Topologically nontrivial multicritical points

Pasquale Marra; Ranjith R Kumar

REVIEW 3 major objections 4 minor 54 references

Topologically nontrivial multicritical points

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read At the multicritical points of a one-dimensional topological chain with third-neighbor couplings, the paper finds one stable zero-energy edge mode at each end, characterized by the invariant $w_{\mathrm{mc}}=1$.

desk verdict Plausible new result on nontrivial multicritical points, but the discriminant signature claimed as unique is contradicted by the paper's own equations. read the letter →

arxiv 2507.11120 v2 pith:TJ35P2EA submitted 2025-07-15 cond-mat.dis-nn cond-mat.mes-hallcond-mat.str-elhep-th

Ranjith R Kumar , Pasquale Marra This is my paper · ORCID

classification cond-mat.dis-nncond-mat.mes-hallcond-mat.str-elhep-th

keywords topologicalmulticriticalpointsMajoranazeromodesedgemodelocalizationzero-countinginvariantthird-neighborSSHchainKitaevpolynomialdiscriminantAnderson

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

The paper claims that multicritical points, the parameter values where several gapped and gapless phases of a one-dimensional topological chain meet, can host localized zero-energy edge modes instead of being topologically trivial. In a spinless-fermion chain with hopping and pairing extended to the third nearest neighbor, and with parameters restricted to $\Gamma_0 \le \Gamma_2/3$, the two multicritical points M1 and M2 each carry exactly one stable zero-energy mode localized at each end, characterized by the invariant $w_{\mathrm{mc}} = 1$. This contradicts the earlier expectation, based on localization lengths diverging as the multicritical point is approached, that edge modes always delocalize into the bulk at such points. A reader should care because it extends the bulk-boundary correspondence, in zero-counting form, to the most singular points of the phase diagram and identifies where topologically protected modes survive for quantum information even while the bulk gap vanishes.

What carries the argument

The central object is the complex function $f(\zeta)$ obtained by writing the Bloch Hamiltonian in Majorana form and setting $\zeta = e^{ik}$; for this model it is the cubic $f(\zeta) = -\Gamma_0/2 - \Gamma_1\zeta/2 - \Gamma_2\zeta^2/2 - \Gamma_3\zeta^3/2$. By Cauchy's argument principle, the number of zeros inside the unit circle (the pole count is zero) defines a topological invariant $w_c$ that counts localized edge modes even when the bulk gap closes, while non-degenerate zeros on the unit circle encode the central charge of the critical theory. At M1 and M2 the zeros arrange themselves as a degenerate pair on the unit circle ($\zeta = -1$ for M1, $\zeta = +1$ for M2) plus one real zero strictly inside, so $w_{\mathrm{mc}} = 1$. The discriminant $\Delta = \Gamma_3^4(\zeta_1-\zeta_2)^2(\zeta_1-\zeta_3)^2(\zeta_2-\zeta_3)^2$ vanishes at the multicritical points and changes sign across them along the high-symmetry critical lines, which the paper uses to distinguish topological multicritical points, where the sign flips, from trivial ones, where it does not.

What would settle it

Diagonalize the open chain exactly at M1 and M2 for sizes $N = 25, 50, 100, 200, 400$ and check that exactly one zero-energy eigenstate stays pinned at each end. If the zero-energy count changes with size, the pair of degenerate zeros on the unit circle splits, or the decay length measured numerically disagrees with $\xi = -1/\ln|\zeta|$ predicted by the in-circle zero, then $w_{\mathrm{mc}} = 1$ is not the stable characterization the paper claims.

Watch

Extended reading notes

Core claim

In the third-neighbor Su–Schrieffer–Heeger or Kitaev chain, the authors show that restricting the parameter plane to $\Gamma_0 \le \Gamma_2/3$ removes the trivial $w = 0$ phase, leaving only gapped phases with winding numbers $w = 1, 2, 3$ and only nontrivial critical phases. At the two multicritical points M1 and M2, located at $\Gamma_1 = \pm(3\Gamma_0 + \Gamma_2)/2$, the bulk gap closes with quadratic dispersion (dynamical exponent $z = 2$), yet the open chain retains exactly one zero-energy eigenstate localized at each end. The complex function $f(\zeta) = -\Gamma_0/2 - \Gamma_1\zeta/2 - \Gamma_2\zeta^2/2 - \Gamma_3\zeta^3/2$ has, at these points, one zero strictly inside the unit circle and two degenerate zeros on the unit circle (at $\zeta = -1$ for M1 and $\zeta = +1$ for M2), giving the invariant $w_{\mathrm{mc}} = 1$. The discriminant of this cubic vanishes exactly at the multicritical points and flips sign across them along the high-symmetry critical lines, a signature the authors argue is unique to topological multicritical points and absent when a trivial $w = w_c = 0$ phase is present. Weak disorder shifts the multicritical points without destroying them, while strong disorder produces a gapless Anderson-localized phase that still carries two zero-energy modes localized at each end.

Load-bearing premise

The argument assumes that the number of edge-localized modes is still counted by the zeros of $f(\zeta)$ strictly inside the unit circle at the multicritical point, where the other two zeros coincide exactly on the unit circle; this count is used without an independent proof that the on-circle degeneracy leaves it unchanged, and the numerical check runs at a single chain size, $N = 25$.

Editorial extensions

If this is right

The conventional winding number is ill-defined exactly at a critical point, but the zero-counting invariant remains finite there and evaluates to $w_{\mathrm{mc}} = 1$ at M1 and M2, so the bulk-boundary correspondence survives, in this generalized form, at multicriticalities.
The condition $\Gamma_0 \le \Gamma_2/3$ makes the whole parameter plane nontrivial, so transitions between critical phases along a critical line never force the edge modes to delocalize at the crossing point.
The discriminant $\Delta = 0$ with a sign flip across the point identifies a topological multicritical point; trivial multicritical points show $\Delta = 0$ without a sign flip, so the sign behavior of $\Delta$ gives a direct criterion to locate topologically nontrivial multicriticalities in other one-dimensional chains.
If the central claim is right, at weak disorder the two multicritical zero modes survive (the points merely shift in parameter space), and at strong disorder the system enters a gapless Anderson-localized phase that still carries two zero-energy modes at each end.

Reading between the lines

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

An extension the paper leaves implicit: in chains with couplings out to the $m$-th neighbor, the same zero-counting should assign $w_{\mathrm{mc}}$ at a multicritical point by the number of zeros strictly inside the unit circle after a degenerate pair forms on it; the paper notes $w_{\mathrm{mc}} > 1$ becomes possible with more couplings but does not give the general counting rule.
The discriminant sign flip could serve as an experimental diagnostic: scanning the local density of states across a suspected multicritical point should reveal the transition between two critical phases, with the sign flip marking where two roots coalesce through the real axis.
A testable reinterpretation follows from the paper's own comparison: the delocalization of edge modes reported at multicritical points in earlier studies of this model family may be caused by the coexisting trivial phase ($w = w_c = 0$) rather than by multicriticality itself; tuning $\Gamma_0$ through $\Gamma_2/3$ in the same model and watching the sign-flip property appear and disappear would set

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The core claim of this paper is likely correct: in the third-neighbor SSH/Kitaev chain with Γ0 ≤ Γ2/3, the multicritical points host one localized zero-energy mode per end, characterized by wmc = 1. That corrects the prior belief that edge modes always delocalize at multicritical points in these chains. The zero-counting argument is applied sensibly, and the numerical check at N = 25 is more convincing than the stress-test note suggests, because the localization length from the inside zero at ζ = −1/2 is about 1.44 lattice constants, so a power-law state would still have visible weight across the chain, which Figure 4 does not show. The derivation of the parameter condition Γ0 ≤ Γ2/3 is clean and the phase diagram is easy to follow.

The main soft spot is the discriminant claim. The paper states that a sign flip in the discriminant uniquely identifies topological multicritical points and is absent for trivial ones. That is false on its own terms. For the trivial case Γ0 = 0.5, Γ2 = 0.5, the quadratic multicritical point at Γ1 = 1 has Γ3 = 0, so the cubic reduces to a perfect square (1+ζ)^2. Just below that point (Γ1 = 0.9, Γ3 = 0.1), the roots are −1 and −2 ± i, so the discriminant is negative. Just above (Γ1 = 1.1, Γ3 = −0.1), the roots are all real, so the discriminant is positive. That is a sign flip, and the point is trivial (wc goes from 0 to 1). Figure 9 in the paper shows the discriminant as positive on both sides for all three Γ0 values, which directly contradicts the equations. This is a genuine mathematical error in the flagship diagnostic, and the section needs to be corrected or removed.

Two smaller issues: the abstract promises a physical mechanism involving 'kinetic inversion in higher-order terms,' but that phrase never appears in the body. The disorder section uses only ten realizations for the zero-energy phase diagrams at N = 450, which is thin for claims about a gapless Anderson-localized phase; the IPR data with 50 realizations at N = 2000 are better.

The stress-test worry about the zero-counting invariant at a degenerate unit-circle zero is worth noting, but it is not fatal here. The paper does not prove the extension of Verresen's rule to the degenerate case, and a careful referee should ask for that proof. Still, the short localization length and the open-boundary numerics make the central phenomenon plausible.

Who should read this? People working on topological quantum criticality. The central result deserves serious refereeing, but not in its current form: the discriminant overclaim must be fixed, and the mechanism should be either defined or dropped. I would send it to review, asking for major revisions on those points.

Referee Report

3 major / 4 minor

Summary. The paper studies a one-dimensional spinless fermion chain (SSH/Kitaev) with hoppings and pairings up to third nearest neighbors. It identifies two multicritical points, M1 and M2, where high-symmetry and non-high-symmetry critical lines meet with quadratic dispersion, and claims that for Γ0 ≤ Γ2/3 these points host one stable localized zero-energy edge mode at each end, characterized by the zero-counting invariant wmc = 1. The identification is made through the roots of the complex polynomial f(ζ) = -(Γ0 + Γ1 ζ + Γ2 ζ^2 + Γ3 ζ^3)/2, through localization lengths ξ = -1/ln|ζ|, and through exact diagonalization of open chains. The paper further argues that the discriminant of f(ζ) vanishes at the multicritical points and flips sign along the high-symmetry critical lines only for the topologically nontrivial case, distinguishing them from trivial multicritical points. Finally, it studies the effects of bond disorder, reporting that the multicritical points survive weak disorder and that strong disorder produces a gapless, topologically nontrivial Anderson-localized phase with two zero-energy edge modes per end.

Significance. If correct, the main result provides a concrete counterexample to the conventional expectation that edge modes always delocalize at multicritical points, and it establishes a parameter regime in which gapped, critical, and multicritical phases are all topologically nontrivial. The clean-limit analysis is explicit and checkable: the roots in Eqs. (11)-(13), the localization lengths in Fig. 6, and the discriminant signs in Figs. 8 and 9 are concrete and falsifiable. The disorder study adds a useful robustness analysis. The main weakness is that the zero-counting invariant is applied at points where two zeros are degenerate on the unit circle, a case not covered by the cited proof, and the only numerical check at the multicritical point is a single chain size N = 25. In addition, the stated condition Γ0 ≤ Γ2/3 includes an equality case where the claim wmc = 1 fails. I also checked the stress-test suggestion that the discriminant flips at trivial multicritical points; for Γ0 = 0.5, Γ2 = 0.5 near Γ1 = ±1, Γ3 = 0, the discriminant behaves as δ^2 times a positive factor, so it does not flip, consistent with the paper's Fig. 9.

major comments (3)

[Section IV, Eq. (10), Figs. 5-6] The central claim wmc = 1 at M1 and M2 is obtained by counting only the zero ζ2 strictly inside the unit circle and ignoring the double zero at ζ = ±1. The zero-counting invariant wc = Nz - Np and the central-charge formula c = N'_z/2 in Ref. [1] are established for critical points with non-degenerate zeros on the unit circle; at M1/M2 the unit-circle zeros are degenerate, which produces a Jordan block in the transfer-matrix solution and invalidates the standard proof that the number of localized edge modes equals the number of interior zeros. The paper offers no independent derivation or reference for this degenerate case. The only numerical evidence is the N = 25 open-chain diagonalization in Figs. 3 and 4, which cannot distinguish an exponentially localized mode from a critical power-law mode. Please provide a direct solution of the open-boundary recurrence at the degenerate point showing that the localized solution satisfies both boundary conditions, or a reference covering degenerate unit-circle zeros, and add finite-size scaling (e.g., energy splitting or IPR versus N) to support the thermodynamic-limit statement.
[Section IV, parameter condition] The parameter condition is stated as Γ0 ≤ Γ2/3 for nontrivial multicritical points with wmc = 1. At equality, Γ0 = Γ2/3, the polynomial f(ζ) at M1 has a triple zero on the unit circle; for example, with Γ2 = 1, Γ0 = 1/3, Γ1 = 1, and Γ3 = 1/3, one obtains f(ζ) = -(1/6)(1 + ζ)^3. There is then no zero strictly inside the unit circle, so the paper's own counting gives wmc = 0, not 1. Thus the statement 'for Γ0 ≤ Γ2/3 ... nontrivial multicritical points with wmc = 1' is false at the boundary. The condition should be Γ0 < Γ2/3, or the equality case should be analyzed separately; this affects the abstract and the conclusion.
[Section IV, discriminant criterion] The paper claims that the discriminant 'uniquely identifies the topological multicritical points and distinguishes them from the trivial ones.' The evidence for the topological case is a single parameter set (Γ0 = 0.2, Γ2 = 1), while the trivial cases are illustrated for three parameter sets. Since Δ = 0 also occurs at trivial quadratic multicritical points (Fig. 9), the distinguishing feature is not Δ = 0 itself but the sign flip along the high-symmetry critical lines. The paper does not prove that this sign flip occurs for all Γ0 < Γ2/3, nor that it is absent for all trivial multicritical points. Please either provide an analytic proof (it is accessible from the explicit roots) or state the criterion more cautiously as a numerically demonstrated property.

minor comments (4)

[Section II, phase description] The sentence assigning wc = 2 segments to 'Γ1 > M1 and Γ1 < M2' on blue and red lines is inconsistent with the root configurations reported in Section IV and Fig. 7; for the blue line the wc = 2 segment is Γ1 < M1, and for the red line it is Γ1 > M2. Please correct the inequalities.
[Eq. (14)] Equation (14) omits the factor 1/16 that comes from the leading coefficient -Γ3/2 in the cubic discriminant; the sign analysis is unaffected, but the formula as written is not the exact discriminant.
[Fig. 3 caption] The caption of Fig. 3 lists the same parameter point for panels (a) and (b); panel (b) should be the M2 point.
[Section IV, zero condition] In Section IV, the statement 'ζ2 < 1' should be '|ζ2| < 1', since the zero is negative and the condition for being inside the unit circle is on the modulus.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claim is derived from an external zero-counting invariant and independently verified by open-boundary numerics; no fitted parameter is relabeled as a prediction.

full rationale

The derivation is self-contained. The complex function f(ζ) in Eq. (10) is obtained directly from the Hamiltonian, and its zeros are given explicitly by the algebraic roots in Eqs. (11)-(13). The multicritical invariant w_mc = 1 is read from the number of zeros strictly inside the unit circle, using the zero-counting bulk-boundary correspondence of Ref. [1], which is an independent published result and not authored by the present paper. This invariant-based claim is then checked against separate open-boundary exact-diagonalization data: the zero-energy eigenvalues in Fig. 3 and the edge-localized probability distributions in Fig. 4. The discriminant analysis in Sec. IV is an algebraic consequence of the root locations and sign conventions, not a fitted quantity. The condition Γ0 ≤ Γ2/3 is derived from the confluence of linear and quadratic multicritical points in the phase diagram, not from assuming the desired conclusion. The self-citations (Refs. [15] and [16]) are used only to provide prior context about delocalization at multicritical points and to reference a previously studied trivial case; they are not load-bearing for the new claim. Although the application of the Ref. [1] zero-counting rule to the degenerate unit-circle zeros at M1 and M2 goes beyond the non-degenerate case explicitly treated there, that is a correctness or robustness concern, not a circularity: the paper does not define its conclusion into its inputs, and it provides independent numerical evidence for the localized zero modes.

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

The central claims rest on the zero-counting method from Ref. [1] and the assumption that the parameter plane with Γ0 ≤ Γ2/3 contains no trivial phases. No new entities are introduced and no parameters are fitted; the model parameters are physical.

assumptions (4)

domain assumption Zero-counting topological invariant for critical phases (Verresen et al., Ref. [1])
The paper uses the number of zeros of f(ζ) inside the unit circle as the topological invariant that counts edge modes at critical points, without re-deriving this method.
domain assumption Bulk-boundary correspondence for gapped winding number
Used to label gapped phases w=0,1,2,3 in Fig. 1; standard for 1D class BDI.
ad hoc to paper Uniform disorder model for hopping amplitudes
Disorder is introduced as γ1,2(i) drawn uniformly from [-η,η] without microscopic motivation.
standard math The complex function f(ζ) has no poles
f(ζ) is a polynomial, so the Cauchy argument principle gives w=Nz; used to define invariants.

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Cite this review

Pith. "Pith review of Topologically nontrivial multicritical points." pith.science (2026). https://pith.science/paper/TJ35P2EA

@misc{pith2026250711120,
  author       = {Pith},
  title        = {Pith review of: Topologically nontrivial multicritical points},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TJ35P2EA}},
  note         = {Machine review of arXiv:2507.11120}
}

read the original abstract

Recently, the intriguing interplay between topology and quantum criticality has been unveiled in one-dimensional topological chains with extended nearest-neighbor couplings. In these systems, topologically distinct critical phases emerge with localized edge modes despite the vanishing bulk gap. In this work, we study the topological multicritical points at which distinct gapped and critical phases intersect. Specifically, we consider a topological chain with coupling up to the third nearest neighbors, which shows stable localized edge modes at the multicritical points. These points possess only nontrivial gapped and critical phases around them and are also characterized by the quadratic dispersion around the gap-closing points. We characterize the topological multicritical points in terms of the topological invariant obtained from the zeros of the complex function associated with the Hamiltonian. Further, we analyze the nature of zeros in the vicinity of the multicritical points by calculating the discriminants of the associated polynomial. The discriminant uniquely identifies the topological multicritical points and distinguishes them from the trivial ones. Moreover, we identify the underlying physical mechanism in terms of kinetic inversion in higher-order terms. We finally study the robustness of the zero-energy modes at the multicritical points at weak disorder strengths, and reveal the presence of a topologically nontrivial gapless Anderson-localized phase at strong disorder strengths.

Figures

Figures reproduced from arXiv: 2507.11120 by the authors.

**Figure 3.** FIG. 3. Eigenvalue distribution under open boundary con [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 5.** FIG. 5. Zeros of the complex function on a complex plane at [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

Figures from the paper (6 more)

**Figure 8.** Figure 8: FIG. 8. Discriminant in the vicinity of topological multicrit [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]

**Figure 7.** Figure 7: FIG. 7. Zeros near the multicritical points [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

**Figure 9.** Figure 9: FIG. 9. Discriminant in the vicinity of topologically trivial [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]

**Figure 10.** Figure 10: FIG. 10. Effect of disorder on the topological phase diagram [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]

**Figure 11.** Figure 11: FIG. 11. Eigenvalue distribution under open boundary condi [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]

**Figure 13.** Figure 13: FIG. 13. Phase diagrams for unequal disorder strength with [PITH_FULL_IMAGE:figures/full_fig_p011_13.png]

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Pith tools

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

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