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REVIEW 2 major objections 5 minor 42 references

Geometrical approach and topological electron density in the $p_x + ip_y$ superconductor

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that the Chern number of the square-lattice $p_x+ip_y$ superconductor equals the local marker $C=-\langle n_{\Gamma}\rangle+2\langle n_{X}\rangle-\langle n_{M}\rangle$, a combination of electron densities measurable with…

desk verdict A clean exact density-to-Chern marker for p_x+ip_y, but it is blind to the sign of the pairing chirality, so ARPES alone does not measure the invariant. read the letter →

arxiv 2608.09415 v1 pith:MIBVDL3C submitted 2026-08-10 cond-mat.str-el cond-mat.mes-hallcond-mat.supr-conmath-phmath.MPquant-ph

Karyn Le Hur , Kabir Narayanan , Óscar Mañas Chambert This is my paper · ORCID
classification cond-mat.str-elcond-mat.mes-hallcond-mat.supr-conmath-phmath.MPquant-ph
keywords p_x+ip_ysuperconductorChernnumberlocaltopologicalmarkerelectrondensityARPESBerrycurvaturephasetransitionVanHovesingularity
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 proposes a way to extract the topological invariant of a two-dimensional $p_x+ip_y$ superconductor from local, momentum-resolved electron densities instead of from edge transport. Its central formula is $C=-\langle n_{\Gamma}\rangle+2\langle n_{X}\rangle-\langle n_{M}\rangle$, where each $\langle n_{\mathbf{k}}\rangle$ is the occupation of a high-symmetry electron state, a quantity accessible to angle-resolved photoemission spectroscopy (ARPES). The paper argues that this combination equals the Chern number, takes the quantized values $-1,0,+1$, and remains a plateau at finite temperature within the BCS model. It derives the marker from two one-dimensional invariants on the Bloch sphere, relates it to a Pfaffian-like sign formula, and links it to a topological superfluid density. If the claim is right, a genuinely global property of the superconducting state becomes a small set of local density measurements.

What carries the argument

The central object is the local topological marker obtained by adding Berry curvatures integrated over small disks centered at the four special points $\Gamma$, $X$, $Y$, $M$; near a transition the disk radius is set by $R^2=m^2/(2|\Delta|^2\sin\zeta)$ and each disk contributes $\pm\frac12\,\mathrm{sgn}(\cdot)$ to the sum. The paper then rewrites the marker in the Anderson pseudo-spin representation, where $\langle S_z\rangle=2\langle n_{\mathbf{k}}\rangle-1$, and shows that the sum is exactly the sum of two 1D invariants $C_x$ and $C_y$ obtained by walking on the Bloch sphere from $\Gamma$ to $X$ and from $X$ to $M$. That rewriting is what converts the geometrical invariant into a density formula, and it also gives the Pfaffian-type expression $C=|C_y|-|C_x|$ as a difference of products of signs of the kinetic energies.

What would settle it

Compute the full Brillouin-zone Chern number and the marker $-\langle n_{\Gamma}\rangle+2\langle n_{X}\rangle-\langle n_{M}\rangle$ in a model with an added next-nearest-neighbor hopping $t'$ large enough to shift the gapless points off the four special momenta; any mismatch between the two values away from the topological transitions would falsify the central claim. An ARPES measurement of the three densities in a candidate chiral superconductor that returns a non-integer combination would likewise rule out the marker as a bulk invariant.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that every value of the chemical potential in the square-lattice model with pairing $\Delta_{\mathbf{k}}=2\Delta(\sin k_x+e^{i\zeta}\sin k_y)$ is captured by the local marker $C=-\langle n_{\Gamma}\rangle+2\langle n_{X}\rangle-\langle n_{M}\rangle$, with $\langle n_{Y}\rangle=\langle n_{X}\rangle$ by symmetry and $\langle n_{\mathbf{k}}\rangle=\int d\omega\,n_F(\omega)A(\mathbf{k},\omega)$ the electron occupation obtained from the spectral function. At the points $\Gamma$, $X$, $Y$, and $M$, where the pairing function vanishes, the occupation is $\langle n_{\mathbf{k}}\rangle=\frac12-\frac12\,\mathrm{sgn}(\xi_{\mathbf{k}})\tanh(\beta E_{\mathbf{k}}/2)$, so the marker reduces to signs of the kinetic energy and changes at $\mu=-4t$, $\mu=0$, and $\mu=+4t$. The paper identifies $C$ with the sum $C_x+C_y$ of two 1D $\mathbb{Z}$ invariants defined on the Bloch sphere, and shows that at half-filling the transition is a protected semimetal characterized by $\langle n_{X}\rangle=1/2$.

Load-bearing premise

The load-bearing premise is that the integrated Berry curvatures on small disks around $\Gamma$, $X$, $Y$, and $M$ add up to the global Chern number; the paper checks this numerically for every $\mu$ but gives no analytic proof of the additivity, so if the additivity failed away from the transitions the marker would deviate from the true invariant.

Editorial extensions

If this is right

  • Within BCS theory, the topological invariant of a $p_x+ip_y$ superconductor can in principle be obtained from bulk ARPES measurements of electron occupation at $\Gamma$, $X$, and $M$, without measuring edge transport.
  • The plateau in $C$ survives finite temperature; the practical upper limit is the superconducting critical temperature, not the thermal smearing of the density step.
  • A finite experimental resolution in momentum space does not destroy the marker: integrating the density over a small disk around $X$ or $Y$ preserves the quantized value while turning the sharp derivative into a logarithmic Van Hove profile.
  • The real-space version of the marker, written with correlation functions $\langle c_i^\dagger c_j\rangle$, is expressed in a form the paper states remains usable when weak disorder is present.

Reading between the lines

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

  • Beyond the paper: if the marker remains equal to the Chern number under perturbations that keep the four special points gapless, the same three-density combination could serve as a fast bulk diagnostic for chiral superconductivity in candidate materials.
  • Beyond the paper: the construction suggests a general recipe for other lattice models and pairing symmetries: whenever the pairing vanishes at a small set of high-symmetry points, the Chern number may be read from a weighted sum of occupations at those points, with weights fixed by kinetic-energy signs.
  • Beyond the paper: the protected value $\langle n_X\rangle=1/2$ at half-filling gives a momentum-space analogue of a half-filled Landau level, and it would be worth testing whether this semimetal description survives interactions beyond BCS mean field.
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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

2 major / 5 minor

Summary. The manuscript proposes a local topological marker for the square-lattice p_x + ip_y superconductor, defined as C = -⟨n_Γ⟩ + 2⟨n_X⟩ - ⟨n_M⟩, where each ⟨n_k⟩ is the BCS electron occupation at the time-reversal-invariant points Γ, X, and M. The authors show that this combination reproduces the Chern-number phase diagram, extend it to finite temperature and finite quasiparticle lifetime, and provide a real-space correlation-function representation. The occupation identity in Eq. (8) is exact within the BCS model, and the paper contains several nontrivial checks. However, the proposed marker depends only on |Δ_k| and ξ_k, so it cannot distinguish the two chiralities p_x + ip_y and p_x - ip_y; this limitation is not stated and conflicts with the abstract's claim of an ARPES-measurable topological invariant.

Significance. If the central claim were fully valid, the paper would provide a remarkably simple ARPES-accessible formula for the Chern number of a chiral superconductor from local momentum-resolved densities alone, together with an exact real-space expression and a finite-temperature stability statement. The exact identity in Eq. (8) is a genuine and useful result within the fixed-chirality model, and the finite-temperature and lifetime calculations are consistent. The main limitation is that the marker is phase-insensitive: it is identical for the two opposite chiralities, so it cannot by itself determine the sign of the topological invariant. The manuscript therefore needs a clear statement of what the marker does and does not measure, and an analytic justification of the disk additivity used in the geometric derivation.

major comments (2)
  1. [Eqs. (7)-(8) and abstract] The central marker is phase-insensitive and cannot distinguish the two chiralities of the pairing. Since ⟨c†_k c_k⟩ = 1/2(1 - ξ_k/E_k) depends only on ξ_k and |Δ_k|, the occupations at Γ, X, and M are identical for Δ_k = 2Δ(sin k_x + i sin k_y) and for its complex conjugate, whereas the BdG Chern number changes sign under ζ → -ζ because the d-vector winding reverses. Consequently, Eq. (8) assigns the same quantized value to p_x + ip_y and p_x - ip_y in every topological phase. The paper should state explicitly that the marker gives the chirality-fixed Chern number (with the chirality supplied by other information) or the absolute value |C|, rather than claiming that the local electron density alone measures the topological invariant. This is a load-bearing issue for the abstract's main claim, though it does not invalidate Eq. (8) as an exact identity within a fixed-chirality convention.
  2. [Sec. I, Eqs. (25)-(26) and Eq. (4)] The disk regularization is tuned to produce the desired half-quantized jumps. The radius is chosen as R² = m²/(2|Δ|² sin ζ), which is not positive definite when sin ζ < 0, so the geometric derivation is not defined for the opposite chirality. In addition, the additivity of the integrated Berry curvatures around X and Y is asserted and checked numerically, but no analytic proof is given. Because Eq. (8) is exact without any Berry-curvature integration, the disk construction is not needed for the main formula; however, the text presents Eqs. (4), (8), (13), and (14) as equivalent definitions, and that equivalence is not established for general ζ. Please supply an analytic derivation of the additivity or explicitly label the disk construction as a heuristic illustration rather than a proof.
minor comments (5)
  1. [Main text after Eq. (2)] There are typographical errors, including 'superfuid' for 'superfluid' and the inconsistent rendering of 'p_x + ip_y' in the title and opening lines.
  2. [Eq. (4)] The symbol ΔS is introduced in Eq. (4) without a definition, and it is later equated to C in Eq. (8); please define the notation clearly before first use.
  3. [Fig. 2] The caption states that the marker is shown for different temperatures but does not identify which curve corresponds to which temperature nor the values of Δ and the line color legend; please add these details.
  4. [Sec. I after Eq. (26)] The sentence 'The factor 1/2 ensures that the jump will always be an integer' is not an explanation; the factor follows from the chosen normalization of the integrated Berry curvature, and this should be stated explicitly.
  5. [Sec. II, Eq. (19)] The 'topological superfluid density' n_s = sin ζ |Δ|²/m² is introduced by analogy with the Karplus-Luttinger velocity; while interesting, this identification is not derived from a response function and should be clearly labeled as a phenomenological analogy.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular dependency: Eq. (8) is an exact BCS identity, with self-citations only motivational.

full rationale

The derivation chain is self-contained. The local Berry curvature F_{kxky}(Γ) is computed from the model eigenstates (Eqs. 22-24); the disk radius in Eq. (25), R^2=m^2/(2|Δ|^2 sinζ), is fixed by the linearized curvature and the half-quantization condition, not by fitting to the final Chern number. Summing the three regularized curvatures gives Eq. (4), and substituting the exact T=0 limit of Eq. (7), <n_k>=1/2-1/2 sgn(ξ_k), into -<n_Γ>+2<n_X>-<n_M> reproduces Eq. (4) algebraically. The identification of this combination with the 2D invariant is a theorem about the model (vortex contributions at the gap nodes), not an input. The many self-citations (e.g., Ref. [14] for the sphere method) are historical/motivational; the needed 1D invariants C_x, C_y are rederived explicitly in Eqs. (12), (36), (41) and the sign formulas are written out, so no load-bearing step rests on an unverified self-citation. A physical caveat—Eq. (8) depends only on |Δ_k| at the nodes and therefore cannot distinguish the opposite chirality ζ=-π/2—is a model-applicability limitation, not a circular reduction of the claimed derivation.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the BCS mean-field Hamiltonian and the standard Chern number of the BdG bands. The only genuine tuning knob in the derivation is the disk radius R used to regularize the local Berry curvature; it is fixed to reproduce the known half-integer jumps. The 'topological superfluid density' n_s is an analogy, not a derived response. No new entities are postulated.

free parameters (1)
  • Disk radius R for Berry curvature integral = R^2 = m^2/(2|Delta|^2 sin zeta)
    Chosen so that each local Berry curvature integral yields exactly a half-integer jump (Eqs. 25-26 in the supplementary). The specific coefficient is fixed by the known Chern number, making the local-marker construction partially circular.
assumptions (5)
  • domain assumption BCS mean-field Hamiltonian (Eq. 1) with xi_k and Delta_k describes the p_x+ip_y superconductor.
    The whole analysis is performed on this BdG Hamiltonian.
  • standard math The global topological invariant is the Chern number of the lower BdG band.
    Used as benchmark in Fig. 2 and in Eqs. (4)-(8).
  • standard math The electron occupation equals the Fermi-weighted integral of the spectral function (Eq. 6).
    Standard Lehmann representation connecting theory to ARPES.
  • domain assumption At Gamma, X, M the pairing Delta_k vanishes, giving Eq. (7).
    Exact for the chosen pairing function and central to the density formula.
  • ad hoc to paper The analogy between the Berry connection and a classical vector potential leading to n_s (Eqs. 18-19).
    The superfluid density n_s = sin(zeta) |Delta|^2/m^2 is asserted by analogy with Karplus-Luttinger; no controlled derivation is provided.

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Pith. "Pith review of Geometrical approach and topological electron density in the $p_x + ip_y$ superconductor." pith.science (2026). https://pith.science/paper/MIBVDL3C

@misc{pith2026260809415,
  author       = {Pith},
  title        = {Pith review of: Geometrical approach and topological electron density in the $p_x + ip_y$ superconductor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MIBVDL3C}},
  note         = {Machine review of arXiv:2608.09415}
}
abstract

We present an analysis on the geometrical and physical nature of the $p_x + ip_y$ superconductor on the square lattice, with an emphasis on the topological phase transition at half-filling. We develop a local topological marker from specific Dirac points within the Brillouin zone, which is introduced via the addition of two one-dimensional (1D) $\mathbb{Z}$ $(\mathbb{Z}_2$) invariants defined on the Bloch sphere. We relate this topological marker to the electron spectral function integrated on frequency through the local momentum-resolved electron density, which may be measured via Angle Resolved Photoemission Spectroscopy (ARPES), and show that it remains well-protected including temperature effects. Integrating on a small area around a specific point in momentum space associated to the measure uncertainty, this also reveals the Van Hove logarithmic profile of the density of states in the derivative of the local marker while preserving the topological information. Topological transitions correspond to a protected semi-metal. We analyse the real space representation of this topological marker from correlation functions. We present physical responses such as the topological superfluid density.

Figures

Figures reproduced from arXiv: 2608.09415 by the authors.

Figure 1
Figure 1. Distribution of the Berry curvature for the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Topological marker in Eq. (8) as a function of µ for different temperatures in units of t = 1 with kB = 1. It is then relevant to relate topological properties with observables such as the local electron density that will question if some topological aspects may be found within the Luttinger theorem or the Fermi function itself i.e. in the way we fill the electron states on the Fermi surface corresponding here to th… view at source ↗
Figure 3
Figure 3. Cartography of the local marker through ⟨c † i cj ⟩(−1 + 2e ikX·(ri−rj ) − e ikM·(ri−rj ) ) for µ = 3.9t. We have fixed ∆ = 0.1 in units of t. Eq. (10) in real space then allows us to identify an almost quantized topological number. derivative of the local electron density also reveals the topological transitions at µ = −4t and µ = +4t through a jump [34]. We show below that the local marker C is in fact equiv￾alent… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: 2D Representation of correlation functions in real space. It is important to emphasize that the scale on the vertical [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: Derivative of the correlation function with [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: Derivative of the local electron density in real space. [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

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