Pith. sign in

REVIEW 4 major objections 5 minor 39 references

Parity-time symmetry phase transition in photonic time-modulated media

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read In a sinusoidally time-modulated medium, the Floquet spectrum is real only when the imaginary modulation depth exceeds the real one ($\gamma>\gamma_0\approx 1$); below that, pulses grow exponentially.

desk verdict Extends temporal PT symmetry to continuous homogeneous time-modulated media with a plausible qualitative transition, but the quantitative gamma>1 criterion rests on an uncontrolled Magnus truncation and the 'necessity' claim is not proven. read the letter →

arxiv 2507.03337 v1 pith:OD2K7W5E submitted 2025-07-04 physics.optics

Rui-Chuan Zhang , Shu Yang , Yixin Sha , Zetao Xie , Yi Yang This is my paper · ORCID
classification physics.optics
keywords parity-timesymmetrytime-modulatedphotonicstemporalphotoniccrystalsFloquetbandstructurenon-HermitianexceptionalpointMagnusexpansioncomplexpermittivitymodulation
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 claims that a homogeneously time-modulated dielectric can host a genuine parity-time symmetry phase transition: the Floquet spectrum is real in one phase and complex, with exponentially growing waves, in the other. Temporal $\mathcal{PT}$ symmetry, defined by $\epsilon(-t)=\epsilon^*(t)$, is necessary but not sufficient for real bands. For the sinusoidal modulation $\epsilon(t)=\epsilon_i+\epsilon_A[\cos(\omega_m t)+i\gamma\sin(\omega_m t)]$, the spectrum is real only when the imaginary modulation depth exceeds the real one, $\gamma>\gamma_0\approx 1$; below that, a non-Hermitian gap opens. If correct, this provides a temporal counterpart of spatial $\mathcal{PT}$ transitions, with the twist that stronger gain/loss modulation restores Hermiticity.

What carries the argument

The argument is carried by the continuous time-evolution matrix $U(t)$ defined by $dU/dt=U R(t)$, with generator $R(t)=\begin{pmatrix}0&i\omega(t)\\ i\omega(t)&\omega'(t)/\omega(t)\end{pmatrix}$ and $\omega(t;k_z)=c k_z/\sqrt{\epsilon(t)}$; the Floquet bands follow from $\Omega(k_z)T=\cos^{-1}[\operatorname{Tr}(U)/2]$. Temporal $\mathcal{PT}$ symmetry is encoded as pseudo-Hermiticity $M^\dagger=P M P^{-1}$ with $P=\sigma_z$, which makes $\operatorname{Tr}(U)$ real and reduces Hermiticity to $|\operatorname{Tr}(U)|\le 2$. To locate the transition, the authors treat $\hat{P}=\frac12(\epsilon'/\epsilon)\partial_t$ as a perturbation to $\hat{O}$ and use a Magnus expansion: the zeroth order gives $u_{11}+u_{22}=2\cos\theta(t)$, the first-order correction vanishes, and the second-order correction $S_2(t)$ in Eq. (12), built from $L(t)=\epsilon'(t)/\epsilon(t)$, lifts the degeneracy and makes the trace depend on $\gamma$. Substituting Eq. (11) yields the trace expansion (13), from which the criterion $\gamma>\gamma_0\approx 1$ emerges.

What would settle it

Directly integrate $dU/dt=U R(t)$ over one period for the permittivity in Eq. (11), scanning $\gamma$ finely around 1 for several values of $\epsilon_i$ and $\epsilon_A$: if the point where $|\operatorname{Tr}(U)|=2$ is not at $\gamma\approx 1$, or if an independent full-wave simulation shows no exponential growth at $\gamma=0.9$, the central criterion is wrong.

Watch

Extended reading notes

Core claim

At the paper's center is the claim that the Floquet spectrum of a temporal $\mathcal{PT}$-symmetric medium is real exactly when the period evolution matrix satisfies $|\operatorname{Tr}(U)|\le 2$; temporal $\mathcal{PT}$ symmetry alone only guarantees that the trace is real. For the sinusoidally modulated permittivity of Eq. (11), this condition is controlled by the depth ratio $\gamma$: for $\gamma<\gamma_0\approx 1$ the system is in the $\mathcal{PT}$-broken phase with complex quasienergies and exponential pulse growth, at $\gamma=\gamma_0$ the exceptional point gives stationary amplitude, and for $\gamma>\gamma_0$ the band structure is real and pulses propagate with an oscillatory envelope. The reversal relative to spatial $\mathcal{PT}$ optics, where weak gain/loss contrast gives the symmetric phase, is identified as a specific feature of temporal modulation.

Load-bearing premise

The paper's analytic placement of the transition at $\gamma_0\approx 1$ comes from stopping a Magnus expansion at second order, without proving that $\epsilon'/\epsilon$ is small or bounding the neglected third-order terms.

Editorial extensions

If this is right

  • A pulse tuned inside the non-Hermitian gap grows exponentially when $\gamma<1$, stays at constant amplitude near $\gamma=1$, and propagates with a non-monotonic oscillatory envelope when $\gamma>1$.
  • The temporal $\mathcal{PT}$-symmetric phase requires stronger imaginary than real modulation, the opposite of the familiar spatial $\mathcal{PT}$ rule; this reverses design intuition for gain/loss engineering.
  • The continuous transfer-matrix framework extends binary temporal-interface transfer-matrix methods to multi-level and continuous complex modulation, so Floquet bands and transition points can be computed without discretizing the modulation waveform.
  • Because temporal $\mathcal{PT}$ symmetry is necessary but not sufficient, any observation of real bands in such media must also satisfy the $|\operatorname{Tr}(U)|\le 2$ condition; the ratio $\gamma$ is a single knob for switching between amplifying and transparent behavior.

Reading between the lines

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

  • A natural testable extension is to replace the single-frequency modulation of Eq. (11) with square-wave, sawtooth, or multi-harmonic complex modulation and check whether the generalized criterion remains a comparison between net imaginary and real modulation strengths rather than a special property of $\gamma$.
  • The paper's threshold at $\gamma_0\approx 1$ is inferred from a second-order Magnus truncation; a dense numerical scan of $\gamma$ near 1 across different $\epsilon_i/\epsilon_A$ would show whether the critical point is exactly 1 or only approximately so.
  • If the reversed phase condition is generic, time-modulated gain media could be operated in the Hermitian phase by increasing the imaginary modulation depth, effectively using the modulation itself to suppress amplification; this is an editorial extrapolation, not a claim of the paper.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The manuscript studies homogeneous photonic media whose permittivity is periodically modulated in time with complex-valued modulation. It introduces a continuous transfer-matrix / differential-operator framework for arbitrary temporal modulation and defines temporal PT-symmetry as ε(−t)=ε*(t). For sinusoidal modulation with a π/2 phase difference between the real and imaginary parts (Eq. 11), it claims that the Floquet band is real (PT-symmetric/Hermitian phase) when γ>γ0≈1, complex (PT-broken phase) when γ<γ0, with the transition at γ≈1. This threshold is supported by a second-order Magnus expansion (Eq. 13), by direct ODE integration of the monodromy, and by full-wave pulse simulations at γ=0.9, 1.0, and 1.1.

Significance. If established, the paper would provide a clean temporal analogue of spatial PT-symmetric photonics, extending prior work on temporal interfaces to continuous modulations of both the real and imaginary parts of permittivity. The paper has real strengths: the continuous differential-operator formulation of the transfer matrix is a useful generalization; the pseudo-Hermiticity argument leading to a real trace under ε(−t)=ε*(t) is elegant; and the predicted distinction between exponential growth (γ<γ0), stationary amplitude (γ≈γ0), and oscillatory decay (γ>γ0) is checked by independent full-wave simulations. The main weaknesses are the quantitative control of the Magnus expansion that fixes γ0≈1 and the absence of a proof of the claimed necessity direction.

major comments (4)
  1. [Magnus expansion / Eq. (13)] The threshold γ0≈1 is derived from a second-order Magnus expansion in which L(t)=ε'(t)/ε(t) is treated as a perturbation. No estimate of the O(L^3) remainder is given, and the perturbation is not small for the parameters actually used in Fig. 3: with ε_i=1.8, ε_A=1, and k_z≈k0=ω_m/(2πc), one has sup_t |L(t)| = ω_m/(ε_i−ε_A) = 1.25 ω_m while the instantaneous frequency is ≈0.1ω_m, so |L|/ω is of order 10. The agreement with the direct ODE solutions in Fig. 3(a) is suggestive, but without a remainder bound, a resummation, or an independent exact computation of γ0, Eq. (13) does not establish the quantitative criterion γ>1 with the claimed precision.
  2. [Fig. 3 and numerical verification] The phase boundary is sampled only at γ=0.9, 1.0, and 1.1. These three points do not discriminate between γ0=1.00 and, say, γ0=1.08: the classification of the shown cases would be unchanged, but the asserted quantitative boundary would be incorrect in the interval (1, γ0). Please report γ0 computed from the exact monodromy (or from a converged high-order expansion) for the parameters used, or state the uncertainty in γ0 explicitly.
  3. [Abstract and Conclusion] The abstract and conclusion claim that temporal PT-symmetry is a necessary condition for a real spectrum, i.e., that real spectra occur only under ε(−t)=ε*(t). The main text demonstrates only the sufficiency direction: Eq. (3) and the surrounding discussion show that ε(−t)=ε*(t) implies Tr M ∈ R, not that every real-spectrum modulation must satisfy this condition. Since 'necessary but insufficient' is a headline claim, please either provide the converse proof, restrict the claim explicitly, or cite a precise theorem and include its proof in an accessible appendix.
  4. [References to SM S3/S5/S6] The version under review repeatedly directs the reader to the Supplemental Material for load-bearing items: the proof of realness of the trace (SM S3), the identity in Eq. (10) (SM S5), and the exact threshold γ0 (SM S6). No Supplemental Material is included in the submitted version. The manuscript is not fully evaluable without these derivations; please include the SM or move the key steps into the main text.
minor comments (5)
  1. [Eq. (2)] The matrix product in Eq. (2) is difficult to parse: the placement of the factor 1/2, the role of the intermediate matrix U, and the final matrix product should be displayed more explicitly, preferably with indices or a short derivation of the n-state case.
  2. [Fig. 2 caption] The phrase 'the first extremum of Tr(U)/2−kz' is unclear; please specify which quantity is extremized (presumably Tr(U) as a function of k_z) and what 'first' refers to.
  3. [γ0 vs γ=1.0] The text uses both γ0≈1 and, in describing Figs. 3 and 4, treats γ=1.0 as the exceptional point. Please state explicitly whether γ0=1 exactly for these parameters or only approximately, and with what numerical accuracy this was determined.
  4. [Eq. (10)] The claim that the operator O annihilates cos(∫_0^t ω(τ)dτ) 'regardless of the specific form of the modulation' is surprising and is deferred to SM S5; a one-line derivation in the main text would improve readability.
  5. [Nomenclature] The term 'temporal PT-symmetry' is defined only as ε(−t)=ε*(t), which involves time reversal and complex conjugation but no spatial parity. The connection to the usual P operator in spatial PT-symmetric photonics should be clarified to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PT transition is derived from the monodromy trace via pseudo-Hermiticity and an independent Magnus/numerical evaluation, with no fitted parameter renamed as a prediction.

full rationale

The central derivation chain is self-contained. Temporal PT symmetry, epsilon(-t)=epsilon*(t), is used to derive the pseudo-Hermiticity relation M†=sigma_z M sigma_z^{-1} (Eq. 3), which yields a real trace Tr(U). The Hermitian regime is then defined by the standard condition |Tr(U)|<=2 (around Eq. 5). The transition threshold gamma0≈1 is obtained by evaluating the second-order Magnus expansion Eq. (13) and checked against direct numerical ODE integration (Fig. 3), not by fitting to the pulse simulation. The full-wave simulations in Fig. 4 are predicted from the independently computed Floquet band structure and then validated, so no target quantity is used as an input. The gamma values {0.9, 1.0, 1.1} are illustrative sample points chosen after the regime classification, not fitted parameters. The main limitation is that the exact threshold derivation is deferred to the non-included SM S6 and the O(L^3) truncation is uncontrolled; that is a rigor and verification concern, not circularity. There are no load-bearing self-citations by the present authors.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new particles, forces, or entities are introduced. The free parameters are model inputs rather than fitted constants; no number is adjusted to force agreement. The main hidden assumption is the validity of the truncated Magnus expansion, which is the most fragile part of the analytic argument.

free parameters (3)
  • gamma (imaginary-to-real modulation depth contrast) = scanned over {0.9, 1.0, 1.1}; not fitted to data
    Central control parameter of the PT transition; the threshold gamma0 approximately 1 is the paper's result, not an input.
  • epsilon_i (background permittivity) = 1.8 in all numerical examples
    Chosen to keep Re epsilon(t) positive for epsilon_A=1; no evidence is provided that the threshold changes qualitatively with this value.
  • epsilon_A (real modulation depth) = set to 1 as normalization
    Normalized without loss of generality; it sets the scale of L(t)=epsilon'/epsilon used in the Magnus expansion.
assumptions (5)
  • domain assumption Maxwell's equations in a homogeneous, nonmagnetic, dispersion-free medium with time-varying complex permittivity, with wavevector kz conserved under temporal modulation.
    Underlies the entire transfer-matrix and differential-operator framework (Eqs. 2-6); standard in photonic time-crystal literature but not derived here.
  • domain assumption Temporal PT symmetry is expressed by the pseudo-Hermiticity relation M† = sigma_z M sigma_z^{-1} (Eq. 3) with P=sigma_z.
    This is the bridge from epsilon(-t)=epsilon*(t) to a real trace; the proof is deferred to SM S3 and not fully shown in the main text.
  • domain assumption The instantaneous frequency omega(t;kz)=c kz/sqrt(epsilon(t)) and the first-order generator R(t) in Eq. (4) correctly represent the continuous limit of the discrete temporal-interface transfer matrix method.
    Required for the differential formulation and for extracting Floquet exponents from Tr(U) via Eq. (5).
  • standard math The Floquet spectrum is Hermitian iff Tr(U) is real and |Tr(U)| <= 2 (Eq. 5).
    Standard monodromy result for a 2x2 transfer matrix; used to define the PT phase transition.
  • ad hoc to paper The second-order Magnus expansion (Eqs. 12-13) is sufficient to locate the transition threshold gamma0 approximately 1, with no proof of convergence or bound on neglected terms.
    The paper's analytic derivation of the phase transition relies on this truncation; the exact threshold is not derived in the main text.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Parity-time symmetry phase transition in photonic time-modulated media." pith.science (2026). https://pith.science/paper/OD2K7W5E

@misc{pith2026250703337,
  author       = {Pith},
  title        = {Pith review of: Parity-time symmetry phase transition in photonic time-modulated media},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OD2K7W5E}},
  note         = {Machine review of arXiv:2507.03337}
}
abstract

Time modulation can cause gain and loss in photonic media, leading to complex modal behaviors and enhanced wave controllability in the non-Hermitian regime. Conversely, we reveal that Hermiticity and parity-time $\mathcal{PT}$-symmetry phase transition are possible under the temporal $\mathcal{PT}$-symmetry in time-modulated photonic media. We prove that, for a homogeneously modulated photonic medium with complex-valued modulation, temporal $\mathcal{PT}$-symmetry is a necessary but insufficient condition for obtaining a real eigenvalue spectrum, giving rise to $\mathcal{PT}$-symmetry phase transition. Specifically, the $\mathcal{PT}$ phase transition critically depends on the contrast between the modulation depth of the real and imaginary parts of permittivity when they are sinusoidally modulated with a $\pi/2$ phase difference. We generalize the discretized temporal-interface transfer matrix method to a continuous differential operator framework, which facilitates the confirmation of the phase transition condition via Magnus expansion analysis. Full-wave simulations and analytical calculations jointly confirm the occurrence of $\mathcal{PT}$-transition by examining the scattering behavior of a propagating pulse in such a type of modulated medium. The findings provide a temporal $\mathcal{PT}$-symmetric paradigm for controlling Hermiticity and non-Hermiticity in spatiotemporal photonic systems.

Figures

Figures reproduced from arXiv: 2507.03337 by the authors.

Figure 1
Figure 1. PT-symmetry in spatial and temporal photonic crys￾tals. (a-b) Sinusoidal spatial (a) and temporal (b) modulation of per￾mittivity that obeys PT-symmetry. The spatial and temporal period￾icity are noted as R and T, respectively. A time-domain version of the PT-symmetry in a homogen￾eous but time-varying photonic medium of permittivity ϵ(t) could be defined as ϵ(−t) = ϵ ∗ (t), where the asterisk denotes arXiv:2507.033… view at source ↗
Figure 2
Figure 2. Temporal PT-symmetric and PT-broken behavior un￾der sinusoidal modulation of ϵ(t). (a-b) Complex-plane trajector￾ies of ϵ(t) under sinusoidal modulation. In the convergent case (a), Re ϵ(t) and Im ϵ(t) oscillate at the same frequency with a phase dif￾ference of π/2, forming PT-symmetric Hermitian (blue ellipse) and PT-broken non-Hermitian (orange ellipse) regions. This behavior corresponds to convergent bounded osci… view at source ↗
Figure 3
Figure 3. Magnus expansion analysis of the PT phase transition and band structures. (a) Open circles, dashed lines, and solid lines are direct numerical ODE solutions, zeroth-order, and second-order Magnus analysis results, respectively. Fixed parameters: ϵi = 1.8, k0 = ωm/(2πc). The horizontal green line indicates Tr(U)/2 = −1. (b-d) Blue solid lines show the real parts of the Ω(kz)T, while red dashed lines represent the ima… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Validation of the appearance of the temporal PT-symmetric transition. (a) Schematic of the simulation setup, showing the propagation of an electromagnetic pulse in a dielectric medium with time-dependent permittivity ϵ(t). The monitor at z0 = 87.3 µm records the wavefo…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

39 extracted references · 34 canonical work pages

  1. [1]

    Yablonovitch, Physical review letters 58, 2059 (1987)

    E. Yablonovitch, Physical review letters 58, 2059 (1987)

    work page 1987

  2. [2]

    John, Physical Review Letters 58, 2486 (1987)

    S. John, Physical Review Letters 58, 2486 (1987)

    work page 1987

  3. [3]

    J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light , 2nd ed. (Princeton University Press, Princeton, NJ, 2008)

    work page 2008

  4. [4]

    C. M. Bender and S. Boettcher, Physical Review Letters 80, 5243 (1998)

    1998

  5. [5]

    C. M. Bender, Reports on Progress in Physics 70, 947 (2007)

    2007

  6. [6]

    Regensburger, M

    A. Regensburger, M. A. Miri, C. Bersch, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature 488, 167 (2012)

    work page 2012

  7. [7]

    L. Feng, R. El-Ganainy, and L. Ge, Nature Photonics 11, 752 (2017)

    2017

  8. [8]

    K.¨Ozdemir, S

    S ¸. K.¨Ozdemir, S. Rotter, F. Nori, and L. Yang, Nature materials 18, 783 (2019)

    work page 2019

Show all 39 references
  1. [9]

    Alaeian and J

    H. Alaeian and J. A. Dionne, Physical Review B 89, 075136 (2014)

    2014

  2. [10]

    Cerjan, A

    A. Cerjan, A. Raman, and S. Fan, Physical Review Letters 116, 203902 (2016)

    2016

  3. [11]

    Mock, Physical Review A 93, 063812 (2016)

    A. Mock, Physical Review A 93, 063812 (2016)

    2016

  4. [12]

    Longhi, Europhysics Letters 120, 64001 (2018)

    S. Longhi, Europhysics Letters 120, 64001 (2018)

    2018

  5. [13]

    S. K. Gupta, Y . Zou, X.-Y . Zhu, M.-H. Lu, L.-J. Zhang, X.-P. Liu, and Y .-F. Chen, Advanced Materials32, 1903639 (2020)

    2020

  6. [14]

    C. Wang, Z. Fu, W. Mao, J. Qie, and A. D. Stone, Advanced Optical Materials (2023), 10.1364 /AOP-15-2-442

    2023

  7. [15]

    Gali ffi, R

    E. Gali ffi, R. Tirole, S. Yin, H. Li, S. Vezzoli, P. A. Huidobro, M. G. Silveirinha, R. Sapienza, A. Al `u, and J. B. Pendry, Ad- vanced Photonics 4, 014002 (2022)

    2022

  8. [16]

    Lustig, O

    E. Lustig, O. Segal, S. Saha, C. Fruhling, V . M. Shalaev, A. Bol- tasseva, and M. Segev, Opt. Express 31, 9165 (2023)

    2023

  9. [17]

    Lustig, Y

    E. Lustig, Y . Sharabi, and M. Segev, Optica 5, 1390 (2018)

    2018

  10. [18]

    Hayran, J

    Z. Hayran, J. B. Khurgin, and F. Monticone, Optical Materials Express 12, 3904 (2022)

    2022

  11. [19]

    Pacheco-Pe˜na, D

    V . Pacheco-Pe˜na, D. M. Sol´ıs, and N. Engheta, Optical Mater- ials Express 12, 3829 (2022)

    2022

  12. [20]

    M. M. Asgari, P. Garg, X. Wang, M. S. Mirmoosa, C. Rock- stuhl, and V . Asadchy, Advances in Optics and Photonics 16, 958 (2024)

    2024

  13. [21]

    Lyubarov, Y

    M. Lyubarov, Y . Lumer, A. Dikopoltsev, E. Lustig, Y . Sharabi, and M. Segev, Science 377, 425 (2022)

    2022

  14. [22]

    Moussa, G

    H. Moussa, G. Xu, S. Yin, E. Gali ffi, Y . Ra’di, and A. Al `u, Nature Physics 19, 863 (2023)

    2023

  15. [23]

    Tirole, S

    R. Tirole, S. Vezzoli, E. Gali ffi, I. Robertson, D. Maurice, B. Tilmann, S. A. Maier, J. B. Pendry, and R. Sapienza, Nature Physics 19, 999 (2023)

    2023

  16. [24]

    Y . Zhou, M. Z. Alam, M. Karimi, J. Upham, O. Reshef, C. Liu, A. E. Willner, and R. W. Boyd, Nature Communications 11, 2180 (2020)

    2020

  17. [25]

    Lustig, O

    E. Lustig, O. Segal, S. Saha, E. Bordo, S. N. Chowdhury, Y . Sharabi, A. Fleischer, A. Boltasseva, O. Cohen, V . M. Shalaev, and M. Segev, Nanophotonics 12, 2221 (2023)

    2023

  18. [26]

    X. Wang, M. S. Mirmoosa, V . S. Asadchy, C. Rockstuhl, S. Fan, and S. A. Tretyakov, Science Advances9, eadg7541 (2023)

    2023

  19. [27]

    S. A. Horsley and J. B. Pendry, Proceedings of the National Academy of Sciences 120, e2302652120 (2023)

    2023

  20. [28]

    Gali ffi, P

    E. Gali ffi, P. Huidobro, and J. B. Pendry, Physical Review Let- ters 123, 206101 (2019)

    2019

  21. [29]

    Franke, J

    S. Franke, J. Ren, M. Richter, A. Knorr, and S. Hughes, Phys- ical Review Letters 127, 013602 (2021)

    2021

  22. [30]

    Tirole, S

    R. Tirole, S. Vezzoli, D. Saxena, S. Yang, T. V . Raziman, E. Galiffi, S. A. Maier, J. B. Pendry, and R. Sapienza, Nature Communications 15, 7752 (2024)

    2024

  23. [31]

    J. Park, H. Cho, S. Lee, K. Lee, K. Lee, H. C. Park, J.-W. Ryu, N. Park, S. Jeon, and B. Min, Science Advances 8, eabo6220 (2022)

    2022

  24. [32]

    Yu and S

    R. Yu and S. Fan, Proceedings of the National Academy of Sci- ences 121, e2401514121 (2024)

    2024

  25. [33]

    J. Park, K. Lee, R.-Y . Zhang, H.-C. Park, J.-W. Ryu, G. Y . Cho, M. Y . Lee, Z. Zhang, N. Park, W. Jeon, et al. , arXiv preprint arXiv:2404.13287 (2024)

    2024

  26. [34]

    J. Bae, K. Lee, B. Min, and K. W. Kim, arXiv preprint arXiv:2501.03106 (2025), 10.48550 /arXiv.2501.03106

    2025 doi

  27. [35]

    J. E. Sustaeta-Osuna, F. J. Garc´ıa-Vidal, and P. Huidobro, ACS photonics 12, 1873 (2025)

    2025

  28. [36]

    H. Li, S. Yin, E. Gali ffi, and A. Al `u, Physical Review Letters 127, 153903 (2021)

    2021

  29. [37]

    Mostafazadeh, Journal of Mathematical Physics 43, 205 (2002)

    A. Mostafazadeh, Journal of Mathematical Physics 43, 205 (2002)

    2002

  30. [38]

    C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, 270401 (2002)

    2002

  31. [39]

    Magnus, Communications on Pure and Applied Mathemat- ics 7, 649 (1954)

    W. Magnus, Communications on Pure and Applied Mathemat- ics 7, 649 (1954)

    1954

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.