Realization of Arbitrary Gauge Fields via Symmetry-Protected Zero Modes

Bingbing Wang; Haoran Xue; Jiangzi Chen; J. X. Dai; Y. X. Zhao

REVIEW 2 major objections 4 minor 37 references

Realization of Arbitrary Gauge Fields via Symmetry-Protected Zero Modes

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper claims that any static O(N) lattice gauge configuration can be realized as an exact spectral block of a positive-coupling tight-binding model, using symmetry-protected zero modes of sublattice-imbalanced bipartite units, and…

desk verdict A genuinely general zero-mode recipe for exact O(N) gauge-field realization with only positive couplings; the main mathematical step is deferred to the SM but appears to hold. read the letter →

arxiv 2608.11609 v1 pith:HI5RMVJG submitted 2026-08-12 cond-mat.mes-hall

J. X. Dai , Bingbing Wang , Jiangzi Chen , Y. X. Zhao , Haoran Xue This is my paper · ORCID

classification cond-mat.mes-hall

keywords gaugefieldssymmetry-protectedzeromodesLieb'stheorempositivecouplingsartificialatomsacoustictopologicalinsulatorsHofstadtermodelnon-Abelian

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

This paper sets out to show that any static O(N) lattice gauge configuration—where each bond carries an orthogonal matrix, possibly continuous and non-Abelian—can be realized exactly in a simple tight-binding model whose couplings are all positive. The engine is a bipartite unit with unequal sublattice populations, whose chiral symmetry forces zero-energy modes; those modes have alternating-sign components, so positive microscopic hoppings produce effective hoppings of either sign between units. The authors further claim that the zero-mode manifold can be decoupled from all other modes, making the target gauge Hamiltonian an exact spectral block separated by a gap, rather than an approximation. They verify the idea in acoustic crystals in three regimes: a Z2 quadrupole topological insulator, an SO(2) Hofstadter model, and a non-Abelian SO(3) topological insulator. A sympathetic reader cares because this would turn prescribed gauge-field models into design rules for real experiments using only positive, short-range couplings.

What carries the argument

The load-bearing object is the sublattice-imbalanced bipartite unit, a finite tight-binding model with more B-sites than A-sites, whose chiral symmetry forces M_B − M_A zero modes by Lieb's theorem; the minimal unit is three sites (A-B-B) with one zero mode (0, 1, −1)/√2. The sign pattern of this zero mode converts positive microscopic couplings into negative effective hoppings: connecting two units in parallel gives +t and crossing gives −t, and for N units the effective matrix entry is [Δ_eff]_jk = 1/2(Δ_{a,b} + Δ_{a+1,b+1} − Δ_{a,b+1} − Δ_{a+1,b}), so parallel and crossing hoppings compete to produce any real number. The construction's exactness comes from the decoupling condition that inter-atom couplings eliminate all matrix elements between the zero-mode manifold and its complement, making the projected Hamiltonian an exact spectral block.

What would settle it

Take an N=2 or N=3 artificial atom and a prescribed real effective hopping matrix, then solve the coupled conditions of Eq. (6) and the decoupling equations for positive coupling strengths; a single target matrix with no positive solution, or an acoustic sample whose low-energy spectrum shows the zero-mode block mixing with higher modes as the intra-unit coupling is lowered, would falsify the exactness claim.

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Extended reading notes

Core claim

The central claim is that one can prescribe any static O(N) gauge configuration on a lattice—each bond carrying an arbitrary N-by-N orthogonal matrix—and realize it exactly in a microscopic model whose only couplings are positive real numbers and short range. The construction replaces every site of the target lattice by an 'artificial atom' made of identical bipartite units whose sublattice imbalance gives one symmetry-protected zero mode per unit. Zero modes live only on B-sites and have sign-changing components, so parallel and crossing inter-atom connections contribute opposite signs to effective hoppings; by choosing coupling strengths, every entry of the effective hopping matrix can be set independently to any real value. The key additional step is choosing inter-atom couplings so that the zero-mode manifold has zero matrix elements with all nonzero modes; the effective target Hamiltonian is then an exact invariant block, not a low-energy approximation. The authors validate this claim numerically and acoustically for Z2, continuously tunable SO(2), and non-Abelian SO(3) link models.

Load-bearing premise

The construction requires that, for any prescribed real N-by-N effective hopping matrix, one can choose non-negative inter-atom couplings that both produce that matrix on the zero-mode manifold and exactly cancel every matrix element linking zero modes to nonzero modes; the general proof of that cancellation is deferred to the Supplemental Material, and the main text only works out the existence for the minimal three-site, N=1 case.

Editorial extensions

If this is right

Any compact gauge group can be covered, since every compact Lie group embeds as a subgroup of O(N) for large enough N; the same positive-coupling construction then realizes its static gauge configurations.
Continuously parameterizable links become possible: for SO(2), changing coupling strengths varies the flux continuously, so the Hofstadter butterfly can be traced as a function of flux without breaking reciprocity or using complex hoppings.
Because the target gauge sector is exactly decoupled and gapped, topological invariants and boundary modes of the target model—corner states, chiral edge states, non-Abelian charges—are reproduced in the microscopic spectrum rather than only in a perturbation limit.
The construction uses only positive short-range couplings, so it transfers directly to other artificial-crystal platforms beyond acoustics, including photonic crystals, mechanical lattices, electric circuits, and cold atoms, and to higher orbital numbers and three dimensions.

Reading between the lines

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

If the decoupling existence proof holds for all N, the scheme is effectively a universal compiler for classical static O(N) gauge theories on any lattice: any bond configuration becomes a set of coupling strengths, which suggests practical reconfigurable simulators in which flux angles are tuned continuously in situ.
The same sign-changing zero-mode mechanism might be exploited for effective complex couplings via pairs of real channels, potentially pushing the positive-coupling construction from O(N) to unitary subgroups beyond O(N); the paper does not claim this.
Because the zero modes are protected by sublattice symmetry rather than fine-tuning, disorder that preserves the bipartite structure may leave the exactness intact; testing robustness of the decoupling to disorder would be a natural follow-up.

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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 paper delivers a general, non-perturbative construction for encoding arbitrary static O(N) gauge links into the zero-mode sector of a bipartite tight-binding model using only positive couplings. That is a real advance over the discrete-group enlarged-lattice approaches in Refs. [15,16], which do not spectrally isolate the gauge sector. The continuous SO(2) and non-Abelian SO(3) acoustic demonstrations go beyond what those constructions were designed to do. I think the central claim is correct: the zero-mode manifold can be made exactly invariant, and the effective hopping matrix can be programmed bond by bond. The stress-test note reconstructs the explicit decoupling block and it works, so the deferred proof in the SM is very likely fine. The minimal three-site example and Eq. (8) make the mechanism transparent, and the three experimental systems give credible proof of principle. Credit where due: the idea of using Lieb's zero modes as sign-changing pseudo-spins to generate negative effective hoppings from positive physical couplings is clean and potentially quite useful. The paper is also honest about sign conventions, including the J=-10 point in the SO(3) section, which is a real but minor wrinkle. Soft spots are mostly archival and quantitative. The exact-decoupling proof is stated in the main text and left to the SM; that is acceptable for a letter but should be verified in any referee process. The experimental data have no error bars, no statistical analysis, and no data availability statement; for a PRL-style letter that is often tolerated, but the edge-state dispersions in Fig. 3(i) are shown as color maps without quantified comparison. On the theory side, the exactness claim is for a tight-binding model, not for the actual acoustic continuum, so the 'exact spectral block' statement is always approximate in the experimental realizations; the authors do not overclaim this, but the abstract's wording could mislead a casual reader. Also, the claim that 'any real effective hopping matrix is accessible' is justified by the parallel/crossing construction, which I agree with, but the detailed proof of the simultaneous decoupling condition for arbitrary N is exactly the part deferred to the SM. I have no serious objection to the physics or the honesty of the presentation. Citation pattern looks fine. The paper deserves a serious referee. It would be a good reading-group paper for people working in artificial gauge fields or topological wave simulation; I would cite it if I needed a clean general construction for exact gauge-sector isolation. My recommendation: send to peer review, ask for the SM decoupling proof and experimental uncertainty estimates before acceptance.

Referee Report

2 major / 4 minor

Summary. This manuscript introduces a general scheme for realizing arbitrary static O(N) lattice gauge configurations using symmetry-protected zero modes of bipartite units with imbalanced sublattices. The target link matrix on each bond is encoded in the connectivity and strengths of positive microscopic couplings between artificial atoms, and the paper claims that the projected Hamiltonian in the zero-mode sector is an exact spectral block of the microscopic Hamiltonian rather than a perturbative approximation. The scheme is illustrated with three acoustic experiments: a Z2 quadrupole topological insulator, an SO(2) Hofstadter model, and an SO(3) non-Abelian topological insulator. The derivation of the effective hopping formula in Eq. (8) is clean, and the mapping from target data to microscopic couplings has no fitted parameters.

Significance. If the exact-block construction holds, this is a valuable and general route to gauge-field physics in artificial platforms, covering discrete Abelian, continuous Abelian, and non-Abelian links with only positive couplings. The experimental demonstrations provide external validation, and the framework is applicable to photonic, mechanical, circuit, and cold-atom systems. The main technical contribution, the parameter-free construction of arbitrary real effective hopping matrices, is significant, though its proof is not fully contained in the main text.

major comments (2)

[General framework, paragraph after Eq. (6)] The exactness claim that 'the couplings can also be chosen to eliminate matrix elements between the zero-mode manifold and its complement' is load-bearing, because it converts the construction from a perturbative projection into an exact spectral block. The proof is deferred to SM Sec. I, but the SM is not included in the arXiv submission. Please make the decoupling construction available, and ideally sketch it in the main text, since without it the central exactness assertion cannot be verified from the submitted manuscript.
[General framework, paragraph after Eq. (8)] The statement that 'any real effective hopping matrix is accessible' is asserted as a mathematical fact but not proved in the main text. Although an explicit per-element construction (parallel couplings for non-negative entries and crossing couplings for negative entries, each with strength |A_jk|) is straightforward and also satisfies the decoupling conditions, the manuscript should include it or cite its location, because this surjectivity is the basis for the claim that arbitrary O(N) configurations can be realized.

minor comments (4)

[SO(3) non-Abelian gauge field, paragraph with J=-10] The value J=-10 appears to conflict with the stated requirement of positive microscopic couplings; the explanation in footnote [35] is adequate but should be moved into the main text or the figure caption, since it is essential for interpreting the constructed model.
[SO(2) Hofstadter model, paragraph with Eq. 'te^{im alpha sigma2}'] Please clarify whether the displayed matrix e^{im alpha sigma2} denotes the hopping block from site m to m+1 or from m+1 to m; Hermiticity requires the opposite block to be its transpose, and the current wording is ambiguous.
[General framework, Eq. (8)] It would improve readability to define 'parallel' and 'crossing' couplings explicitly in terms of the indices a,b,a+1,b+1 before stating Eq. (8), since these terms carry the sign mechanism of the construction.
[Supplemental Material reference [30]] The manuscript refers to SM Sec. IX and Sec. X for the 2D SO(3) model and acoustic structural details, but the SM is not provided in the arXiv submission; this limits verification of the experimental implementation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the target zero-mode Hamiltonian is constructed directly from positive microscopic couplings, with no fitted parameters and no load-bearing self-citation.

full rationale

The central claim is that any real N x N effective hopping matrix can be realized as an exact spectral block of a microscopic tight-binding model with only positive couplings. The mapping is constructive rather than definitional: Eq. (8) expresses each effective matrix element as a combination of parallel and crossing couplings, and the paper inverts this relation by choosing the microscopic inter-atom coupling block Delta to produce a prescribed Delta_eff, while the decoupling condition after Eq. (6) is an additional constraint on Delta rather than a fitted input or a restatement of the target. The zero modes themselves come from Lieb's theorem, an external mathematical result, and the experimental acoustic spectra are independent validations, not inputs to the construction. The only self-citation in the load-bearing vicinity, [36], defines a gauge-dressed inversion operation and is not used to force the gauge-field realization; the non-Abelian classification is cited to external work [37]. The deferred proof of exact decoupling in SM Sec. I is a completeness or support issue, not a circular one, because the claim is explicitly asserted as a construction step with the proof relegated to supplemental material rather than being assumed as the target result. No equation in the paper reduces by construction to its own input, no fitted parameter is renamed as a prediction, and no self-citation chain carries the central argument. Therefore the analysis finds no significant circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The construction introduces no fitted free parameters: all coupling strengths are directly prescribed by the target model. The theoretical input is standard bipartite zero-mode theory; the practical input is the assumption that acoustic couplings can be tuned as designed. No new physical entities are postulated.

assumptions (3)

standard math Lieb's theorem: a finite bipartite tight-binding model with M_B > M_A has M_B - M_A zero-energy eigenstates protected by chiral symmetry.
Used in the General Framework section to establish the existence and form of the zero modes.
standard math The singular value decomposition Q = V Sigma U^T provides orthonormal bases for the null spaces of Q and Q^T.
Used to construct the zero-mode basis |psi_j> in the paragraph following Eq. (3).
domain assumption Acoustic resonators and coupling tubes are accurately described by a tight-binding model with tunable positive hopping amplitudes.
The experimental realization in Figs. 2-4 assumes the fabricated acoustic crystal maps onto the constructed tight-binding model; fabrication tolerances and finite-size effects are not quantified in the main text.

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Pith. "Pith review of Realization of Arbitrary Gauge Fields via Symmetry-Protected Zero Modes." pith.science (2026). https://pith.science/paper/HI5RMVJG

@misc{pith2026260811609,
  author       = {Pith},
  title        = {Pith review of: Realization of Arbitrary Gauge Fields via Symmetry-Protected Zero Modes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HI5RMVJG}},
  note         = {Machine review of arXiv:2608.11609}
}

abstract

Gauge fields are fundamental to modern physics, but prescribed gauge configurations are often difficult to implement in artificial systems. Here, we present a general scheme for realizing arbitrary static $\mathrm{O}(N)$ lattice gauge configurations using symmetry-protected zero modes of sublattice-imbalanced bipartite units. The target $\mathrm{O}(N)$ link on each bond is encoded in the connectivity and strengths of positive microscopic couplings. By decoupling the zero-mode manifold from the remaining modes, the target gauge Hamiltonian forms an exact spectral block of the microscopic tight-binding model rather than a perturbative approximation. We experimentally demonstrate this framework in acoustic crystals through a $\mathbb{Z}_2$ quadrupole topological insulator, an $\mathrm{SO}(2)$ Hofstadter model, and an $\mathrm{SO}(3)$ non-Abelian topological insulator. Our results provide a general and accessible route to gauge-field physics in artificial systems.

Figures

Figures reproduced from arXiv: 2608.11609 by the authors.

**Figure 2.** FIG. 2. A quadrupole topological insulator with [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. An SO(2) Hofstadter model. (a) The target SO(2) Hofstadter lattice model, with hopping phases indicated. (b) The [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. A non-Abelian SO(3) gauge field. (a) The target lattice model featuring an SO(3) gauge field. Each site ( [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

37 extracted references · 22 canonical work pages

[1]

C. N. Yang and R. L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev.96, 191 (1954)

1954
[2]

Aharonov and D

Y. Aharonov and D. Bohm, Significance of electromag- netic potentials in the quantum theory, Phys. Rev.115, 485 (1959)

1959
[3]

F. D. M. Haldane, Model for a quantum Hall effect with- out Landau levels: Condensed-matter realization of the ”parity anomaly”, Phys. Rev. Lett.61, 2015 (1988). 6

work page 1988
[4]

C. L. Kane and E. J. Mele, Quantum spin Hall effect in graphene, Phys. Rev. Lett.95, 226801 (2005)

2005
[5]

B. A. Bernevig and S.-C. Zhang, Quantum spin Hall ef- fect, Phys. Rev. Lett.96, 106802 (2006)

work page 2006
[6]

Y. X. Zhao, Y.-X. Huang, and S. A. Yang,Z 2-projective translational symmetry protected topological phases, Phys. Rev. B102, 161117 (2020)

work page 2020
[7]

Z. Chen, S. A. Yang, and Y. Zhao, Brillouin Klein bot- tle from artificial gauge fields, Nat. Commun.13, 2215 (2022)

work page 2022
[8]

Y. Yang, B. Yang, G. Ma, J. Li, S. Zhang, and C. Chan, Non-Abelian physics in light and sound, Science383, eadf9621 (2024)

work page 2024

Show all 37 references

[9]

Serra-Garcia, V

M. Serra-Garcia, V. Peri, R. S¨ usstrunk, O. R. Bilal, T. Larsen, L. G. Villanueva, and S. D. Huber, Obser- vation of a phononic quadrupole topological insulator, Nature555, 342 (2018)

2018
[10]

C. W. Peterson, W. A. Benalcazar, T. L. Hughes, and G. Bahl, A quantized microwave quadrupole insulator with topologically protected corner states, Nature555, 346 (2018)

2018
[11]

H. Xue, Y. Ge, H.-X. Sun, Q. Wang, D. Jia, Y.-J. Guan, S.-Q. Yuan, Y. Chong, and B. Zhang, Observation of an acoustic octupole topological insulator, Nat. Commun. 11, 2442 (2020)

2020
[12]

Y. Yang, C. Peng, D. Zhu, H. Buljan, J. D. Joannopoulos, B. Zhen, and M. Soljaˇ ci´ c, Synthesis and observation of non-Abelian gauge fields in real space, Science365, 1021 (2019)

2019
[13]

J. Wu, Z. Wang, Y. Biao, F. Fei, S. Zhang, Z. Yin, Y. Hu, Z. Song, T. Wu, F. Song,et al., Non-Abelian gauge fields in circuit systems, Nat. Electron.5, 635 (2022)

2022
[14]

Cheng, K

D. Cheng, K. Wang, C. Roques-Carmes, E. Lustig, O. Y. Long, H. Wang, and S. Fan, Non-Abelian lattice gauge fields in photonic synthetic frequency dimensions, Nature 637, 52 (2025)

2025
[15]

Marciani, Translation groups for arbitrary gauge fields in synthetic crystals with real hopping amplitudes (2025), arXiv:2508.08461 [cond-mat.mes-hall]

M. Marciani, Translation groups for arbitrary gauge fields in synthetic crystals with real hopping amplitudes (2025), arXiv:2508.08461 [cond-mat.mes-hall]

2025 arXiv
[16]

Guba, R.-J

Z. Guba, R.-J. Slager, L. K. Upreti, and T. Bzduˇ sek, Topological non-Abelian gauge structures in cayley- schreier lattices, Nat. Commun.17, 4669 (2026)

2026
[17]

R¨ ontgen, M

M. R¨ ontgen, M. Pyzh, C. Morfonios, N. Palaiodimopou- los, F. Diakonos, and P. Schmelcher, Latent symmetry in- duced degeneracies, Phys. Rev. Lett.126, 180601 (2021)

2021
[18]

Hamanaka, T

S. Hamanaka, T. Yoshida, and K. Kawabata, Non- Hermitian topology in Hermitian topological matter, Phys. Rev. Lett.133, 266604 (2024)

2024
[19]

L. Lu, J. D. Joannopoulos, and M. Soljaˇ ci´ c, Topological photonics, Nat. Photon.8, 821 (2014)

2014
[20]

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
[21]

H. Xue, Y. Yang, and B. Zhang, Topological acoustics, Nat. Rev. Mater.7, 974 (2022)

2022
[22]

S. D. Huber, Topological mechanics, Nat. Phys.12, 621 (2016)

2016
[23]

G. Ma, M. Xiao, and C. T. Chan, Topological phases in acoustic and mechanical systems, Nat. Rev. Phys.1, 281 (2019)

2019
[24]

Sahin, M

H. Sahin, M. Jalil, and C. H. Lee, Topolectrical circuits- Recent experimental advances and developments, APL Electron. Devices1, 10.1063/5.0265293 (2025)

2025 doi
[25]

Zhang, Y.-Q

D.-W. Zhang, Y.-Q. Zhu, Y. Zhao, H. Yan, and S.-L. Zhu, Topological quantum matter with cold atoms, Adv. Phys.67, 253 (2018)

2018
[26]

N. R. Cooper, J. Dalibard, and I. B. Spielman, Topo- logical bands for ultracold atoms, Rev. Mod. Phys.91, 015005 (2019)

2019
[27]

E. H. Lieb, Two theorems on the Hubbard model, Phys. Rev. Lett.62, 1201 (1989)

1989
[28]

P. W. Brouwer, E. Racine, A. Furusaki, Y. Hatsugai, Y. Morita, and C. Mudry, Zero modes in the random hopping model, Phys. Rev. B66, 014204 (2002)

2002
[29]

Such a system possesses a sublattice (chiral) symmetry operatorS, which anticom- mutes with the Hamiltonian, namely,{H,S}= 0

A bipartite system is characterized by a HamiltonianH that couples two distinct sublattices,AandB, withM A andM B sites, respectively. Such a system possesses a sublattice (chiral) symmetry operatorS, which anticom- mutes with the Hamiltonian, namely,{H,S}= 0. This symmetry di...
[30]

See Supplemental Material
[31]

W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Quantized electric multipole insulators, Science357, 61 (2017)

2017
[32]

Xiao, W.-J

M. Xiao, W.-J. Chen, W.-Y. He, and C. T. Chan, Syn- thetic gauge flux and Weyl points in acoustic systems, Nat. Phys.11, 920 (2015)

2015
[33]

H. Xue, Y. Yang, F. Gao, Y. Chong, and B. Zhang, Acoustic higher-order topological insulator on a kagome lattice, Nat. Mater.18, 108 (2019)

2019
[34]

D. R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B14, 2239 (1976)

1976
[35]

3(g) and (h) plot−E/t

To facilitate direct comparison with the acoustic exper- iment, Figs. 3(g) and (h) plot−E/t. In the SO(3) ex- ample, the tight-binding parameters are written with an overall negative sign to match the spectral convention used for the acoustic realization. These sign convention...
[36]

Y. X. Zhao, C. Chen, X.-L. Sheng, and S. A. Yang, Switching spinless and spinful topological phases with projectivePTsymmetry, Phys. Rev. Lett.126, 196402 (2021)

2021
[37]

Q. Wu, A. A. Soluyanov, and T. Bzduˇ sek, Non-Abelian band topology in noninteracting metals, Science365, 1273 (2019)

2019

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

C. N. Yang and R. L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev.96, 191 (1954)

1954

[2] [2]

Aharonov and D

Y. Aharonov and D. Bohm, Significance of electromag- netic potentials in the quantum theory, Phys. Rev.115, 485 (1959)

1959

[3] [3]

F. D. M. Haldane, Model for a quantum Hall effect with- out Landau levels: Condensed-matter realization of the ”parity anomaly”, Phys. Rev. Lett.61, 2015 (1988). 6

work page 1988

[4] [4]

C. L. Kane and E. J. Mele, Quantum spin Hall effect in graphene, Phys. Rev. Lett.95, 226801 (2005)

2005

[5] [5]

B. A. Bernevig and S.-C. Zhang, Quantum spin Hall ef- fect, Phys. Rev. Lett.96, 106802 (2006)

work page 2006

[6] [6]

Y. X. Zhao, Y.-X. Huang, and S. A. Yang,Z 2-projective translational symmetry protected topological phases, Phys. Rev. B102, 161117 (2020)

work page 2020

[7] [7]

Z. Chen, S. A. Yang, and Y. Zhao, Brillouin Klein bot- tle from artificial gauge fields, Nat. Commun.13, 2215 (2022)

work page 2022

[8] [8]

Y. Yang, B. Yang, G. Ma, J. Li, S. Zhang, and C. Chan, Non-Abelian physics in light and sound, Science383, eadf9621 (2024)

work page 2024