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REVIEW 4 major objections 4 minor 41 references

Realization of complex conjugate media using non-PT-symmetric photonic crystals

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A photonic crystal without PT symmetry can have real bands and act as a complex conjugate medium.

desk verdict A clean design recipe for real-spectra non-PT photonic crystals, but the 'always' universality claim is broader than the evidence shown. read the letter →

arxiv 1908.04492 v2 pith:DNQVGE4P submitted 2019-08-13 physics.optics

Xiaohan Cui , Kun Ding , Jianwen Dong , C. T. Chan This is my paper · ORCID
classification physics.optics
keywords non-Hermitianphotonicspseudo-HermiticityDirac-likeconephotoniccrystalsexceptionalpointscomplexconjugatemediumeffectivetheorylasing
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 shows that a non-Hermitian photonic crystal need not be PT-symmetric to have real frequency bands. In a two-component crystal whose band structure has a Dirac-like cone at the Brillouin-zone center, one can choose a single loss-gain ratio so that the average non-Hermiticity of each participating eigenstate inside the unit cell is zero; the two-band Hamiltonian then becomes pseudo-Hermitian and its eigenvalues are real outside a ring of exceptional points. In the long-wavelength limit the crystal is described as a complex conjugate medium: the effective refractive index is real, while the effective permittivity and permeability are complex numbers. The same real bands can produce $S$-matrix poles, i.e. a lasing effect, at particular slab thicknesses, and flipping the sign of loss and gain changes the effective parameters by complex conjugation, leading to different scattering behavior.

What carries the argument

The load-bearing object is the two-band non-Hermitian Hamiltonian built from the Hermitian Bloch modes of the photonic crystal. The odd flat band decouples from the two even bands by symmetry, leaving a $2\times2$ matrix whose diagonal entries carry $\tau_\pm$ and whose off-diagonal entries carry $\kappa$. The condition $\tau_\pm=0$ makes the prefactor $\beta$ real and renders the Hamiltonian pseudo-Hermitian with $\eta H(\gamma)\eta^{-1}=H(\gamma)^\dagger=H(-\gamma)$, where $\eta=\mathrm{diag}(W_-^{(0)},-W_+^{(0)})$. The Dirac-like cone does the constructive work: band inversion is claimed to force the overlap functions $F_{\Omega,++}$ and $F_{\Omega,--}$ to cross away from $\Gamma$, so Eq. (12) fixes a real loss-gain ratio $\ell_r$; the eigenvalues then take the square-root form whose branch cut is a ring of exceptional points, with real spectrum outside the ring.

What would settle it

Compute $F_{\Omega,++}-F_{\Omega,--}$ along a $k$-line away from $\Gamma$ for a family of two-component photonic crystals that all have monopole/dipole Dirac-like cones, varying filling fraction and permittivity contrast; a crystal in which this difference never changes sign breaks the 'always' claim. Alternatively, realize the rod-in-air crystal with $\ell_r=-0.15235$ and $\gamma=0.367$ and look for the predicted real band and the lasing singularity at $\omega a/2\pi c=0.5416$ with $d/a=17$; if transmission and reflection stay finite while the effective parameters match, the complex-conjugate-medium mapping is wrong.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that pseudo-Hermiticity, not PT symmetry, is the operative condition for real spectra in a non-Hermitian photonic crystal. Expanding the non-Hermitian operator in the basis of Hermitian Bloch modes and keeping only the two linearly dispersive bands gives a $2\times2$ Hamiltonian whose diagonal average non-Hermiticity $\tau_\pm$ and off-diagonal overlap $\kappa$ control the spectrum. Setting $\tau_\pm=0$ makes the Hamiltonian pseudo-Hermitian, and the eigenvalues are $W_\pm=(1/\beta)(W_d\pm\sqrt{(C_gk)^2(1+\gamma^2|\kappa|^2)-\gamma^2|\kappa|^2W_d^2})$, which are real outside a ring of exceptional points of radius $k_c=k_b(1+|\gamma\kappa|^{-2})^{-1/2}\le k_b$. For a rod-type square-lattice photonic crystal with rod permittivity $12.5+i\gamma$ in air with permittivity $1+i\ell_r\gamma$, choosing $\ell_r=-0.15235$ realizes $\tau_\pm\approx0$ near the Dirac-like cone, and the numerically computed bands match the model. The retrieved effective medium has $n_e^2=\varepsilon_e\mu_e$ real while $\varepsilon_e$ and $\mu_e$ are complex, and a slab of this medium reproduces the lasing poles expected of a complex conjugate medium.

Load-bearing premise

The broad 'always' conclusion depends on the assumption that for every two-component photonic crystal whose Dirac-like cone comes from monopolar and dipolar resonances, the overlap functions $F_{\Omega,++}$ and $F_{\Omega,--}$ cross somewhere away from the $\Gamma$ point so that a single real loss-gain ratio makes $\tau_\pm=0$; the paper says this follows from band inversion and defers the proof to a supplement.

Editorial extensions

If this is right

  • A two-component photonic crystal with a Dirac-like cone formed by monopolar and dipolar modes has a real spectral region once the loss-gain ratio is tuned so that $\tau_\pm=0$, and this does not require PT symmetry in space.
  • In the long-wavelength limit the crystal behaves as a complex conjugate medium with real $n_e$ and complex $\varepsilon_e,\mu_e$; the real refractive index is guaranteed by the real spectrum rather than by simple loss-gain compensation.
  • A slab of thickness $d/a=17$ for $\gamma_+$ (or $d/a=9$ for $\gamma_-$) shows transmission and reflection singularities at $\omega a/2\pi c=0.5416$, which is the predicted lasing signature.
  • The conjugate-paired systems $\gamma_+$ and $\gamma_-$ share nearly identical real bands but have complex-conjugate effective parameters, so their scattering, impedance and lensing behavior differ even though $n_e$ is the same.

Reading between the lines

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

  • If the overlap-function crossing is genuinely forced by band inversion for every monopole/dipole Dirac-like cone, the recipe would work across many lattice geometries and filling ratios; the main text states this and leaves the full proof to the supplement.
  • The same pseudo-Hermitian construction should carry over to acoustic or elastic metamaterials with monopole/dipole Dirac-like cones, since the argument rests on mode symmetry and the zero-average-non-Hermiticity condition rather than on the Maxwell equations specifically.
  • A clean experimental check would be to measure the lasing peak at the predicted frequency and slab thickness, or to map the complex band structure by angle-resolved reflectance; success would validate the effective complex-conjugate-medium picture without requiring a direct extraction of $\varepsilon_e$ and $\mu_e$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies two-dimensional photonic crystals (PCs) with a loss-gain distribution that is not PT-symmetric. The authors build a two-band non-Hermitian Hamiltonian for TM polarization near a Dirac-like cone, show that when the average non-Hermiticity in the unit cell vanishes the Hamiltonian becomes pseudo-Hermitian, and derive a real band spectrum outside a ring of exceptional points. They then extract effective permittivity and permeability from the real bands, finding a complex conjugate medium with real refractive index, and demonstrate S-matrix poles (lasing) for a slab of this effective medium, with COMSOL simulations for one geometry. The central assertion is that such real spectra can always be obtained in two-component PCs carrying Dirac-like cones formed by monopolar and dipolar resonances, because the overlap functions F_Omega,++, F_Omega,-- necessarily cross near the Gamma point, enabling a real loss-gain ratio l_r through Eq. (12).

Significance. If the universality claim holds, the paper offers a conceptually new route to real spectra in non-Hermitian photonic systems beyond PT symmetry and connects it to an effective complex-conjugate medium with real refractive index. The analytical Hamiltonian model and the boundary-field-averaging effective medium calculation agree with full-wave COMSOL results for the presented structure, and the reproduction of S-matrix poles by the effective slab is a clean demonstration. The connection between pseudo-Hermiticity, Dirac-like cones, and CCM behavior is potentially useful for designing gain-loss photonic devices. However, the paper's main theoretical novelty is the 'always' statement, and that statement is the part that is least supported in the main text.

major comments (4)
  1. [Section III A, Eq. (12) and paragraph after Fig. 1B] The universality claim rests on the assertion that F_Omega,++ and F_Omega,-- necessarily cross as k moves away from the Gamma point for any two-component Dirac-like cone PC whose bands arise from monopolar/dipolar resonances. The main text shows a single crossing for rc=0.1999a and epsilon_A=12.5 at kxa/2pi=0.019, and the general proof is deferred to Supplementary Note 6. Since Eq. (12) and the pseudo-Hermitian Hamiltonian (7) require this crossing to realize tau_plus=tau_minus=0 simultaneously for both bands, the 'always' claim is load-bearing. The main text should present the proof or at least the precise hypotheses of the supplementary proof, and an independent numerical example with different material parameters would substantially strengthen the claim.
  2. [Section III A, Eq. (12)] The condition for a single real l_r to make both tau_+ and tau_- vanish is F_A,++/F_B,++ = F_A,--/F_B,--, which is a ratio equality. The text states that 'we must tune the system parameters so that the system satisfies F_Omega,++ = F_Omega,--' and then uses Eq. (12). Equality of the F's is sufficient (given the orthonormality relation, Eq. (3)) but is not necessary. The manuscript should clarify whether the supplementary proof establishes the stronger equality or only the ratio condition, because the universality statement concerns whichever condition is actually required.
  3. [Section III A, paragraph after Fig. 1C; Eq. (9)] The analytical band structure Eq. (9) is derived under the exact conditions tau_plus=tau_minus=0 and k-independent tau_m and kappa. In the actual system l_r is fixed at the crossing point, so at k values away from that point tau_plus and tau_minus are only approximately zero; the text calls them 'almost constant' and 'approximately fulfilled' without a quantitative error bound. The COMSOL agreement in Fig. 2 for gamma=+0.367 supports the approximation for this one structure, but the k-independence and the small-tau assumption are assumptions, not derived results. Please state the range of k and gamma over which the approximation is expected to hold and, ideally, provide an error estimate for the real parts of the eigenfrequencies.
  4. [Abstract and Section IV] The abstract's claim that such PCs 'can always exhibit real spectra as long as the average non-Hermiticity strength... is zero' is conditional: the real spectra follow by construction once l_r is chosen from Eq. (12) to force tau_plus=tau_minus=0. The nontrivial assertion is the existence of a real l_r that realizes this condition for any Dirac-like cone PC of the stated class. The manuscript should be explicit that the prediction is the existence of this parameter and the occurrence of real spectra in a finite k-region, rather than real spectra appearing without parameter tuning. As written, a reader could mistake a designed condition for an emergent phenomenon.
minor comments (4)
  1. [Fig. 1 caption and Section III A] The vertical dashed line in Fig. 1C is described as denoting the intersection F_Omega,++ = F_Omega,--, but the figure shows the average non-Hermiticity tau_m; please clarify on the figure itself which quantity crosses and at which k value, and define the domain Omega in the caption.
  2. [Throughout] There is a typo 'tunning' in Section III A near 'by tunning l_r'; it should read 'tuning'.
  3. [Section II, Eq. (4)] The definition of H as H_1^{-1} H_2 and the role of the scalar beta would be clearer if the text explicitly noted that beta is complex in general but positive real when tau_plus=tau_minus=0, since the pseudo-Hermitian discussion uses that case.
  4. [Fig. 4 caption] In the caption of Fig. 4, the labels for panels (D) and (F) are correct, but the main text refers to 'Figure 4B' for the PC slab schematic while Fig. 4B is also used for the EM/PC transmission and reflection plot; please renumber or refer to the schematic explicitly to avoid ambiguity.

Circularity Check

2 steps flagged · score 4.0 of 10

Real spectra are engineered by tuning 𝓁r to force τ± = 0 by construction, and the 'always' universality is deferred to a self-cited band-inversion guarantee plus Supplementary Note 6; independent COMSOL verification of the single example keeps circularity partial.

  1. fitted input called prediction [Sec. III A, Eqs. (5), (7)-(9), (12) and Fig. 1C]
    "τm =FA,mm +𝓁rFB,mm, (m =±) (5) ... Substituting the orthonormal relationship Eq. (3) into the pseudo-Hermitian criteria τ± = 0, we solve a specific value of the loss-gain ratio 𝓁r [Eq. (12)] ... which is found to be 𝓁r = −0.15235."

    Equation (12) is solved for the loss-gain ratio 𝓁r, and by the definition in Eq. (5) this choice forces τm = FA,mm + 𝓁r FB,mm ≡ 0 identically for m = ±. The pseudo-Hermitian Hamiltonian (7) and the real eigenvalues (9) then follow as direct algebraic corollaries of the imposed zero, so the claimed real-spectrum result is the input condition restated rather than a free prediction. The paper is explicit that the parameter is solved for, which mitigates the charge of a concealed fit, and the COMSOL full-wave bands, EP ring at kc = 0.019, effective εe/µe, and S-matrix lasing poles are computed independently with the same 𝓁r and match the model, so the phenomenon is demonstrated in one realized geometry; hence the circularity is partial, not total.

  2. uniqueness imported from authors [Sec. III A, paragraph after Eq. (12); abstract 'always' claim; Sec. IV Discussion]
    "We can see that there is an intersection (FΩ,++ = FΩ,−−) at kxa/2π = 0.019 (marked by the dashed line) ... Physically, the existence of such an intersection is guaranteed by the band inversion in the Dirac-like cone 25. The condition FΩ,++ = FΩ,−− can always be achieved near the Dirac-like cone (see Supplementary Note 6 and Supplementary Figure S1 for details)."

    The abstract's headline universality claim ('can always exhibit real spectra') rests on the assertion that for any two-component Dirac-like-cone PC the eigenmode overlaps must cross, FΩ,++ = FΩ,−−, so that Eq. (12) can always be applied. The main text exhibits one intersection at kxa/2π = 0.019 for one geometry (rc = 0.1999a, εA = 12.5) and then asserts the crossing is 'guaranteed by the band inversion in the Dirac-like cone', citing ref. [25] (Huang, Lai, Hang, Zheng, and Chan, Nature Materials 2011) — prior work of co-author C. T. Chan — with the actual proof deferred to Supplementary Note 6, which is not in the main text.

full rationale

Score 4, not higher, because the paper contains a genuine independent core: the pseudo-Hermitian mechanism is derived from the two-band model, the realness of Eq. (9) outside the EP ring is an ordinary conditional theorem (external ref. 13 supplies pseudo-Hermiticity), and — decisively — the real bands, EP-ring radius kc = 0.019(2π/a), complex k(ω), effective permittivity/permeability, and lasing poles are all reproduced by independent COMSOL full-wave simulations (Figs. 2-4), so the central phenomenon is not merely a restatement of the model's input in the concrete example. The circularity that remains is: (1) the model-level 'real spectra' is by construction, since 𝓁r = −0.15235 is solved from Eq. (12) exactly to make τ± = 0 (Eq. (5)), and Eqs. (7)-(9) then return the real spectrum as the direct image of that imposed condition; the paper is transparent about the tuning, which is why this is a fitted-input flag rather than a concealed fit; (2) the 'always' universality in the abstract is load-bearing and is justified by a self-citation (ref. [25], co-authored by C. T. Chan) plus a proof deferred to Supplementary Note 6 that is not in the main text, with only one tuned geometry shown. If Supplementary Note 6 contains a complete proof covering all monopole/dipole Dirac-like-cone PCs, finding (2) becomes a benign citation, but as printed the universal claim is imported rather than demonstrated. The k-independence assumption for τm and κ ('almost constant' near the cone) is stated openly and is an approximation risk, not circularity. Weighing the transparent tuning and the extensive independent COMSOL verification, the appropriate score is 4.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central construction depends on one fitted design ratio (l_r), one hand-chosen strength (gamma), and two modeling assumptions (k-independence and the universality of the F-crossing) that are not fully proved in the main text. No new entities are postulated.

free parameters (3)
  • l_r (gain-to-loss ratio) = -0.15235
    Solved from Eq. (12) to enforce the pseudo-Hermitian condition tau_plus = 0 at kx a/2pi = 0.019; the real-spectrum result depends on this choice.
  • gamma (non-Hermitian strength) = +0.367
    Chosen as a representative value for the demonstration; the real spectra exist outside the EP ring for a range of gamma, so it does not encode the result but is a hand-picked parameter.
  • slab thickness d/a = 17.4 for gamma+; 8.9 for gamma-
    Chosen to satisfy the lasing condition Eq. (19); not part of the core band-structure claim but used in the scattering demonstration.
assumptions (4)
  • standard math Pseudo-Hermiticity theorem: a Hamiltonian with a real spectrum is pseudo-Hermitian, and certain pseudo-Hermitian Hamiltonians have real spectra.
    Used to interpret the model Hamiltonian; cited from Mostafazadeh (ref 13).
  • domain assumption The non-Hermitian eigenproblem can be expanded in the complete set of Hermitian (gamma = 0) Bloch eigenfunctions.
    The model Hamiltonian Eq. (1) is built from Hermitian-system Bloch states; completeness and validity at gamma = 0.367 are assumed.
  • ad hoc to paper The overlap quantities tau_m and kappa are independent of k near the Dirac-like cone.
    Stated in Section III.A to simplify the model; the eigenmode profiles are said to be almost unchanged for small k, but no quantitative bound is given.
  • ad hoc to paper For Dirac-like cones from monopolar/dipolar resonances, F_Omega,++ and F_Omega,-- always cross near Gamma due to band inversion.
    Main text asserts this and defers the proof to Supplementary Note 6; it underpins the 'always' claim.

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Cite this review

Pith. "Pith review of Realization of complex conjugate media using non-PT-symmetric photonic crystals." pith.science (2026). https://pith.science/paper/DNQVGE4P

@misc{pith2026190804492,
  author       = {Pith},
  title        = {Pith review of: Realization of complex conjugate media using non-PT-symmetric photonic crystals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DNQVGE4P}},
  note         = {Machine review of arXiv:1908.04492}
}
read the original abstract

While parity-time (PT)-symmetric systems can exhibit real spectra in the exact PT-symmetry regime, the PT-symmetry is actually not a necessary condition for the real spectra. Here we show that non-PT-symmetric photonic crystals carrying Dirac-like cone dispersions can always exhibit real spectra as long as the average non-Hermiticity strength within the unit cell for the eigenstates is zero. By building a non-Hermitian Hamiltonian model, we find that the real spectra of the non-PT-symmetric system can be explained using the concept of pseudo-Hermiticity. We demonstrate using effective medium theories that in the long-wavelength limit, such non-PT-symmetric photonic crystals behave like the so-called complex conjugate medium (CCM) whose refractive index is real but whose permittivity and permeability are complex numbers. The real refractive index for this effective CCM is guaranteed by the real spectrum of the photonic crystals and the complex permittivity and permeability comes from the non-PT-symmetric loss-gain distributions. We show some interesting phenomena associated with the CCM, such as the lasing effect.

Figures

Figures reproduced from arXiv: 1908.04492 by the authors.

Figure 1
Figure 1. FIG. 1. Properties of the Dirac-like cone. (A) Band disper [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Band dispersions of the pseudo-Hermitian bands. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Effective parameters obtained from the real spectra [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Scattering properties of the CCM with effective pa [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.