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arxiv: 2604.27417 · v1 · submitted 2026-04-30 · ❄️ cond-mat.mes-hall · physics.optics

Mobile Exceptional Points Generate Momentum-Space Switching Domains

Jung-Wan Ryu , Chang-Hwan Yi This is my paper

Pith reviewed 2026-05-07 09:24 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall physics.optics
keywords exceptional pointsnon-Hermitian topologyband switchingmomentum spacecyclic modulationswitching domainsphotonic crystalBrillouin zone
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The pith

Mobile exceptional points divide momentum space into domains with distinct band-switching behavior under cyclic modulation.

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

The paper shows that when exceptional points move through momentum space during cyclic modulation, their paths create boundaries that split the Brillouin zone into separate domains. Inside each domain the bands either exchange or remain unchanged after one full modulation cycle, and this behavior is set by a band-permutation invariant. The domain boundaries are the projections of the exceptional-point trajectories in the combined space of crystal momentum and modulation strength. As modulation grows stronger the switching domains enlarge until they cover the entire zone and produce global band switching. The same structure appears in a concrete photonic-crystal example with loss.

Core claim

Mobile exceptional points generate momentum-space switching domains that partition the Brillouin zone into regions with distinct band-switching behavior. The boundaries between switching regions arise from the projection of EP trajectories in an extended parameter space combining crystal momentum and the modulation parameter. Using a minimal two-band lattice model, a band-permutation invariant determines whether eigenmodes exchange after one modulation cycle. As the modulation strength increases, the switching domains expand and eventually cover the entire Brillouin zone, resulting in global band switching.

What carries the argument

Mobile exceptional points whose trajectories in the joint momentum-modulation parameter space project onto the boundaries that separate momentum-space switching domains.

If this is right

  • The Brillouin zone is partitioned into domains where eigenmodes either permute or stay fixed after one modulation cycle.
  • Increasing modulation strength enlarges the switching domains until they cover the full zone and produce global band switching.
  • The band-permutation invariant classifies the switching behavior in each domain and can be measured experimentally.
  • The domain structure appears in a photonic crystal containing lossy materials.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same trajectory-projection mechanism could be used to move domain boundaries on demand and thereby toggle topological band properties by changing only the modulation amplitude.
  • The construction may generalize beyond two-band models to higher-dimensional or multi-band non-Hermitian systems where EP motion can be engineered.
  • Photonic or acoustic lattices could be designed so that the switching domains act as momentum-space filters or switches whose response is controlled by the modulation strength.
  • Direct imaging of EP trajectories in the extended parameter space would provide an independent test of the projected-boundary picture.

Load-bearing premise

A minimal two-band lattice model suffices to capture the essential topology of mobile exceptional points and the band-permutation invariant remains well-defined and measurable under cyclic modulation.

What would settle it

Mapping the locations of band-switching boundaries in momentum space and checking whether they coincide with the projected trajectories of exceptional points as the modulation amplitude is varied; any systematic mismatch would falsify the claim that mobile EPs generate the domains.

Figures

Figures reproduced from arXiv: 2604.27417 by Chang-Hwan Yi, Jung-Wan Ryu.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic illustration of momentum-space switching view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Momentum-space switching domains generated by mobile EPs. The Brillouin zone is partitioned into regions char view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Eigenvalue braiding associated with the switching view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Photonic realization of momentum-space switching domains. (a) Schematic of the photonic-crystal unit cell consisting view at source ↗
read the original abstract

Exceptional points (EPs), non-Hermitian degeneracies where both eigenvalues and eigenvectors coalesce, play a central role in the topology of non-Hermitian spectra. Recent advances have enabled the controlled creation and manipulation of EPs in a wide range of physical systems, raising the question of what new band topology emerges when EPs become mobile under cyclic modulation. Here we show that mobile EPs generate momentum-space switching domains that partition the Brillouin zone into regions with distinct band-switching behavior. Using a minimal two-band lattice model, we introduce a band-permutation invariant that determines whether eigenmodes exchange after one modulation cycle. The boundaries between switching regions arise from the projection of EP trajectories in an extended parameter space combining crystal momentum and the modulation parameter. As the modulation strength increases, the switching domains expand and eventually cover the entire Brillouin zone, resulting in global band switching. The predicted switching-domain structure is further demonstrated in a photonic crystal with lossy materials. These results open a new avenue within non-Hermitian topology by enabling the engineering of EP-driven phenomena through their controlled motion.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that mobile exceptional points under cyclic modulation generate momentum-space switching domains partitioning the Brillouin zone into regions of distinct band-switching behavior. A minimal two-band lattice model is used to define a band-permutation invariant from the projection of EP trajectories in the extended (k, modulation) parameter space; increasing modulation strength expands these domains until global switching occurs. The structure is demonstrated in a photonic crystal with lossy materials.

Significance. If the central claim holds, the work introduces a new EP-driven mechanism for engineering non-Hermitian band topology and controllable mode switching, with the minimal model providing clear analytic insight and the photonic-crystal example offering a concrete experimental platform. The explicit construction of the invariant and the demonstration of domain expansion are strengths.

major comments (2)
  1. [§3.2] §3.2 (definition of band-permutation invariant): The invariant is constructed by projecting EP trajectories onto the modulation cycle for each k and counting encirclements. This construction assumes the system remains in instantaneous eigenstates, but the manuscript does not analyze the non-adiabatic regime when a modulation path passes arbitrarily close to a mobile EP (where the gap vanishes). A concrete counter-example or numerical check of the final state after one cycle for such paths is needed to confirm the invariant still classifies the switching domains correctly.
  2. [§4] §4 (photonic-crystal demonstration): The numerical results for the lossy photonic crystal show domain formation, but no quantitative comparison is given between the predicted boundaries from the lattice-model invariant and the simulated switching regions. Without this, it is unclear whether the minimal-model invariant survives the additional degrees of freedom and disorder present in the realistic structure.
minor comments (2)
  1. [Figure 2] Figure 2 caption: the color scale for the switching indicator is not defined in the figure or caption; please add an explicit legend or equation reference.
  2. [Eq. (7)] Eq. (7): the symbol for the modulation parameter is introduced without prior definition in the text; a short sentence linking it to the earlier lattice Hamiltonian would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the positive assessment of our work and the constructive suggestions for improvement. Below we provide point-by-point responses to the major comments, indicating the revisions made to the manuscript.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (definition of band-permutation invariant): The invariant is constructed by projecting EP trajectories onto the modulation cycle for each k and counting encirclements. This construction assumes the system remains in instantaneous eigenstates, but the manuscript does not analyze the non-adiabatic regime when a modulation path passes arbitrarily close to a mobile EP (where the gap vanishes). A concrete counter-example or numerical check of the final state after one cycle for such paths is needed to confirm the invariant still classifies the switching domains correctly.

    Authors: We thank the referee for highlighting this important aspect regarding non-adiabatic dynamics near mobile EPs. The band-permutation invariant is topologically defined via the projection of EP trajectories in the extended parameter space and classifies the switching domains based on encirclement. To rigorously address the concern about paths passing close to EPs, we have performed additional numerical simulations of the time-dependent Schrödinger equation in the minimal two-band model for modulation paths that approach the EP arbitrarily closely. These checks, now included in the revised §3.2, demonstrate that the final band permutation after one cycle aligns with the invariant's prediction, provided the modulation is not infinitely fast. This confirms the validity of the domain partitioning even in the presence of potential non-adiabatic effects. revision: yes

  2. Referee: [§4] §4 (photonic-crystal demonstration): The numerical results for the lossy photonic crystal show domain formation, but no quantitative comparison is given between the predicted boundaries from the lattice-model invariant and the simulated switching regions. Without this, it is unclear whether the minimal-model invariant survives the additional degrees of freedom and disorder present in the realistic structure.

    Authors: We agree with the referee that a quantitative comparison is essential to demonstrate the robustness of the invariant in the realistic photonic crystal setting. In the revised manuscript, we have added a direct comparison in §4: the switching domain boundaries predicted by the lattice-model invariant (for parameters calibrated to the photonic crystal) are overlaid on the simulated switching regions obtained from full-wave numerical calculations. The agreement is good, with small discrepancies attributable to the additional photonic degrees of freedom and material losses. This new analysis strengthens the connection between the minimal model and the experimental platform. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation of switching domains

full rationale

The paper's central claim rests on analysis of an explicit minimal two-band lattice model where a band-permutation invariant is introduced to classify eigenmode exchange after modulation cycles. The switching domains are delineated by projections of EP trajectories in the extended parameter space, which follows directly from the model's topology without any reduction to fitted inputs or self-citations. The derivation is self-contained, proceeding from the lattice Hamiltonian to the partitioning of the Brillouin zone, and is further validated in a photonic crystal implementation. No load-bearing steps reduce by construction to the inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The paper rests on the standard definition of exceptional points and a minimal two-band model; the switching domains are derived entities without external falsifiable evidence beyond the model itself.

free parameters (1)
  • modulation strength
    Parameter varied to observe expansion of switching domains until global coverage.
axioms (1)
  • standard math Exceptional points are non-Hermitian degeneracies at which both eigenvalues and eigenvectors coalesce.
    Invoked at the opening to define the objects whose motion is studied.
invented entities (1)
  • momentum-space switching domains no independent evidence
    purpose: Regions of the Brillouin zone distinguished by whether eigenmodes exchange after one modulation cycle.
    New concept introduced as the central observable consequence of mobile EPs.

pith-pipeline@v0.9.0 · 5486 in / 1258 out tokens · 61067 ms · 2026-05-07T09:24:50.766862+00:00 · methodology

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Works this paper leans on

59 extracted references · 59 canonical work pages

  1. [1]

    Kato, Perturbation theory for linear operators; 2nd ed., Grundlehren der mathematischen Wissenschaften : a se- ries of comprehensive studies in mathematics (Springer, Berlin, 1976)

    T. Kato, Perturbation theory for linear operators; 2nd ed., Grundlehren der mathematischen Wissenschaften : a se- ries of comprehensive studies in mathematics (Springer, Berlin, 1976)

    work page 1976

  2. [2]

    W. D. Heiss and A. L. Sannino, Avoided level crossing and exceptional points, J. Phys. A: Math. Gen.23, 1167 (1990)

    work page 1990

  3. [3]

    W. D. Heiss, Exceptional points of non-hermitian oper- ators, Journal of Physics A: Mathematical and General 37, 2455 (2004)

    work page 2004

  4. [4]

    Ryu, S.-Y

    J.-W. Ryu, S.-Y. Lee, and S. W. Kim, Coupled noniden- tical microdisks: Avoided crossing of energy levels and unidirectional far-field emission, Phys. Rev. A79, 053858 (2009)

    work page 2009

  5. [5]

    Dietz, H

    B. Dietz, H. L. Harney, O. N. Kirillov, M. Miski-Oglu, A. Richter, and F. Sch¨ afer, Exceptional points in a mi- crowave billiard with time-reversal invariance violation, Phys. Rev. Lett.106, 150403 (2011)

    work page 2011

  6. [6]

    H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, Topo- logical energy transfer in an optomechanical system with exceptional points, Nature537, 80 (2016)

    work page 2016

  7. [7]

    P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, Orbital angular momentum mi- crolaser, Science353, 464 (2016)

    work page 2016

  8. [8]

    K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, Emergence, coalescence, and topological properties of multiple exceptional points and their experimental re- alization, Phys. Rev. X6, 021007 (2016)

    work page 2016

  9. [9]

    W. Chen, S. K. ¨Ozdemir, G. Zhao, J. Wiersig, and L. Yang, Exceptional points enhance sensing in an op- tical microcavity, Nature548, 192 (2017)

    work page 2017

  10. [10]

    Heiss, Phases of wave functions and level repulsion, The European Physical Journal D7, 1 (1999)

    W. Heiss, Phases of wave functions and level repulsion, The European Physical Journal D7, 1 (1999)

    work page 1999

  11. [11]

    T. Gao, E. Estrecho, K. Y. Bliokh, M. D. Liew, T. C. H. andFraser, S. Brodbeck, M. Kamp, C. Schneider, S. H¨ ofling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, Observa- tion of non-hermitian degeneracies in a chaotic exciton- polariton billiard, Nature526, 554 (2015)

    work page 2015

  12. [12]

    B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljaˇ ci´ c, Spawning rings of exceptional points out of dirac cones, Nature525, 354–358 (2015)

    work page 2015

  13. [13]

    H. Shen, B. Zhen, and L. Fu, Topological band theory for non-hermitian hamiltonians, Phys. Rev. Lett.120, 146402 (2018)

    work page 2018

  14. [14]

    D. J. Luitz and F. Piazza, Exceptional points and the topology of quantum many-body spectra, Phys. Rev. Res.1, 033051 (2019)

    work page 2019

  15. [15]

    Chen, H.-Y

    H.-Z. Chen, H.-Y. L. T. Liu, R.-J. Liu, X.-Y. Wang, X.-F. Zhu, Y.-B. Li, Z.-M. Gu, S.-J. Liang, H. Gao, L. Lu, L. Ge, S. Zhang, J. Zhu, and R.-M. Ma, Revealing the missing dimension at an exceptional point, Nature Physics16, 571 (2020)

    work page 2020

  16. [16]

    W. Tang, X. Jiang, K. Ding, Y.-X. Xiao, Z.-Q. Zhang, C. T. Chan, and G. Ma, Exceptional nexus with a hybrid topological invariant, Science370, 1077 (2020)

    work page 2020

  17. [17]

    E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Ex- ceptional topology of non-hermitian systems, Rev. Mod. Phys.93, 015005 (2021)

    work page 2021

  18. [18]

    Z. Yang, A. P. Schnyder, J. Hu, and C.-K. Chiu, Fermion doubling theorems in two-dimensional non-hermitian sys- tems for fermi points and exceptional points, Phys. Rev. Lett.126, 086401 (2021)

    work page 2021

  19. [19]

    K. Wang, A. Dutt, C. C. Wojcik, and S. Fan, Topological complex-energy braiding of non-hermitian bands, Nature 598, 59 (2021)

    work page 2021

  20. [20]

    Schumer, Y

    A. Schumer, Y. G. N. Liu, J. Leshin, L. Ding, Y. Alah- 6 madi, A. U. Hassan, H. Nasari, S. Rotter, D. N. Christodoulides, P. LiKamWa, and M. Khajavikhan, Topological modes in a laser cavity through exceptional state transfer, Science375, 884 (2022)

    work page 2022

  21. [21]

    Zhang, Y

    Q. Zhang, Y. Li, H. Sun, X. Liu, L. Zhao, X. Feng, X. Fan, and C. Qiu, Observation of acoustic non- hermitian bloch braids and associated topological phase transitions, Phys. Rev. Lett.130, 017201 (2023)

    work page 2023

  22. [22]

    I. I. Arkhipov, F. Minganti, A. Miranowicz, S. K. ¨Ozdemir, and F. Nori, Restoring adiabatic state trans- fer in time-modulated non-hermitian systems, Phys. Rev. Lett.133, 113802 (2024)

    work page 2024

  23. [23]

    J. Chen, Z. Wang, Y.-T. Tan, C. Wang, and J. Ren, Machine learning of knot topology in non-hermitian band braids, Communications Physics7, 209 (2024)

    work page 2024

  24. [24]

    J. Erb, N. Shaibe, R. Calvo, D. P. Lathrop, T. M. Antonsen, T. Kottos, and S. M. Anlage, Topology and manipulation of scattering singularities in complex non- hermitian systems: Two-channel case, Phys. Rev. Res.7, 023090 (2025)

    work page 2025

  25. [25]

    Yi, J.-W

    C.-H. Yi, J.-W. Ryu, T. S. Rodemund, and M. Hentschel, Robust exceptional point chains and chirality switch in a vast optical parameter space, Phys. Rev. A112, L031501 (2025)

    work page 2025

  26. [26]

    A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Observation ofPT-symmetry breaking in complex optical potentials, Phys. Rev. Lett.103, 093902 (2009)

    work page 2009

  27. [27]

    C. E. R¨ uter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Observation of parity–time symmetry in optics, Nature Physics6, 192 (2010)

    work page 2010

  28. [28]

    El-Ganainy, K

    R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non- hermitian physics and pt symmetry, Nature Physics14, 11 (2018)

    work page 2018

  29. [29]

    J. Park, H. Cho, S. Lee, K. Lee, K. Lee, H. C. Park, J.-W. Ryu, N. Park, S. Jeon, and B. Min, Revealing non-hermitian band structure of photonic floquet media, Science Advances8, eabo6220 (2022)

    work page 2022

  30. [30]

    M. S. Ergoktas, S. Soleymani, N. Kakenov, K. Wang, T. B. Smith, G. Bakan, S. Balci, A. Principi, K. S. Novoselov, S. K. Ozdemir, and C. Kocabas, Topological engineering of terahertz light using electrically tunable exceptional point singularities, Science376, 184 (2022)

    work page 2022

  31. [31]

    Shen, Y.-L

    Z. Shen, Y.-L. Zhang, Y. Chen, Y.-F. Xiao, C.-L. Zou, G.-C. Guo, and C.-H. Dong, Nonreciprocal frequency conversion and mode routing in a microresonator, Phys. Rev. Lett.130, 013601 (2023)

    work page 2023

  32. [32]

    W. Zhao, Y. Zhang, Z. Gao, D. Peng, J. long Kou, Y. qing Lu, R. El-Ganainy, S ¸ahin Kaya ¨Ozdemir, and Q. Zhong, Exceptional points induced by unidirectional coupling in electronic circuits, Nature Communications 15, 9907 (2024)

    work page 2024

  33. [33]

    Schindler, A

    J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kot- tos, Experimental study of active lrc circuits withPT symmetries, Phys. Rev. A84, 040101 (2011)

    work page 2011

  34. [34]

    Peng, S ¸ahin Kaya¨Ozdemir, F

    B. Peng, S ¸ahin Kaya¨Ozdemir, F. Lei, F. Monifi, M. Gi- anfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Parity–time-symmetric whispering-gallery mi- crocavities, Nature Physics10, 394–398 (2014)

    work page 2014

  35. [35]

    C. Shi, M. Dubois, Y. Chen, L. Cheng, H. Ramezani, Y. Wang, and X. Zhang, Accessing the exceptional points of parity-time symmetric acoustics, Nat. Commun.7, 11110 (2016)

    work page 2016

  36. [36]

    Brandenbourger, X

    M. Brandenbourger, X. Locsin, E. Lerner, and C. Coulais, Non-reciprocal robotic metamaterials, Nature Communications10, 4608 (2019)

    work page 2019

  37. [37]

    Helbig, T

    T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, Generalized bulk–boundary correspondence in non-hermitian topolectrical circuits, Nature Physics16, 747–750 (2020)

    work page 2020

  38. [38]

    Zhang and C

    X.-L. Zhang and C. T. Chan, Dynamically encircling ex- ceptional points in a three-mode waveguide system, Com- mun. Phys.2, 63 (2019)

    work page 2019

  39. [39]

    Miri and A

    M.-A. Miri and A. Al` u, Exceptional points in optics and photonics, Science363, eaar7709 (2019)

    work page 2019

  40. [40]

    S. K. ¨Ozdemir, S. Rotter, F. Nori, and L. Yang, Par- ity–time symmetry and exceptional points in photonics, Nat. Mater.18, 783 (2019)

    work page 2019

  41. [41]

    Q. Liu, S. Li, B. Wang, S. Ke, C. Qin, K. Wang, W. Liu, D. Gao, P. Berini, and P. Lu, Efficient mode transfer on a compact silicon chip by encircling moving exceptional points, Phys. Rev. Lett.124, 153903 (2020)

    work page 2020

  42. [42]

    Z. Gu, H. Gao, P.-C. Cao, T. Liu, X.-F. Zhu, and J. Zhu, Controlling sound in non-hermitian acoustic systems, Phys. Rev. Appl.16, 057001 (2021)

    work page 2021

  43. [43]

    Yu, X.-L

    F. Yu, X.-L. Zhang, Z.-N. Tian, Q.-D. Chen, and H.-B. Sun, General rules governing the dynamical encircling of an arbitrary number of exceptional points, Phys. Rev. Lett.127, 253901 (2021)

    work page 2021

  44. [44]

    Dembowski, H.-D

    C. Dembowski, H.-D. Gr¨ af, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, Experimen- tal observation of the topological structure of exceptional points, Phys. Rev. Lett.86, 787 (2001)

    work page 2001

  45. [45]

    Dembowski, B

    C. Dembowski, B. Dietz, H.-D. Gr¨ af, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, Encircling an exceptional point, Phys. Rev. E69, 056216 (2004)

    work page 2004

  46. [46]

    Ryu, S.-Y

    J.-W. Ryu, S.-Y. Lee, and S. W. Kim, Analysis of multi- ple exceptional points related to three interacting eigen- modes in a non-hermitian hamiltonian, Phys. Rev. A85, 042101 (2012)

    work page 2012

  47. [47]

    Lee, J.-W

    S.-Y. Lee, J.-W. Ryu, S. W. Kim, and Y. Chung, Geo- metric phase around multiple exceptional points, Phys. Rev. A85, 064103 (2012)

    work page 2012

  48. [48]

    Doppler, A

    J. Doppler, A. A. Mailybaev, J. B¨ ohm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moi- seyev, and S. Rotter, Dynamically encircling an excep- tional point for asymmetric mode switching, Nature537, 76 (2016)

    work page 2016

  49. [49]

    Zhong, M

    Q. Zhong, M. Khajavikhan, D. N. Christodoulides, and R. El-Ganainy, Winding around non-hermitian singular- ities, Nat. Commun.9, 4808 (2018)

    work page 2018

  50. [50]

    J. W. Yoon, Y. Choi, C. Hahn, G. Kim, S. H. Song, K.-Y. Yang, J. Y. Lee, Y. Kim, C. S. Lee, J. K. Shin, H.-S. Lee, and P. Berini, Time-asymmetric loop around an exceptional point over the full optical communications band, Nature562, 86 (2018)

    work page 2018

  51. [51]

    A. Li, J. Dong, J. Wang, Z. Cheng, J. S. Ho, D. Zhang, J. Wen, X.-L. Zhang, C. T. Chan, A. Al` u, C.-W. Qiu, and L. Chen, Hamiltonian hopping for efficient chiral mode switching in encircling exceptional points, Phys. Rev. Lett.125, 187403 (2020)

    work page 2020

  52. [52]

    Ryu, J.-H

    J.-W. Ryu, J.-H. Han, C.-H. Yi, H. C. Park, and 7 M. J. Park, Pseudo-hermitian topology in multiband non-hermitian systems, Phys. Rev. A111, 042205 (2025)

    work page 2025

  53. [53]

    Cerjan, A

    A. Cerjan, A. Raman, and S. Fan, Exceptional contours and band structure design in parity-time symmetric pho- tonic crystals, Phys. Rev. Lett.116, 203902 (2016)

    work page 2016

  54. [54]

    S. Chen, W. Zhang, B. Yang, T. Wu, and X. Zhang, Tai- loring exceptional points with one-dimensional graphene- embedded photonic crystals, Scientific Reports9, 5551 (2019)

    work page 2019

  55. [55]

    Zhong, C

    J. Zhong, C. C. Wojcik, D. Cheng, and S. Fan, Numer- ical and theoretical study of eigenenergy braids in two- dimensional photonic crystals, Phys. Rev. B108, 195413 (2023)

    work page 2023

  56. [56]

    Wiersig, Boundary element method for resonances in dielectric microcavities, Journal of Optics A: Pure and Applied Optics5, 53 (2002)

    J. Wiersig, Boundary element method for resonances in dielectric microcavities, Journal of Optics A: Pure and Applied Optics5, 53 (2002)

    work page 2002

  57. [57]

    Li, Y.-S

    F.-L. Li, Y.-S. Wang, and C. Zhang, Boundary element method for bandgap computation of photonic crystals, Optics Communications285, 527 (2012)

    work page 2012

  58. [58]

    C.-H. Yi, H. C. Park, and M. J. Park, Strong interlayer coupling and stable topological flat bands in twisted bi- layer photonic moir´ esuperlattices, Light: Science & Ap- plications11, 289 (2022)

    work page 2022

  59. [59]

    C.-H. Yi, H. C. Park, and M. J. Park, Bloch theorem dic- tated wave chaos in microcavity crystals, Light: Science & Applications12, 106 (2023). 8 SUPPLEMENT AL MA TERIAL A. Analytic condition and geometric interpretation of the switching boundary The boundary separating switching and nonswitching regions can be obtained analytically from Eqs. (1) and (2) ...

    work page 2023