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REVIEW 4 major objections 5 minor 97 references

Exceptional horns in $n$-root graphene and Lieb photonic ring lattices

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Pith's one-line read Dirac cones collapse into exceptional horns in n-root lattices

desk verdict Genuinely new n-root construction with exceptional horns and root Landau-level scaling, but the magnetic-flux block structure and the Jordan decomposition are asserted rather than proven. read the letter →

arxiv 2602.16670 v2 pith:6GPCMI7S submitted 2026-02-18 cond-mat.mes-hall cond-mat.other

A. M. Marques , D. Viedma , V. Ahufinger , R. G. Dias This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.other
keywords non-Hermitianlatticesn-rootmodelsexceptionalpointsDiracconesLieblatticegrapheneLandaulevelsphotonicringresonators
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 lattice can be built so that its Hamiltonian is an nth root of a Hermitian parent lattice — graphene or the Lieb lattice — and that this root operation transforms the parent's Dirac cones in a controlled way. Replacing every parent hopping with a loop module of unidirectional couplings makes the nth power of the Hamiltonian block-diagonal, with the parent model as the first block; the full spectrum then consists of n rotated branches whose nth powers reproduce the parent's real bands, plus flat bands at zero energy. The central result is that a carefully chosen phase configuration on the couplings can cancel the parent's energy offset, turning each Dirac point into a zero-energy exceptional point of order n (or higher) around which energy scales as E ~ |q|^{1/n}, a sublinear regime the authors call an exceptional horn. The same algebraic recipe predicts Landau levels that scale with magnetic flux as φ^{1/(2n)}, with an exceptional zeroth Landau level for the n-root Lieb lattice. The models are matched to a photonic ring-resonator design, so the predicted horns and Landau scaling are experimentally addressable.

What carries the argument

The loop module — a replacement of each parent hopping by n unidirectional couplings of strength J^{1/n} passing through n−1 intermediate sites — is the construction that makes H^n block-diagonal, with the Hermitian parent as the first block. The generalized chiral symmetry C_n (defined by C_n H C_n^{-1} = ω_n^{-1} H, ω_n = e^{2πi/n}) is what forces the n-branch rotation of the spectrum. The phase configuration of the loop couplings is the control knob: it either leaves the parent's energy offset (case i of the perturbation theory in Appendix A, giving Dirac points with renormalized Fermi velocity) or cancels it (case ii, giving zero-energy exceptional points and |q|^{1/n} dispersion). The n

What would settle it

In the photonic 3-root graphene design, zoom into the p=0 branch near the K point: the exceptional-horn claim requires the two approaching bands to separate as |q|^{1/3}; observing any different power law (e.g., linear) would falsify it. Separately, compute the Hofstadter spectrum with flux also threading the loop modules: if the nth power of the butterfly no longer collapses onto the parent graphene (or Lieb) butterfly, the block-diagonal assumption that all Landau-level results rest on is broken.

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Extended reading notes

Core claim

For n≥3, the n-root graphene and n-root Lieb lattices are formed by replacing each hopping of the parent with a closed loop of n unidirectional couplings of magnitude J^{1/n}. Because the generalized chiral symmetry C_n imposes that every finite-energy eigenvalue appears in the n-tuple {ω_n^p E}, the spectrum is a union of n branches whose nth powers coincide with the parent's real bands, plus zero-energy flat bands protected by sublattice imbalance. The paper's central discovery is that the fate of the parent's Dirac cone depends on a phase configuration on the loop couplings. If the parent block retains a finite energy offset (3J for graphene), the n-root spectrum has an n-fold set of Dira

Load-bearing premise

The whole magnetic-field analysis assumes the Peierls phase string places flux only on the large plaquettes of the n-root lattice, leaving the loop modules flux-free, so that the nth power of the flux-inserted Hamiltonian still has the Hermitian parent as a block; if that ansatz fails, the derived Landau-level scaling E ~ φ^{1/(2n)} and the exceptional zeroth Landau level are unsupported.

Editorial extensions

If this is right

  • The n-root construction produces n rotated branches in the complex energy plane whose nth powers match the parent's real spectrum, so the parent's physics is recovered after an nth-power operation.
  • With the parent energy offset intact, each Dirac point survives as an n-tuple of Dirac points with a renormalized Fermi velocity; with the offset cancelled, it becomes a zero-energy exceptional point of order n or higher.
  • The dispersion E ~ |q|^{1/n} around the exceptional horns implies a formally infinite Fermi velocity as q→0, in contrast to the constant velocity of a Dirac cone.
  • Landau levels follow E ~ φ^{1/(2n)} around the exceptional region, and the n-root Lieb lattice has an exceptional zeroth Landau level.
  • Because the generalized chiral symmetry is unbroken by the flux, the zero-energy flat bands remain pinned and mix with the exceptional-point eigenvectors only at the high-symmetry point.

Reading between the lines

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

  • If the photonic implementation is realized, the |q|^{1/n} dispersion near an exceptional horn should make the local density of optical states diverge at zero energy, which could be probed as an enhanced emission or absorption signal at the K point.
  • The same loop-module recipe should transplant the mechanism to other parent models with Dirac or nodal degeneracies, such as Weyl semimetals or the Haldane model; the authors list these targets but leave them unexplored.
  • Because the exceptional Landau level is pinned at zero energy by symmetry, it may offer a magnetic-field-tunable, symmetry-protected degeneracy that is robust to disorder — a testable alternative to fine-tuned EP sensors.
  • The predicted dependence of the zeroth LL on flux (flat at zero) could be checked in circuits or coupled-ring arrays without needing perfect unidirectionality, since the flatness only relies on the block-diagonal structure.
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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 / 5 minor

Summary. This paper constructs n-th root tight-binding models (n>=3) for two 2D parent lattices, graphene and the Lieb lattice, built from loop modules of unidirectional couplings. The authors show that the Bloch spectrum consists of n rotated copies of the n-th root of the parent spectrum plus zero-energy flat bands protected by a generalized chiral symmetry. With a suitable phase configuration, the finite-energy Dirac points of the parent can be shifted to zero energy, converting the Dirac cones into 'exceptional horns': zero-energy exceptional points of order n (or higher) with dispersion E ~ |q|^(1/n). The same n-root relation is used to predict Landau-level energies scaling as phi^(1/(2n)) under a uniform magnetic field; for the n-root Lieb lattice a zero-energy exceptional Landau level is identified. A photonic ring-resonator implementation for n=3 is proposed and studied with finite-element simulations.

Significance. Should the results hold, the paper provides a clean, parameter-free algebraic route to high-order exceptional points in two-dimensional lattices, with distinctive sublinear dispersion and unusual Landau-level flux scaling. The spectrum-rooting construction is exact, and the low-energy perturbation theory in Appendix A is analytic and general. The explicit photonic implementation, including quantitative modeling of imperfections in Appendix D, is a strong practical contribution. The main limitations are the unproven Peierls-substitution ansatz and the unproven Jordan-form statements, which gate the Landau-level and EP-order claims.

major comments (4)
  1. [Sec. II B, first paragraph; Sec. III A, first paragraph] The statement that imposing no flux through the loop modules ensures that the n-th power of the flux-inserted Hamiltonian has the parent Hofstadter Hamiltonian as its H1 block is an assertion, not a derivation. For each loop module the zero-flux condition fixes only the product of Peierls phases around that module, not the phases of the individual h_l entries. The matrix elements of H1 = h1...hn can therefore pick up k-independent or flux-dependent corrections that would change the parent Landau-level spectrum. The LL energies in (21) and (33) and the exceptional LL in Sec. III A depend on this block structure. Please provide an explicit check (e.g., compute H^n for a Peierls-substituted n=3 or n=4 model and verify the H1 block) or derive the gauge constraints under which the identity holds. The numerics in Figs. 5 and 8 support the ansatz only for low flux and specific n.
  2. [Sec. II A (Jordan form after Eq. (18)); Sec. III (Jordan form before Fig. 7)] The Jordan decompositions J = J_n(0) ⊕ J_{2n-1}(0) for n-root graphene and J = J_1(0) ⊕ J_{2n-1}(0) ⊕ J_{2n-1}(0) for n-root Lieb are central to the claimed EP orders, but they are stated without proof. Without a derivation or construction of generalized eigenvectors, the order of the exceptional points is an assertion rather than a demonstrated result. Please provide a proof or a direct computation of the Jordan form for general n.
  3. [Sec. II A (Fig. 2(a)) and paragraph after Eq. (18)] The phase configuration that removes the 3J offset in n-root graphene is explicitly shown only for n=3. Since the exceptional-horn phenomenon and the Landau-level analysis are claimed for general n, an explicit general-n phase assignment (analogous to Eq. (24) for the Lieb case) or a proof of its existence is required. The statement that the self-loop contributions cancel for general n is not demonstrated.
  4. [Sec. III A (around Eq. (36))] The exceptional Landau level result rests on the uniqueness of the zero-energy solution at each LL: 'it is possible to show that there is a single exact solution at each LL'. This is not proven. If there are multiple linearly independent zero-energy states at the m-th LL, they do not coalesce and the level is not exceptional. Please provide the proof of uniqueness or state the result as a conjecture. The lower-bound statement 'at least order n' is acceptable, but the underlying uniqueness must be established.
minor comments (5)
  1. [Eq. (12)] The eigenvector formula uses E_± in the denominators and is singular at E=0; this is appropriate since these eigenvectors apply to non-zero eigenvalues, but a brief remark would avoid confusion.
  2. [Sec. II B, Eq. (14)] The Landau-level substitution is applied to h_n and h_1; the ordering of b and b^† for the K' valley should be checked or briefly explained.
  3. [Sec. II A] The name 'exceptional horn' is evocative but the connection to Gabriel's horn could be clarified: the horn refers to the divergent derivative (infinite group velocity) at the EP, not to a geometric shape of the spectrum.
  4. [Appendix D, Eq. (D1)] The notation H(phi) -> H^†(-phi) is clear in context, but a one-sentence derivation would improve readability.
  5. [General] The paper relies heavily on the authors' own generalized index theorem [47]. A short recapitulation of the theorem's statement would make the paper more self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central scalings are algebraic consequences of the explicit n-root construction, with design conditions and external benchmarks rather than fitted predictions.

full rationale

The claimed derivation chain is self-contained. The spectral relation E^(p)_±(k)=ω^p_n E^{1/n}_{1,±}(k) (Eq. 11) follows from the explicitly written H^n(k)=diag(H1,...,Hn) (Eqs. 6-8), which is an algebraic identity for the h_j blocks (Eqs. 2-4), not from any fitted parameter. The exceptional-horn dispersion E±(q)=±ω_n^{p/2}(v_F|q|)^{1/n} (Eq. 18, App. A case ii) is the Taylor expansion of (E0±v_F|q|)^{1/n} around E0=0; the zero of E0 is engineered by the explicit phase configuration whose self-energy contributions sum to J(1+e^{i2π/3}+e^{-i2π/3})=0 (Sec. II A). The LL formulas (21) and (33) are nth roots of the standard graphene [80] and Lieb [82] LL spectra, and are cross-checked numerically in Figs. 5(c) and 8(b) rather than fitted. The only assumption that could invalidate the LL scaling is the Peierls condition 'no flux goes through the loop modules' (Sec. II B and III A); this is an explicit, testable ansatz, not a circular reuse of the target result — if the bracket structure fails, the parent-block correspondence is lost, but the paper does not define the parent block in terms of the LL energies. Self-citations [38] and [47] supply the loop-module construction and the generalized index theorem; both are prior published results with stated assumptions and are used as ingredients (flat-band counting, chiral symmetry), not as the conclusion being claimed. No step reduces a prediction to an input by construction beyond the intended algebraic n-root identity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central spectrum-rooting is an exact algebraic property of the loop-module construction, so the paper contributes the phase-engineering mechanism and the LL/ELL analysis. No free parameters enter the ideal-model claims; the only fitted parameters appear in the deviation modeling for photonic simulations.

free parameters (2)
  • α (backward coupling ratio) = ≈0.008 (graphene), ≈0.007 (Lieb)
    Chosen to model imperfect unidirectionality in the photonic simulations; not a parameter of the ideal model.
  • J_cc (cross-circulation coupling) = 4×10^{-3} J
    Hand-picked in Appendix D to qualitatively reproduce the deviations seen in COMSOL spectra.
assumptions (4)
  • domain assumption Generalized index theorem for non-Hermitian n-partite lattices (Ref. [47])
    Used to count the n−1 flat bands and set the lower bound on zero-energy states; taken from the authors' prior work without re-derivation.
  • standard math The nth power of the n-root Hamiltonian is block diagonal with the parent Hamiltonian as the first block (Eqs. (6)-(9))
    Follows from the loop-module construction in Sec. II, but is ultimately an algebraic identity for the idealized unidirectional model.
  • ad hoc to paper Peierls substitution on the n-root lattice with flux only through plaquettes yields the parent Hamiltonian with uniform flux in H1
    Invoked in Sec. II B and III A to justify using the known parent LL solutions; not derived from the microscopic tight-binding Hamiltonian.
  • standard math Perturbation theory for n-partite lattices (Appendix A) assuming a single massless symmetric cone in the parent
    Generalizes Ref. [96]; valid for graphene and Lieb parents.

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Pith. "Pith review of Exceptional horns in $n$-root graphene and Lieb photonic ring lattices." pith.science (2026). https://pith.science/paper/6GPCMI7S

@misc{pith2026260216670,
  author       = {Pith},
  title        = {Pith review of: Exceptional horns in $n$-root graphene and Lieb photonic ring lattices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6GPCMI7S}},
  note         = {Machine review of arXiv:2602.16670}
}
abstract

We present a systematic construction of non-Hermitian tight-binding lattices whose Bloch spectra are $n$th roots of those of Hermitian parent two-dimensional (2D) lattices, namely graphene and the Lieb lattice. The $n$-roots of these models are constructed from connecting loop modules of unidirectional couplings whose geometrical arrangements match that of the corresponding parent system. Their energy spectrum is shown to consist of $n$ rotated and equivalent branches in the complex energy plane, each matching the real spectrum of the parent model when raised to the $n$th power, together with extra zero-energy flat bands (FBs) accounted for by the generalized index theorem. We show how the low-energy Dirac cones of the parent models translate, for an appropriate choice of phase configuration for the couplings of the $n$-root lattices, as what we call an "exceptional horn" appearing at each branch, with the central Dirac point (DP) converted into zero-energy exceptional points (EPs) of order $n$ or higher at high-symmetry momenta. These exceptional horns reflect the behavior of low-lying excitations that scale with momentum as $E\sim\vert \mathbf{q}\vert^{\frac{1}{n}}$, with $n\geq 3$, as opposed to the linear massless modes that characterize a Dirac cone. Moreover, we derive analytic expressions for the associated Landau levels (LLs), whose energies scale with magnetic flux as $E\sim\phi^{\frac{1}{2n}}$. For the case of the $n$-root Lieb lattice, the zeroth LL is shown to be exceptional. These results are analytically derived for both $n$-root models and numerically demonstrated for certain values of $n$. Finally, we propose a realistic photonic implementation based on coupled ring resonators with a split configuration of optical gain and loss.

Figures

Figures reproduced from arXiv: 2602.16670 by the authors.

Figure 1
Figure 1. Illustration of the n-root graphene model. Dashed limited square region encloses a unit cell. The primitive vec￾tors are a1 = a  1 2 , √ 3 2  and a2 = a  − 1 2 , √ 3 2  , with a ≡ 1 the lattice constant. The general form of the loop modules is depicted at the bottom, with the arrows indicating the direc￾tion of the unidirectional couplings of magnitude √n J. The color scheme at the right indicates the SL to whic… view at source ↗
Figure 2
Figure 2. (a) Unit cell of the 3-root graphene model with a [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Bulk complex energy spectrum of branch p = 0, 1, 2, with Ep=0 standing for the energies of the real zero branch, for the 3-root graphene model, with ω p 3 = e i 2π 3 p and J = 1. A zoomed plot of the exceptional horn delimited by the dashed orange oval is shown at the right, where the zero-energy EP is highlighted. [47], there are n − 1 extra flat bands coming from sub￾lattice imbalance of each SLj>1 with SL1 that c… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Bulk complex energy spectrum of the 3-root [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: (a) Complex energy spectrum of the periodic 3-root graphene lattice as a function of [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: Illustration of the n-root Lieb lattice. Dashed lim￾ited square region encloses a unit cell. The primitive vectors are ax = (a, 0) and ay = (0, a), with a ≡ 1 the lattice con￾stant. The blue and magenta loop modules have the configu￾ration depicted at the bottom of [P…
Figure 7
Figure 7. Figure 7: (a) Bulk complex energy spectrum of branch [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: (a) Complex energy spectrum of the periodic 4-root Lieb lattice as a function of [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: Complex spectrum of eigenfrequencies for the cubic [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: (a) Real part of the bulk energy spectrum for the [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: Complex spectrum of eigenfrequencies for the 3- [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: (a) Unit cell of the 3-root Lieb model with a mod [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]
Figure 14
Figure 14. Figure 14: (a) Energy spectrum for the tight-binding Hamiltonian of 3-root graphene, including the relevant phase factors, [PITH_FULL_IMAGE:figures/full_fig_p015_14.png]
Figure 15
Figure 15. Figure 15: (a) Energy spectrum for the tight-binding Hamiltonian of the 3-root Lieb model, including the relevant phase [PITH_FULL_IMAGE:figures/full_fig_p015_15.png]

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