-diagonalization of the four-band tight-binding Hamiltonian as A1⊕A1⊕E2 at Γ, with real spati

What would settle it

Compute the Γ-point eigenmodes of the full Maxwell operator for the C6v structure (e.g., via a converged plane-wave or finite-element solver) and project the radiative linewidths onto the IR sectors: if any A2, B2, or E2 mode acquires a nonzero imaginary frequency at Γ, the selection rule is false. Alternatively, measure the far-field emission from a fabricated C6v photonic-crystal slab; observation of radiation from a mode classified as dark would also refute it.

Watch

Extended reading notes

Core claim

The paper's central claim is a symmetry-based selection rule for open photonic crystals. At the Γ point of a triangular lattice with C6v symmetry, the first shell of reciprocal lattice vectors, after imposing the transverse condition G·E=0, carries the reducible representation A2⊕B2⊕E1⊕E2. The far-field radiation continuum transforms as E1. By the standard direct-product criterion (only modes whose IR appears in the reduction of Γ_in-plane ⊗ Γ_rad can couple), the paper concludes that A2, B2, and E2 modes are symmetry-protected BICs with vanishing radiative linewidth, while E1 modes are the bright modes that source optical vortex beams. The same decomposition, tabulated for other point group

Load-bearing premise

The truncated minimal models (six plane waves for the TE slab, four bands for the tight-binding lattice) are assumed to be homeomorphic to the full photonic problem, preserving the symmetry classification and topological content; if higher diffraction orders or longer-range hoppings mix symmetry sectors or shift degeneracies, the predicted dark/bright inventory and Berry phases need not hold.

Editorial extensions

If this is right

In C6v-symmetric 2D photonic crystals, the symmetry classification alone identifies A2, B2, and E2 modes as symmetry-protected BICs with zero radiative coupling at Γ, and E1 modes as vortex-beam emitters.
Mirror-symmetry breaking (C6v → C6) turns the dark B2 mode into a quasi-BIC whose linewidth can be tuned by spatial perturbations, and de-quantizes the effective angular momentum of the E1 doublet as the mirror-breaking angle θ grows.
Targeted real (spatial) perturbations shift individual IR bands without inter-block mixing in C6v, but in C6 they hybridize symmetry blocks, enabling controlled leakage engineering.
For the four-band triangular lattice, tuning the A1 IR coefficient reverses the band-connectivity pattern and inverts the sign of Berry curvature at the K/K′ points; adding a time-reversal-breaking coupling then yields Chern numbers 0, ±1, 0 in the four bands.
The same IR-based classification extends to other point groups (C4v, C4, C2v, C3v, C2, C3), giving a general dark/bright inventory for high-symmetry points above the light cone.

Reading between the lines

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

A natural test that the paper does not run: compute the Γ-point eigenmodes of the full Maxwell operator (not the truncated basis) and project them onto the IR sectors; the selection rule predicts exactly zero linewidth for A2/B2/E2 in the exact theory, which would be a sharper statement than the truncated-model result.
The same shell-based decomposition should apply to higher diffraction orders whenever shells remain energetically separated; the paper's own appendix notes the classification breaks down when inter-shell coupling becomes comparable to shell spacing, so the framework implicitly predicts a crossover from symmetry-protected to quasi-dark modes at high frequencies.
Because the selection rule is purely group-theoretic, it should transfer to other classical wave systems (acoustic, mechanical, or plasmonic) whose fields have vector or scalar character and obey a transversality-like constraint; the paper's 'n-band' generalization hints at this but does not demonstrate it.
For the Berry-phase part, the inversion of Berry curvature at K/K′ under A1 tuning implies that valley-contrasting transport in multi-band photonic crystals can be engineered without breaking time-reversal symmetry, only by spatial distortion — an untested route to valley routing in flat optics.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The honest summary: the C6v Γ-point result (A2⊕B2⊕E1⊕E2; only E1 radiates) is not new, and the authors say so, citing Refs [18,38,40–42]. What is new and worth taking seriously is the transversality-adapted PWE basis that puts transversality into the group action (Eq. 13), the first-shell table for other point groups (Table I, including off-Γ groups), and the C6 mirror-breaking analysis leading to dequantized leff. The Schur-lemma formulation in Sec. IV.B is clean and standard, and the IR decompositions check out. The four-band tight-binding model with COMSOL band structures in Fig. 8 is a nice concrete anchor. I don't see circularity: the classification follows from characters, not from the BIC/vortex phenomenology it explains.

The soft spots are real but fixable. First, the dark/bright conclusion in Sec. IV.B is stated without the condition that only the G=0 diffraction order is propagating. Eq. (14) treats the radiation Hilbert space as a single copy of E1; that is true only at normal incidence below the first diffraction threshold. If any reciprocal vector G with |G|<ω/c propagates, those channels transform as A2⊕B2⊕E1⊕E2, same as the in-plane modes, and Schur's lemma no longer forbids A2/B2/E2 from radiating. The statement should be conditional on opening only the G=0 channel. That is a one-paragraph fix, but it matters for Table I too.

Second, the quantitative extension is thinner than the classification. leff=cos(2Θ) with Θ=θ+φ appears in Appendix IXA.4 with the angle θ defined in Eq. (27), but there's no real derivation, and the factor of two in the argument is suspect for the 2×2 mixing. The reader flagged a possible missing factor of two; I agree it needs to be checked. Relatedly, the paper asserts rather than proves that the first-shell PWE basis and nearest-neighbor tight-binding model are homeomorphic to the full Maxwell problem. Appendix IXA.2 shows higher shells refine the same IR sector for an individual eigenmode, and Appendix IXB proves C6v symmetry for isotropic long-range couplings, but neither establishes topological equivalence of the truncated models. So the linewidth and Chern-number predictions in Secs. IV–V should be read as model-based illustrations, not theorems about the COMSOL structure.

The self-citations to Refs [32,43] are used appropriately—they supply the non-Hermitian effective model and local Berry-curvature results—but a referee will want enough of that content restated so the paper stands alone.

Who's it for: people doing group-theoretic design of BIC lasers and vortex emitters, and topological photonics folks who want a compact table of dark/bright channels. I'd send it to peer review with the expectation of a conditional acceptance. The classification part is solid; the quantitative parts need tightening.

Referee Report

4 major / 4 minor

Summary. The paper proposes a group-theoretical framework for classifying Bloch modes in two-dimensional photonic crystals, with emphasis on local k-space topology (BICs, optical vortex beams) and Berry phase. The framework uses irreducible representations to block-diagonalize Bloch Hamiltonians, applies it to a plane-wave expansion for TE modes in a triangular lattice, deriving the Γ-point selection rule that only E1 in-plane modes couple to the far field under C6v, and to a tight-binding model for TM modes, where it studies Berry curvature and Chern phases. The paper includes character tables, symmetry-adapted bases, tables of dark/bright modes for various point groups, and full-wave COMSOL simulations for the tight-binding model.

Significance. If the central selection rule holds, the framework provides a simple symmetry-based inventory of dark and bright modes that can guide BIC and vortex-beam engineering. The explicit character decomposition for the first-shell PWE basis and its extension to other point groups are useful and internally consistent. The paper also connects the group-theoretic selection rule to the non-Hermitian coupling matrix W, which is pedagogically valuable. However, the quantitative claims (effective angular momentum formula, homeomorphism between truncated models and the full Maxwell problem) are not rigorously established, and the central BIC claim is stated without an important applicability condition.

major comments (4)

[Sec. IV.B, Eq. (14)] The conclusion that only E1 in-plane modes couple to the far field rests on the radiation Hilbert space being a single copy of the E1 vector representation. This is valid only when the zeroth diffraction order (G=0) is the sole propagating channel at normal incidence. The manuscript does not state this condition. If higher propagating orders exist (|G| < ω/c), each contributes a vector channel; the first-shell radiation functions transform as A2⊕B2⊕E1⊕E2, and Schur's lemma no longer forbids coupling of A2/B2/E2 modes. The unqualified claim in Table I is therefore overgeneralized. Please add the below-first-diffraction-threshold condition or generalize the analysis to all propagating orders.
[Appendix IXA.4, Eq. (29)] The formula leff = cos(2Θ) is central to the de-quantization result in Fig. 4, but it is stated without derivation. The definition of the 'intrinsic basis' |h±⟩ and the geometric angle φ are not given explicitly, and the step from the rotation (Eq. 26) to the observable (Eq. 29) is not shown. Please provide a derivation or a reference.
[Appendix IXA.2, Eqs. (24)-(25)] The statement that higher shells merely refine the same symmetry sector is not consistent with Eq. (24), which shows that the second shell introduces B1 in addition to the first-shell IRs. Eq. (25) then ignores the inter-shell coupling ϵ, which is an uncontrolled approximation. This matters because Table I and the BIC classification are derived from the first-shell representation. Either prove that the inter-shell coupling is symmetry-forbidden in the relevant sector, or restrict the dark/bright table to a specified shell.
[Sec. VI.A and Appendix IXB] The claim that the nearest-neighbor tight-binding model (Eq. 21) is 'homomorphic' to the COMSOL-simulated structure is asserted but not demonstrated. The four isolated bands in Fig. 8 are encouraging, but no quantitative comparison of the band topology (e.g., Berry curvature, Wilson loops, or IR labels) between the model and the full-wave simulation is provided. Since the Berry phase predictions of Sec. V are based on this homeomorphism, it is load-bearing. Please provide evidence or soften the claim.

minor comments (4)

[Sec. V.B] Typo: 'regrading' should be 'regarding'.
[Fig. 4 caption] The caption says 'fixed θ=0.05' but the text describes varying θ; please clarify which parameter is swept.
[Sec. IV.C] The notation leff is introduced with inconsistent subscript formatting; please define it once with a clear notation.
[Eq. (27)] The quantity H_E in Eq. (27) is not defined; please define the 2×2 E-block of the effective Hamiltonian.

Circularity Check

0 steps flagged · score 1.0 of 10

Selection rules derived from IR characters and Schur's lemma; no fitted or self-citation-forced predictions.

full rationale

The central Γ-point selection rule is not circular. It is derived from independent inputs: the characters of the first-shell in-plane basis χ_shell = (6,0,0,0,−2,0), the characters of the far-field vector representation χ_far-field = (2,1,−1,−2,0,0) identified with E1, and the standard IR product criterion Eq. (15). The Schur-lemma argument with the intertwining condition D_rad(g)W = W D_mode(g) is a mathematical consequence of the symmetry setup, not an input fitted to the BIC/vortex outcome. The non-Hermitian Hamiltonian in Sec. IV.C is derived in the text (following Refs. [43,44]), and the statement that it is "in accordance with" the group-theoretic result is corroborative rather than load-bearing. Refs. [32] and [43] are self-authored but are used for supporting context (e.g., "Similar mechanisms for the formulation of the Hamiltonian in square lattices can be found in [32]" and "net Berry curvature concentration ... as shown in Ref. [43] and Sec. V"), not as the derivation of the main selection rule. The tight-binding Berry-phase and Chern-number results in Sec. V are explicit computations from Eq. (21) with chosen perturbations; Chern numbers are read off computed curvature, not fitted. The COMSOL band structure in Fig. 8 additionally provides an external check on the minimal tight-binding model. The assumptions a skeptic might question—that the radiative Hilbert space is a single E1 copy (i.e., only the G = 0 diffraction order is propagating at Γ) and that the truncated PWE/tight-binding models are homeomorphic to the full Maxwell problem—are genuine modeling limitations and correctness risks. In particular, Sec. IV.B does not state the below-first-diffraction-threshold condition, so the unqualified claim that A2, B2, E2 are symmetry-protected BICs is overgeneralized if additional diffraction orders propagate. But this is an incompleteness or overgeneralization, not a circular reduction: the paper's predictions are not defined in terms of the results they explain, and no fitted parameter is renamed as a prediction. No circular step can be exhibited on the quoted text.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical particles, forces, or dimensions; the load is carried by symmetry/truncation assumptions and hand-set model parameters. The central classification is standard IR mathematics, but the extension to open photonic systems relies on the transversality projection and the unproven homeomorphism between truncated and full models.

free parameters (4)

Hermitian coupling parameters v, w, u = v=0.1, w=0.05, u=0.01 (C6v); w=0.05 e^{iπ/8} (C6)
Chosen by hand to set band geometry and hybridization; directly determine leff values, band ordering, and linewidth behavior in Fig. 3.
Loss rate γ = γ=-0.005
Sets the non-Hermitian linewidths and imaginary parts in Tables II–III; no derivation from material or radiative parameters is given.
Mirror-breaking dynamic angle θ = θ=π/8 in Fig. 3(b); θ=0.05 in Fig. 4
Parameterizes the degree of C6v→C6 mirror-symmetry breaking and is the main control knob for leff dequantization, but its connection to a concrete geometry is not derived.
Perturbation coefficients c_B/c_B2 and c_A1, c_{σz} = c_B=c_B2=-0.025; c_A1=±0.1; c_{σz}=∓0.05i
Hand-set strengths of targeted IR perturbations; the reported Chern numbers and connectivity patterns depend on their signs and magnitudes.

assumptions (6)

domain assumption The first shell of reciprocal-lattice vectors, after imposing transversality (Eq. 10), captures the symmetry and topological content of the guided modes; higher shells only refine the same IR sector.
Invoked in Sec. IV.A and Appendix IXA.2 to justify truncating to six plane waves. No convergence or topological-equivalence proof is provided.
domain assumption For the TM/tight-binding case, band separability is well-defined across the whole BZ, so little groups at high-symmetry points are fixed by the Γ-point IRs.
Stated in Secs. II and V.A and required for the Berry-phase/band-connectivity analysis to be globally valid.
standard math The effective non-Hermitian Hamiltonian H_tot = H_H + H_NH with H_NH = iγ[...] follows from Refs [43,44] and respects the crystal symmetry; W is an intertwining operator so Schur's lemma applies.
Used in Sec. IV.B–C to justify coupling selection and block structure; the explicit H_NH matrix entries are imported without full derivation.
ad hoc to paper The four-band nearest-neighbor tight-binding model (Eq. 21) is homomorphic to the COMSOL-simulated 2×2 triangular unit cell and preserves the topological properties.
Assumed in Secs. V.B and VI.A; Appendix IXB only proves that an isotropic long-range Hamiltonian has C6v symmetry, not homomorphism or topological equivalence.
domain assumption TRS breaking by a magnetic field can be represented by adding c_{σz} σ_z to the E2 block of the symmetry-adapted Hamiltonian.
Used in Sec. V.C to obtain Chern numbers 0,±1; no microscopic model of the magneto-optic response is given.
ad hoc to paper leff = cos(2Θ) with Θ=θ+φ correctly quantifies the optical angular momentum/polarization of eigenmodes.
Stated in Appendix IXA.4 without derivation; standard two-level mixing would involve a factor 1/2 in the mixing angle, so this formula is not obviously correct.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A Group-Theoretical Framework for Local k-Space Topology and Berry Phase in 2D Photonic Systems." pith.science (2026). https://pith.science/paper/WZCHSEJD

@misc{pith2026260729356,
  author       = {Pith},
  title        = {Pith review of: A Group-Theoretical Framework for Local k-Space Topology and Berry Phase in 2D Photonic Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WZCHSEJD}},
  note         = {Machine review of arXiv:2607.29356}
}

abstract

Two-dimensional photonic crystals (2D PhCs) enable fine-grained control over a broad set of Bloch modes without the constraints of band occupancy and natural crystal structures, and, as intrinsically open systems, serve as versatile platforms for exploring diverse topological phenomena. Here, we develop a theoretical framework inspired by the irreducible-representation formalism in solid-state physics, while explicitly incorporating key characteristics of photonic Bloch systems, such as radiative coupling and transverse condition. Within this framework, we study the symmetry origins of local k-space topology, e.g., bound-states in the continuum and optical vortex beams, and Berry phase in two representative systems. We further analyze, from a group-theory perspective, how tailored structural designs and targeted symmetry perturbations can be exploited to manipulate these topological features. In particular, we showcase the application of the formalism to Bloch modes from distinct truncation approaches and specify the preferable regimes for each, both under a generic $n$-band configuration. The analysis can thereby be readily extended to a wide range of artificial wave crystals beyond scalar Schr\"odinger-like operators and two-level treatment.

Figures

Figures reproduced from arXiv: 2607.29356 by the authors.

**Figure 1.** FIG. 1. Representative site geometries in real space of photonic crystals, annotated with the little groups of the corresponding [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** FIG. 2. Schematic of the minimal constituent of the trian [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) Real and imaginary parts for the energy bands of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: FIG. 4. The influence of the structure dependent angle [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. The model of a [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6. Energy dispersions of the illustrative system under continuous tuning of [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7. Results for Berry curvature distribution under representative parameter setting of applying an external magnetic field, [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

**Figure 8.** Figure 8: FIG. 8. Full-wave COMSOL band structure of a 2D PhC with four circular dielectric sites per unit cell. The refractive index [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

**Figure 9.** Figure 9: FIG. 9. Schematics showing the primary plane wave space of commonly seen point groups appearing at the high symmetry [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

61 extracted references · 2 canonical work pages

[1]

Extension to Commonly Seen Point Groups We start with a6×6identity matrixI 6, with each col- umn representing{ϕ i |i= 1,2, . . . ,6}. By sequentially apply the symmetry operationsgof aC 6v group, i.e.,E, {C k 6 |k= 1,2, . . . ,5},σv,σ d, one gets the permutation matrixD(g)of each. For rotation operationC 6, we have Gn 7→G n+1,t n 7→t n+1, withG n andt n d...
[2]

One fixes a given high-symmetry pointkin the first Brillouin zone and organizes the plane-wave components according to the magnitude of |k+G|

Inclusion of Higher Diffraction Orders The analysis of higher-energy Bloch modes can be sys- tematically extended using the same shell-based frame- work, as shown above. One fixes a given high-symmetry pointkin the first Brillouin zone and organizes the plane-wave components according to the magnitude of |k+G|. Enlarging the representation space and allow...
[3]

jqOtnWrMH3bRX+iTNlPbsp7mQZs=

Purity Check of Right Eigenvectors The right eigenvectors of the total Hamiltonian can be expressed in the symmetry adapted basis as|ψi⟩=P α,i cα,i|ψα⟩. Weuse|c α,i|2 asameasuretoassesstheIR 11 ! K M ! Frequency(THz) 0 50 100 150 200 250 300 350 ! K M ! Frequency(THz) 0 50 100 150 200 250 300 350 ! K M ! Frequency(THz) 0 50 100 150 200 250 300 350 <latexi...
[4]

3 are as shown in Table II and Table III, respectively

The results for the purity check of the eigenstates got in Fig. 3 are as shown in Table II and Table III, respectively. Since the complex term introduced by the non-Hermitian part acts only on the diagonal elements of theE 1 block, the total Hamiltonian ofC6v remain fully block-diagonalizable. As a result, all the eigenstates are pure states. On the contr...
[5]

Derivation of the effective optical angular momentum The IR basis is rotated by a geometric angleϕrelative to this intrinsic basis, where the Hamiltonian is diagonal. The relation between the 2D block IR bases{|E±⟩and its corresponding intrinsic bases|h±⟩can be expressed as |E+⟩ |E−⟩ = cosϕ−sinϕ sinϕcosϕ |h+⟩ |h−⟩ .(26) Thus,ϕcan be calculated once the po...
[6]

F. D. M. Haldane and S. Raghu, Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry, Physical Review Letters 100, 013904 (2008)

2008
[7]

Raghu and F

S. Raghu and F. D. M. Haldane, Analogs of quantum- Hall-effect edge states in photonic crystals, Physical Re- view A78, 033834 (2008)

2008
[8]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, Reflection-free one-way edge modes in a gyromagnetic photonic crystal, Physical Review Letters100, 013905 (2008)

2008

Show all 61 references

[9]

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

2009
[10]

L. Lu, J. D. Joannopoulos, and M. Soljačić, Topological photonics, Nature photonics8, 821 (2014)

2014
[11]

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, Bound states in the continuum, Nature Reviews Materials1, 1 (2016)

2016
[12]

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,et al.,Topologicalphotonics,ReviewsofModern Physics91, 015006 (2019)

2019
[13]

Price, Y

H. Price, Y. Chong, A. Khanikaev, H. Schomerus, L. J. Maczewsky, M. Kremer, M. Heinrich, A. Szameit, O. Zil- berberg, Y. Yang,et al., Roadmap on topological pho- tonics, Journal of Physics: Photonics4, 032501 (2022)

2022
[14]

Tang, X.-T

G.-J. Tang, X.-T. He, F.-L. Shi, J.-W. Liu, X.-D. Chen, and J.-W. Dong, Topological photonic crystals: physics, designs, and applications, Laser & photonics reviews16, 2100300 (2022)

2022
[15]

C. T. Chan, Essay: Photonic crystals as a platform to ex- plore new physics, Physical Review Letters135, 080001 (2025)

2025
[16]

Q. Wang, M. Xiao, H. Liu, S. Zhu, and C. Chan, Mea- surement of the zak phase of photonic bands through the interface states of a metasurface/photonic crystal, Phys- ical Review B93, 041415 (2016)

2016
[17]

Vaidya, A

S. Vaidya, A. Ghorashi, T. Christensen, M. C. Rechts- man, and W. A. Benalcazar, Topological phases of pho- tonic crystals under crystalline symmetries, Physical Re- view B108, 085116 (2023)

2023
[18]

S. J. Palmer and V. Giannini, Berry bands and pseudo- spin of topological photonic phases, Physical Review Re- search3, L022013 (2021)

2021
[19]

Cuerda, J

J. Cuerda, J. M. Taskinen, N. Källman, L. Grabitz, and P. Törmä, Observation of quantum metric and non- Hermitian Berry curvature in a plasmonic lattice, Phys- ical Review Research6, L022020 (2024)

2024
[20]

Cuerda, J

J. Cuerda, J. M. Taskinen, N. Källman, L. Grabitz, and P. Törmä, Pseudospin-orbit coupling and non-Hermitian effects in the quantum geometric tensor of a plasmonic lattice, Physical Review B109, 165439 (2024)

2024
[21]

Lehikoinen, R

J. Lehikoinen, R. Heilmann, A. J. Dahlberg, E. Härmä, M. Mahmoudi, A. Dutta, K. S. Daskalakis, and P. Törmä, Flat bands from diffraction in periodic systems, arXiv:2602.21830 10.48550/arXiv.2602.21830 (2026)

2026 doi
[22]

Iwahashi, Y

S. Iwahashi, Y. Kurosaka, K. Sakai, K. Kitamura, N. Takayama, and S. Noda, Higher-order vector beams produced by photonic-crystal lasers, Optics Express19, 11963 (2011)

2011
[23]

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Sol- jačić, Topological nature of optical bound states in the continuum, Physical Review Letters113, 257401 (2014)

2014
[24]

Kodigala, T

A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, Lasing action from photonic bound states 14 Perturbation Implementation Representation IR Channel Primary Observables cir Uniform Scaling Uniform hole size adjustment, e.g. [52] Diagonal energy shift A2 (Full...

2017
[25]

H. M. Doeleman, F. Monticone, W. den Hollander, A. Alù, and A. F. Koenderink, Experimental Observa- tion of a Polarization Vortex at an Optical Bound State in the Continuum, Nature Photonics12, 397 (2018)

2018
[26]

Berguiga, X

N.D.Le, P.Bouteyre, A.Kheir-Aldine, F.Dubois, S.Cu- eff, L. Berguiga, X. Letartre, P. Viktorovitch, T. Beny- attou, and H. S. Nguyen, Super bound states in the con- tinuum on a photonic flatband: concept, experimental realization, and optical trapping demonstration, Physi- cal...

2024
[27]

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, Physical Review Letters125, 053902 (2020)

2020
[28]

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, Nature Pho- tonics14, 623 (2020)

2020
[29]

Zhang, K

T. Zhang, K. Dong, J. Li, F. Meng, J. Li, S. Muna- gavalasa, C.P.Grigoropoulos, J.Wu,andJ.Yao,Twisted moiré photonic crystal enabled optical vortex generation through bound states in the continuum, Nature Commu- nications14, 6014 (2023)

2023
[30]

C. Han, J. He, C. Tong, C. Liu, M. Yang, and B. Wang, Generating first-order optical vortex beams by photonic crystal slabs, Optics Express32, 27591 (2024)

2024
[31]

Y. Shen, Q. Zhang, P. Shi, L. Du, X. Yuan, and A. V. Zayats, Optical skyrmions and other topological quasi- particles of light, Nature Photonics18, 15 (2024)

2024
[32]

R. Deng, T. Li, W. Liu, J. Wang, L. Shi, and J. Zi, Inverse-designed photonic crystals for tailored oam beam generation and multiplexing in momentum space, Pho- tonics Research14, 834 (2026)

2026
[33]

Xiao, M.-C

D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Reviews of modern physics82, 1959 (2010)

1959
[34]

H. C. Po, A. Vishwanath, and H. Watanabe, Symmetry- based indicators of band topology in the 230 space groups, Nature communications8, 50 (2017)

2017
[35]

Blanco de Paz, C

M. Blanco de Paz, C. Devescovi, G. Giedke, J. J. Saenz, M.G.Vergniory, B.Bradlyn, D.Bercioux,andA.García- Etxarri, Tutorial: computing topological invariants in 2d photonic crystals, Advanced Quantum Technologies3, 1900117 (2020)

2020
[36]

B. J. Wieder, B. Bradlyn, J. Cano, Z. Wang, M. G. Vergniory, L. Elcoro, A. A. Soluyanov, C. Felser, T. Neu- pert, N. Regnault,et al., Topological materials discovery from crystal symmetry, Nature Reviews Materials7, 196 (2022)

2022
[37]

Arjas, G

K. Arjas, G. Salerno, and P. Törmä, Topological invari- ants and topological charges in photonic systems, Physi- cal Review B112, 235428 (2025)

2025
[38]

Slager, A

R.-J. Slager, A. Mesaros, V. Juričić, and J. Zaanen, The space group classification of topological band-insulators, Nature Physics9, 98 (2013)

2013
[39]

Kruthoff, J

J. Kruthoff, J. De Boer, J. Van Wezel, C. L. Kane, and 15 R.-J. Slager, Topological classification of crystalline in- sulators through band structure combinatorics, Physical Review X7, 041069 (2017)

2017
[40]

Bradlyn, L

B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Topological quantum chemistry, Nature547, 298 (2017)

2017
[41]

Dong and C.-X

X.-Y. Dong and C.-X. Liu, Classification of topologi- cal crystalline insulators based on representation theory, Physical Review B93, 045429 (2016)

2016
[42]

Petralanda, Y

U. Petralanda, Y. Jiang, B. A. Bernevig, N. Reg- nault, and L. Elcoro, Two-dimensional topological quan- tum chemistry and catalog of topological materials, arXiv:2411.08950 10.48550/arXiv.2411.08950 (2024)

2024 doi arXiv
[43]

C. F. Doiron, I. Brener, and A. Cerjan, Realizing symmetry-guaranteed pairs of bound states in the contin- uum in metasurfaces, Nature Communications13, 7534 (2022)

2022
[44]

A groupGis cyclic if it is generated by a single element g∈G, i.e.,G=⟨g⟩={g n |n∈Z}
[45]

Heilmann, G

R. Heilmann, G. Salerno, J. Cuerda, T. K. Hakala, and P. Torma, Quasi-BIC mode lasing in a quadrumer plas- monic lattice, ACS photonics9, 224 (2022)

2022
[46]

Salerno, R

G. Salerno, R. Heilmann, K. Arjas, K. Aronen, J.-P. Martikainen, and P. Törmä, Loss-driven topological tran- sitions in lasing, Physical Review Letters129, 173901 (2022)

2022
[47]

Arjas, J

K. Arjas, J. M. Taskinen, R. Heilmann, G. Salerno, and P. Törmä, High topological charge lasing in quasicrystals, Nature Communications15, 9544 (2024)

2024
[48]

X. Yuan, L. Malgrey, H. Sigurðsson, H. S. Nguyen, and G. Salerno, Breakdown of bulk-radiation correspondence in radiative photonic lattices, Physical Review Research 7, 043141 (2025)

2025
[49]

V. A. Nguyen, H. S. Nguyen, Z. Yuan, D. X. Nguyen, C. Dang, S. T. Ha, X. Letartre, Q. Le-Van, and H. S. Nguyen, Generalized non-hermitian hamiltonian for guided resonances in photonic crystal slabs, Nanophoton- ics14, 5229 (2025)

2025
[50]

Such symmetry breaking mechanism, i.e., representing mirror symmetry breaking with the complex entries, is not valid forC 4v→C 4 due to the equivalence between C2 operation and inversion symmetry, as discussed in Sec. IIIA
[51]

Wu and X

L.-H. Wu and X. Hu, Scheme for achieving a topologi- cal photonic crystal by using dielectric material, Physical Review Letters114, 223901 (2015)

2015
[52]

Specific to this approach, the relative strength between cA1 andc(σ z)needs to be scrutinized to ensure the influ- ence of magnetic field doesn’t change the prototype after spatial perturbation
[53]

Kawabata, K

K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Sym- metry and topology in non-Hermitian physics, Physical Review X9, 041015 (2019), publisher: APS

2019
[54]

J. Jin, L. He, J. Lu, L. Chang, C. Shang, J. E. Bowers, E. J. Mele, and B. Zhen, Towards floquet chern insulators of light, Nature Nanotechnology20, 1574 (2025)

2025
[55]

L. Wang, H. Liu, J. Liu, A. Liu, J. Huang, Q. Li, H. Dai, C. Zhang, J. Wu, K. Fan,et al., Photoswitchable excep- tionalpointsderivedfromboundstatesinthecontinuum, Light: Science & Applications14, 377 (2025)

2025
[56]

X. Yan, M. Tang, Z. Zhou, L. Ma, Y. Vaynzof, J. Yao, H. Dong, and Y. S. Zhao, Topologically reconfigurable room-temperature polariton condensates from bound states in the continuum in organic metasurfaces, Nature Communications16, 2375 (2025)

2025
[57]

Chern, Y.-C

R.-L. Chern, Y.-C. Kao, and R. R. Hwang, Dirac bound states in the continuum in honeycomb photonic crys- tal slabs, Scientific Reports 10.1038/s41598-026-37156-z (2026)

2026 doi
[58]

S. Li, L. Huang, H. Zhong, M. Ning, L.-E. Zhang, Y. Yin, Y.Cheng,andL.Li,Observationofmultiplequasi-bound states in the continuum by symmetry breaking in a pho- tonic crystal slab, Photonics Research13, 968 (2025)

2025
[59]

Sagnes, A

O.Jamadi, E.Rozas, G.Salerno, M.Milićević, T.Ozawa, I. Sagnes, A. Lemaître, L. L. Gratiet, A. Harouri, I. Caru- sotto, J. Bloch, and A. Amo, Direct observation of pho- tonic Landau levels and helical edge states in strained honeycomb lattices, Light Sci. Appl.9, 144 (2020)

2020
[60]

Kim, M.-S

K.-H. Kim, M.-S. Hwang, H.-R. Kim, J.-H. Choi, Y.- S. No, and H.-G. Park, Direct observation of exceptional pointsincoupledphotonic-crystallaserswithasymmetric optical gains, Nature communications7, 13893 (2016)

2016
[61]

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

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Extension to Commonly Seen Point Groups We start with a6×6identity matrixI 6, with each col- umn representing{ϕ i |i= 1,2, . . . ,6}. By sequentially apply the symmetry operationsgof aC 6v group, i.e.,E, {C k 6 |k= 1,2, . . . ,5},σv,σ d, one gets the permutation matrixD(g)of each. For rotation operationC 6, we have Gn 7→G n+1,t n 7→t n+1, withG n andt n d...

[2] [2]

One fixes a given high-symmetry pointkin the first Brillouin zone and organizes the plane-wave components according to the magnitude of |k+G|

Inclusion of Higher Diffraction Orders The analysis of higher-energy Bloch modes can be sys- tematically extended using the same shell-based frame- work, as shown above. One fixes a given high-symmetry pointkin the first Brillouin zone and organizes the plane-wave components according to the magnitude of |k+G|. Enlarging the representation space and allow...

[3] [3]

jqOtnWrMH3bRX+iTNlPbsp7mQZs=

Purity Check of Right Eigenvectors The right eigenvectors of the total Hamiltonian can be expressed in the symmetry adapted basis as|ψi⟩=P α,i cα,i|ψα⟩. Weuse|c α,i|2 asameasuretoassesstheIR 11 ! K M ! Frequency(THz) 0 50 100 150 200 250 300 350 ! K M ! Frequency(THz) 0 50 100 150 200 250 300 350 ! K M ! Frequency(THz) 0 50 100 150 200 250 300 350 <latexi...

[4] [4]

3 are as shown in Table II and Table III, respectively

The results for the purity check of the eigenstates got in Fig. 3 are as shown in Table II and Table III, respectively. Since the complex term introduced by the non-Hermitian part acts only on the diagonal elements of theE 1 block, the total Hamiltonian ofC6v remain fully block-diagonalizable. As a result, all the eigenstates are pure states. On the contr...

[5] [5]

Derivation of the effective optical angular momentum The IR basis is rotated by a geometric angleϕrelative to this intrinsic basis, where the Hamiltonian is diagonal. The relation between the 2D block IR bases{|E±⟩and its corresponding intrinsic bases|h±⟩can be expressed as |E+⟩ |E−⟩ = cosϕ−sinϕ sinϕcosϕ |h+⟩ |h−⟩ .(26) Thus,ϕcan be calculated once the po...

[6] [6]

F. D. M. Haldane and S. Raghu, Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry, Physical Review Letters 100, 013904 (2008)

2008

[7] [7]

Raghu and F

S. Raghu and F. D. M. Haldane, Analogs of quantum- Hall-effect edge states in photonic crystals, Physical Re- view A78, 033834 (2008)

2008

[8] [8]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, Reflection-free one-way edge modes in a gyromagnetic photonic crystal, Physical Review Letters100, 013905 (2008)

2008